Difference between revisions of "Results of Gamma Function"

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| [https://dlmf.nist.gov/5.4.E18 5.4.E18] || [[Item:Q2056|<math>\imagpart@@{\digamma@{1+iy}} = -\frac{1}{2y}+\frac{\pi}{2}\coth@{\pi y}</math>]] || <code>Im(Psi(1 + I*y))= -(1)/(2*y)+(Pi)/(2)*coth(Pi*y)</code> || <code>Im[PolyGamma[1 + I*y]]= -Divide[1,2*y]+Divide[Pi,2]*Coth[Pi*y]</code> || Failure || Failure || Successful || Successful  
 
| [https://dlmf.nist.gov/5.4.E18 5.4.E18] || [[Item:Q2056|<math>\imagpart@@{\digamma@{1+iy}} = -\frac{1}{2y}+\frac{\pi}{2}\coth@{\pi y}</math>]] || <code>Im(Psi(1 + I*y))= -(1)/(2*y)+(Pi)/(2)*coth(Pi*y)</code> || <code>Im[PolyGamma[1 + I*y]]= -Divide[1,2*y]+Divide[Pi,2]*Coth[Pi*y]</code> || Failure || Failure || Successful || Successful  
 
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| [https://dlmf.nist.gov/5.4.E19 5.4.E19] || [[Item:Q2057|<math>\digamma@{\frac{p}{q}} = -\EulerConstant-\ln@@{q}-\frac{\pi}{2}\cot@{\frac{\pi p}{q}}+\frac{1}{2}\sum_{k=1}^{q-1}\cos@{\frac{2\pi kp}{q}}\ln@{2-2\cos@{\frac{2\pi k}{q}}}</math>]] || <code>Psi((p)/(q))= - gamma - ln(q)-(Pi)/(2)*cot((Pi*p)/(q))+(1)/(2)*sum(cos((2*Pi*k*p)/(q))*ln(2 - 2*cos((2*Pi*k)/(q))), k = 1..q - 1)</code> || <code>PolyGamma[Divide[p,q]]= - EulerGamma - Log[q]-Divide[Pi,2]*Cot[Divide[Pi*p,q]]+Divide[1,2]*Sum[Cos[Divide[2*Pi*k*p,q]]*Log[2 - 2*Cos[Divide[2*Pi*k,q]]], {k, 1, q - 1}]</code> || Failure || Failure || Skip || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>DirectedInfinity[] <- {Rule[k, 1], Rule[p, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>DirectedInfinity[] <- {Rule[k, 2], Rule[p, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>DirectedInfinity[] <- {Rule[k, 3], Rule[p, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>DirectedInfinity[] <- {Rule[k, 1], Rule[p, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>DirectedInfinity[] <- {Rule[k, 2], Rule[p, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>DirectedInfinity[] <- {Rule[k, 3], Rule[p, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>DirectedInfinity[] <- {Rule[k, 1], Rule[p, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>DirectedInfinity[] <- {Rule[k, 2], Rule[p, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>DirectedInfinity[] <- {Rule[k, 3], Rule[p, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>DirectedInfinity[] <- {Rule[k, 1], Rule[p, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>DirectedInfinity[] <- {Rule[k, 2], Rule[p, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>DirectedInfinity[] <- {Rule[k, 3], Rule[p, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br></div></div>  
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| [https://dlmf.nist.gov/5.4.E19 5.4.E19] || [[Item:Q2057|<math>\digamma@{\frac{p}{q}} = -\EulerConstant-\ln@@{q}-\frac{\pi}{2}\cot@{\frac{\pi p}{q}}+\frac{1}{2}\sum_{k=1}^{q-1}\cos@{\frac{2\pi kp}{q}}\ln@{2-2\cos@{\frac{2\pi k}{q}}}</math>]] || <code>Psi((p)/(q))= - gamma - ln(q)-(Pi)/(2)*cot((Pi*p)/(q))+(1)/(2)*sum(cos((2*Pi*k*p)/(q))*ln(2 - 2*cos((2*Pi*k)/(q))), k = 1..q - 1)</code> || <code>PolyGamma[Divide[p,q]]= - EulerGamma - Log[q]-Divide[Pi,2]*Cot[Divide[Pi*p,q]]+Divide[1,2]*Sum[Cos[Divide[2*Pi*k*p,q]]*Log[2 - 2*Cos[Divide[2*Pi*k,q]]], {k, 1, q - 1}]</code> || Failure || Failure || Skip || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>DirectedInfinity[] <- {Rule[k, 1], Rule[p, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>DirectedInfinity[] <- {Rule[k, 2], Rule[p, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>DirectedInfinity[] <- {Rule[k, 3], Rule[p, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>DirectedInfinity[] <- {Rule[k, 1], Rule[p, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  
 
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| [https://dlmf.nist.gov/5.5.E1 5.5.E1] || [[Item:Q2059|<math>\EulerGamma@{z+1} = z\EulerGamma@{z}</math>]] || <code>GAMMA(z + 1)= z*GAMMA(z)</code> || <code>Gamma[z + 1]= z*Gamma[z]</code> || Successful || Successful || - || -  
 
| [https://dlmf.nist.gov/5.5.E1 5.5.E1] || [[Item:Q2059|<math>\EulerGamma@{z+1} = z\EulerGamma@{z}</math>]] || <code>GAMMA(z + 1)= z*GAMMA(z)</code> || <code>Gamma[z + 1]= z*Gamma[z]</code> || Successful || Successful || - || -  
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| [https://dlmf.nist.gov/5.6.E8 5.6.E8] || [[Item:Q2075|<math>\left|\frac{\EulerGamma@{z+a}}{\EulerGamma@{z+b}}\right| <= \frac{1}{|z|^{b-a}}</math>]] || <code>abs((GAMMA(z + a))/(GAMMA(z + b)))< =(1)/((abs(z))^(b - a))</code> || <code>Abs[Divide[Gamma[z + a],Gamma[z + b]]]< =Divide[1,(Abs[z])^(b - a)]</code> || Failure || Failure || Error || Successful  
 
| [https://dlmf.nist.gov/5.6.E8 5.6.E8] || [[Item:Q2075|<math>\left|\frac{\EulerGamma@{z+a}}{\EulerGamma@{z+b}}\right| <= \frac{1}{|z|^{b-a}}</math>]] || <code>abs((GAMMA(z + a))/(GAMMA(z + b)))< =(1)/((abs(z))^(b - a))</code> || <code>Abs[Divide[Gamma[z + a],Gamma[z + b]]]< =Divide[1,(Abs[z])^(b - a)]</code> || Failure || Failure || Error || Successful  
 
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| [https://dlmf.nist.gov/5.6.E9 5.6.E9] || [[Item:Q2076|<math>|\EulerGamma@{z}| <= (2\pi)^{1/2}|z|^{x-(1/2)}e^{-\pi|y|/2}\exp@{\tfrac{1}{6}|z|^{-1}}</math>]] || <code>abs(GAMMA(z))< =(2*Pi)^(1/ 2)*(abs(z))^(x -(1/ 2))* exp(- Pi*abs(y)/ 2)*exp((1)/(6)*(abs(z))^(- 1))</code> || <code>Abs[Gamma[z]]< =(2*Pi)^(1/ 2)*(Abs[z])^(x -(1/ 2))* Exp[- Pi*Abs[y]/ 2]*Exp[Divide[1,6]*(Abs[z])^(- 1)]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>.3896047846 <= .1665021267 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 2}</code><br><code>.3896047846 <= .3461239156e-1 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 3}</code><br><code>.3896047846 <= .3330042534 <- {z = 2^(1/2)+I*2^(1/2), x = 2, y = 2}</code><br><code>.3896047846 <= .6922478312e-1 <- {z = 2^(1/2)+I*2^(1/2), x = 2, y = 3}</code><br><code>.3896047846 <= .1384495662 <- {z = 2^(1/2)+I*2^(1/2), x = 3, y = 3}</code><br><code>.3896047846 <= .1665021267 <- {z = 2^(1/2)-I*2^(1/2), x = 1, y = 2}</code><br><code>.3896047846 <= .3461239156e-1 <- {z = 2^(1/2)-I*2^(1/2), x = 1, y = 3}</code><br><code>.3896047846 <= .3330042534 <- {z = 2^(1/2)-I*2^(1/2), x = 2, y = 2}</code><br><code>.3896047846 <= .6922478312e-1 <- {z = 2^(1/2)-I*2^(1/2), x = 2, y = 3}</code><br><code>.3896047846 <= .1384495662 <- {z = 2^(1/2)-I*2^(1/2), x = 3, y = 3}</code><br><code>.9483190752e-1 <= .3461239156e-1 <- {z = -2^(1/2)-I*2^(1/2), x = 1, y = 3}</code><br><code>.9483190752e-1 <= .6922478312e-1 <- {z = -2^(1/2)-I*2^(1/2), x = 2, y = 3}</code><br><code>.9483190752e-1 <= .3461239156e-1 <- {z = -2^(1/2)+I*2^(1/2), x = 1, y = 3}</code><br><code>.9483190752e-1 <= .6922478312e-1 <- {z = -2^(1/2)+I*2^(1/2), x = 2, y = 3}</code><br></div></div> || Successful  
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| [https://dlmf.nist.gov/5.6.E9 5.6.E9] || [[Item:Q2076|<math>|\EulerGamma@{z}| <= (2\pi)^{1/2}|z|^{x-(1/2)}e^{-\pi|y|/2}\exp@{\tfrac{1}{6}|z|^{-1}}</math>]] || <code>abs(GAMMA(z))< =(2*Pi)^(1/ 2)*(abs(z))^(x -(1/ 2))* exp(- Pi*abs(y)/ 2)*exp((1)/(6)*(abs(z))^(- 1))</code> || <code>Abs[Gamma[z]]< =(2*Pi)^(1/ 2)*(Abs[z])^(x -(1/ 2))* Exp[- Pi*Abs[y]/ 2]*Exp[Divide[1,6]*(Abs[z])^(- 1)]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>.3896047846 <= .1665021267 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 2}</code><br><code>.3896047846 <= .3461239156e-1 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 3}</code><br><code>.3896047846 <= .3330042534 <- {z = 2^(1/2)+I*2^(1/2), x = 2, y = 2}</code><br><code>.3896047846 <= .6922478312e-1 <- {z = 2^(1/2)+I*2^(1/2), x = 2, y = 3}</code><br>... skip entries to safe data<br></div></div> || Successful  
 
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| [https://dlmf.nist.gov/5.7.E1 5.7.E1] || [[Item:Q2077|<math>\frac{1}{\EulerGamma@{z}} = \sum_{k=1}^{\infty}c_{k}z^{k}</math>]] || <code>(1)/(GAMMA(z))= sum(c[k]*(z)^(k), k = 1..infinity)</code> || <code>Divide[1,Gamma[z]]= Sum[Subscript[c, k]*(z)^(k), {k, 1, Infinity}]</code> || Failure || Failure || Skip || Skip  
 
| [https://dlmf.nist.gov/5.7.E1 5.7.E1] || [[Item:Q2077|<math>\frac{1}{\EulerGamma@{z}} = \sum_{k=1}^{\infty}c_{k}z^{k}</math>]] || <code>(1)/(GAMMA(z))= sum(c[k]*(z)^(k), k = 1..infinity)</code> || <code>Divide[1,Gamma[z]]= Sum[Subscript[c, k]*(z)^(k), {k, 1, Infinity}]</code> || Failure || Failure || Skip || Skip  
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| [https://dlmf.nist.gov/5.7.E8 5.7.E8] || [[Item:Q2084|<math>\imagpart@@{\digamma@{1+\iunit y}} = \sum_{k=1}^{\infty}\frac{y}{k^{2}+y^{2}}</math>]] || <code>Im(Psi(1 + I*y))= sum((y)/((k)^(2)+ (y)^(2)), k = 1..infinity)</code> || <code>Im[PolyGamma[1 + I*y]]= Sum[Divide[y,(k)^(2)+ (y)^(2)], {k, 1, Infinity}]</code> || Failure || Failure || Skip || Successful  
 
| [https://dlmf.nist.gov/5.7.E8 5.7.E8] || [[Item:Q2084|<math>\imagpart@@{\digamma@{1+\iunit y}} = \sum_{k=1}^{\infty}\frac{y}{k^{2}+y^{2}}</math>]] || <code>Im(Psi(1 + I*y))= sum((y)/((k)^(2)+ (y)^(2)), k = 1..infinity)</code> || <code>Im[PolyGamma[1 + I*y]]= Sum[Divide[y,(k)^(2)+ (y)^(2)], {k, 1, Infinity}]</code> || Failure || Failure || Skip || Successful  
 
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| [https://dlmf.nist.gov/5.8.E2 5.8.E2] || [[Item:Q2086|<math>\frac{1}{\EulerGamma@{z}} = ze^{\EulerConstant z}\prod_{k=1}^{\infty}\left(1+\frac{z}{k}\right)e^{-z/k}</math>]] || <code>(1)/(GAMMA(z))= z*exp(gamma*z)*product((1 +(z)/(k))* exp(- z/ k), k = 1..infinity)</code> || <code>Divide[1,Gamma[z]]= z*Exp[EulerGamma*z]*Product[(1 +Divide[z,k])* Exp[- z/ k], {k, 1, Infinity}]</code> || Successful || Failure || - || Error
+
| [https://dlmf.nist.gov/5.8.E2 5.8.E2] || [[Item:Q2086|<math>\frac{1}{\EulerGamma@{z}} = ze^{\EulerConstant z}\prod_{k=1}^{\infty}\left(1+\frac{z}{k}\right)e^{-z/k}</math>]] || <code>(1)/(GAMMA(z))= z*exp(gamma*z)*product((1 +(z)/(k))* exp(- z/ k), k = 1..infinity)</code> || <code>Divide[1,Gamma[z]]= z*Exp[EulerGamma*z]*Product[(1 +Divide[z,k])* Exp[- z/ k], {k, 1, Infinity}]</code> || Successful || Failure || - || Successful
 
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| [https://dlmf.nist.gov/5.8.E3 5.8.E3] || [[Item:Q2087|<math>\left|\frac{\EulerGamma@{x}}{\EulerGamma@{x+\iunit y}}\right|^{2} = \prod_{k=0}^{\infty}\left(1+\frac{y^{2}}{(x+k)^{2}}\right)</math>]] || <code>(abs((GAMMA(x))/(GAMMA(x + I*y))))^(2)= product(1 +((y)^(2))/((x + k)^(2)), k = 0..infinity)</code> || <code>(Abs[Divide[Gamma[x],Gamma[x + I*y]]])^(2)= Product[1 +Divide[(y)^(2),(x + k)^(2)], {k, 0, Infinity}]</code> || Failure || Failure || Skip || Successful  
 
| [https://dlmf.nist.gov/5.8.E3 5.8.E3] || [[Item:Q2087|<math>\left|\frac{\EulerGamma@{x}}{\EulerGamma@{x+\iunit y}}\right|^{2} = \prod_{k=0}^{\infty}\left(1+\frac{y^{2}}{(x+k)^{2}}\right)</math>]] || <code>(abs((GAMMA(x))/(GAMMA(x + I*y))))^(2)= product(1 +((y)^(2))/((x + k)^(2)), k = 0..infinity)</code> || <code>(Abs[Divide[Gamma[x],Gamma[x + I*y]]])^(2)= Product[1 +Divide[(y)^(2),(x + k)^(2)], {k, 0, Infinity}]</code> || Failure || Failure || Skip || Successful  
 
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| [https://dlmf.nist.gov/5.9.E1 5.9.E1] || [[Item:Q2090|<math>\frac{1}{\mu}\EulerGamma@{\frac{\nu}{\mu}}\frac{1}{z^{\nu/\mu}} = \int_{0}^{\infty}\exp@{-zt^{\mu}}t^{\nu-1}\diff{t}</math>]] || <code>(1)/(mu)*GAMMA((nu)/(mu))*(1)/((z)^(nu/ mu))= int(exp(- z*(t)^(mu))*(t)^(nu - 1), t = 0..infinity)</code> || <code>Divide[1,\[Mu]]*Gamma[Divide[\[Nu],\[Mu]]]*Divide[1,(z)^(\[Nu]/ \[Mu])]= Integrate[Exp[- z*(t)^(\[Mu])]*(t)^(\[Nu]- 1), {t, 0, Infinity}]</code> || Failure || Failure || Skip || Error
+
| [https://dlmf.nist.gov/5.9.E1 5.9.E1] || [[Item:Q2090|<math>\frac{1}{\mu}\EulerGamma@{\frac{\nu}{\mu}}\frac{1}{z^{\nu/\mu}} = \int_{0}^{\infty}\exp@{-zt^{\mu}}t^{\nu-1}\diff{t}</math>]] || <code>(1)/(mu)*GAMMA((nu)/(mu))*(1)/((z)^(nu/ mu))= int(exp(- z*(t)^(mu))*(t)^(nu - 1), t = 0..infinity)</code> || <code>Divide[1,\[Mu]]*Gamma[Divide[\[Nu],\[Mu]]]*Divide[1,(z)^(\[Nu]/ \[Mu])]= Integrate[Exp[- z*(t)^(\[Mu])]*(t)^(\[Nu]- 1), {t, 0, Infinity}]</code> || Failure || Failure || Skip || Successful
 
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| [https://dlmf.nist.gov/5.9.E2 5.9.E2] || [[Item:Q2091|<math>\frac{1}{\EulerGamma@{z}} = \frac{1}{2\pi i}\int_{-\infty}^{(0+)}e^{t}t^{-z}\diff{t}</math>]] || <code>(1)/(GAMMA(z))=(1)/(2*Pi*I)*int(exp(t)*(t)^(- z), t = - infinity..(0 +))</code> || <code>Divide[1,Gamma[z]]=Divide[1,2*Pi*I]*Integrate[Exp[t]*(t)^(- z), {t, - Infinity, (0 +)}]</code> || Error || Failure || - || Error  
 
| [https://dlmf.nist.gov/5.9.E2 5.9.E2] || [[Item:Q2091|<math>\frac{1}{\EulerGamma@{z}} = \frac{1}{2\pi i}\int_{-\infty}^{(0+)}e^{t}t^{-z}\diff{t}</math>]] || <code>(1)/(GAMMA(z))=(1)/(2*Pi*I)*int(exp(t)*(t)^(- z), t = - infinity..(0 +))</code> || <code>Divide[1,Gamma[z]]=Divide[1,2*Pi*I]*Integrate[Exp[t]*(t)^(- z), {t, - Infinity, (0 +)}]</code> || Error || Failure || - || Error  
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| [https://dlmf.nist.gov/5.9.E4 5.9.E4] || [[Item:Q2093|<math>\EulerGamma@{z} = \int_{1}^{\infty}t^{z-1}e^{-t}\diff{t}+\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(z+k)k!}</math>]] || <code>GAMMA(z)= int((t)^(z - 1)* exp(- t), t = 1..infinity)+ sum(((- 1)^(k))/((z + k)* factorial(k)), k = 0..infinity)</code> || <code>Gamma[z]= Integrate[(t)^(z - 1)* Exp[- t], {t, 1, Infinity}]+ Sum[Divide[(- 1)^(k),(z + k)* (k)!], {k, 0, Infinity}]</code> || Failure || Successful || Skip || -  
 
| [https://dlmf.nist.gov/5.9.E4 5.9.E4] || [[Item:Q2093|<math>\EulerGamma@{z} = \int_{1}^{\infty}t^{z-1}e^{-t}\diff{t}+\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(z+k)k!}</math>]] || <code>GAMMA(z)= int((t)^(z - 1)* exp(- t), t = 1..infinity)+ sum(((- 1)^(k))/((z + k)* factorial(k)), k = 0..infinity)</code> || <code>Gamma[z]= Integrate[(t)^(z - 1)* Exp[- t], {t, 1, Infinity}]+ Sum[Divide[(- 1)^(k),(z + k)* (k)!], {k, 0, Infinity}]</code> || Failure || Successful || Skip || -  
 
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| [https://dlmf.nist.gov/5.9.E5 5.9.E5] || [[Item:Q2094|<math>\EulerGamma@{z} = \int_{0}^{\infty}t^{z-1}\left(e^{-t}-\sum_{k=0}^{n}\frac{(-1)^{k}t^{k}}{k!}\right)\diff{t}</math>]] || <code>GAMMA(z)= int((t)^(z - 1)*(exp(- t)- sum(((- 1)^(k)* (t)^(k))/(factorial(k)), k = 0..n)), t = 0..infinity)</code> || <code>Gamma[z]= Integrate[(t)^(z - 1)*(Exp[- t]- Sum[Divide[(- 1)^(k)* (t)^(k),(k)!], {k, 0, n}]), {t, 0, Infinity}]</code> || Failure || Failure || Skip || Error
+
| [https://dlmf.nist.gov/5.9.E5 5.9.E5] || [[Item:Q2094|<math>\EulerGamma@{z} = \int_{0}^{\infty}t^{z-1}\left(e^{-t}-\sum_{k=0}^{n}\frac{(-1)^{k}t^{k}}{k!}\right)\diff{t}</math>]] || <code>GAMMA(z)= int((t)^(z - 1)*(exp(- t)- sum(((- 1)^(k)* (t)^(k))/(factorial(k)), k = 0..n)), t = 0..infinity)</code> || <code>Gamma[z]= Integrate[(t)^(z - 1)*(Exp[- t]- Sum[Divide[(- 1)^(k)* (t)^(k),(k)!], {k, 0, n}]), {t, 0, Infinity}]</code> || Failure || Failure || Skip || Skip
 
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| [https://dlmf.nist.gov/5.9.E6 5.9.E6] || [[Item:Q2095|<math>\EulerGamma@{z}\cos@{\tfrac{1}{2}\pi z} = \int_{0}^{\infty}t^{z-1}\cos@@{t}\diff{t}</math>]] || <code>GAMMA(z)*cos((1)/(2)*Pi*z)= int((t)^(z - 1)* cos(t), t = 0..infinity)</code> || <code>Gamma[z]*Cos[Divide[1,2]*Pi*z]= Integrate[(t)^(z - 1)* Cos[t], {t, 0, Infinity}]</code> || Successful || Failure || - || Error
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| [https://dlmf.nist.gov/5.9.E6 5.9.E6] || [[Item:Q2095|<math>\EulerGamma@{z}\cos@{\tfrac{1}{2}\pi z} = \int_{0}^{\infty}t^{z-1}\cos@@{t}\diff{t}</math>]] || <code>GAMMA(z)*cos((1)/(2)*Pi*z)= int((t)^(z - 1)* cos(t), t = 0..infinity)</code> || <code>Gamma[z]*Cos[Divide[1,2]*Pi*z]= Integrate[(t)^(z - 1)* Cos[t], {t, 0, Infinity}]</code> || Successful || Failure || - || Successful
 
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| [https://dlmf.nist.gov/5.9.E7 5.9.E7] || [[Item:Q2096|<math>\EulerGamma@{z}\sin@{\tfrac{1}{2}\pi z} = \int_{0}^{\infty}t^{z-1}\sin@@{t}\diff{t}</math>]] || <code>GAMMA(z)*sin((1)/(2)*Pi*z)= int((t)^(z - 1)* sin(t), t = 0..infinity)</code> || <code>Gamma[z]*Sin[Divide[1,2]*Pi*z]= Integrate[(t)^(z - 1)* Sin[t], {t, 0, Infinity}]</code> || Successful || Failure || - || Error
+
| [https://dlmf.nist.gov/5.9.E7 5.9.E7] || [[Item:Q2096|<math>\EulerGamma@{z}\sin@{\tfrac{1}{2}\pi z} = \int_{0}^{\infty}t^{z-1}\sin@@{t}\diff{t}</math>]] || <code>GAMMA(z)*sin((1)/(2)*Pi*z)= int((t)^(z - 1)* sin(t), t = 0..infinity)</code> || <code>Gamma[z]*Sin[Divide[1,2]*Pi*z]= Integrate[(t)^(z - 1)* Sin[t], {t, 0, Infinity}]</code> || Successful || Failure || - || Successful
 
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| [https://dlmf.nist.gov/5.9.E8 5.9.E8] || [[Item:Q2097|<math>\EulerGamma@{1+\frac{1}{n}}\cos@{\frac{\pi}{2n}} = \int_{0}^{\infty}\cos@{t^{n}}\diff{t}</math>]] || <code>GAMMA(1 +(1)/(n))*cos((Pi)/(2*n))= int(cos((t)^(n)), t = 0..infinity)</code> || <code>Gamma[1 +Divide[1,n]]*Cos[Divide[Pi,2*n]]= Integrate[Cos[(t)^(n)], {t, 0, Infinity}]</code> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[ConditionalExpression[0, And[Greater[Re[n], 1], Equal[Im[n], 0]]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[ConditionalExpression[0, And[Greater[Re[n], 1], Equal[Im[n], 0]]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[ConditionalExpression[0, And[Greater[Re[n], 1], Equal[Im[n], 0]]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[ConditionalExpression[0, And[Greater[Re[n], 1], Equal[Im[n], 0]]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br></div></div>
+
| [https://dlmf.nist.gov/5.9.E8 5.9.E8] || [[Item:Q2097|<math>\EulerGamma@{1+\frac{1}{n}}\cos@{\frac{\pi}{2n}} = \int_{0}^{\infty}\cos@{t^{n}}\diff{t}</math>]] || <code>GAMMA(1 +(1)/(n))*cos((Pi)/(2*n))= int(cos((t)^(n)), t = 0..infinity)</code> || <code>Gamma[1 +Divide[1,n]]*Cos[Divide[Pi,2*n]]= Integrate[Cos[(t)^(n)], {t, 0, Infinity}]</code> || Successful || Failure || - || Successful
 
|-
 
|-
| [https://dlmf.nist.gov/5.9.E9 5.9.E9] || [[Item:Q2098|<math>\EulerGamma@{1+\frac{1}{n}}\sin@{\frac{\pi}{2n}} = \int_{0}^{\infty}\sin@{t^{n}}\diff{t}</math>]] || <code>GAMMA(1 +(1)/(n))*sin((Pi)/(2*n))= int(sin((t)^(n)), t = 0..infinity)</code> || <code>Gamma[1 +Divide[1,n]]*Sin[Divide[Pi,2*n]]= Integrate[Sin[(t)^(n)], {t, 0, Infinity}]</code> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[ConditionalExpression[0, And[Greater[Re[n], 1], Equal[Im[n], 0]]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[ConditionalExpression[0, And[Greater[Re[n], 1], Equal[Im[n], 0]]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[ConditionalExpression[0, And[Greater[Re[n], 1], Equal[Im[n], 0]]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[ConditionalExpression[0, And[Greater[Re[n], 1], Equal[Im[n], 0]]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br></div></div>
+
| [https://dlmf.nist.gov/5.9.E9 5.9.E9] || [[Item:Q2098|<math>\EulerGamma@{1+\frac{1}{n}}\sin@{\frac{\pi}{2n}} = \int_{0}^{\infty}\sin@{t^{n}}\diff{t}</math>]] || <code>GAMMA(1 +(1)/(n))*sin((Pi)/(2*n))= int(sin((t)^(n)), t = 0..infinity)</code> || <code>Gamma[1 +Divide[1,n]]*Sin[Divide[Pi,2*n]]= Integrate[Sin[(t)^(n)], {t, 0, Infinity}]</code> || Successful || Failure || - || Successful
 
|-
 
|-
| [https://dlmf.nist.gov/5.9.E10 5.9.E10] || [[Item:Q2099|<math>\Ln@@{\EulerGamma@{z}} = \left(z-\tfrac{1}{2}\right)\ln@@{z}-z+\tfrac{1}{2}\ln@{2\pi}+2\int_{0}^{\infty}\frac{\atan@{t/z}}{e^{2\pi t}-1}\diff{t}</math>]] || <code>ln(GAMMA(z))=(z -(1)/(2))* ln(z)- z +(1)/(2)*ln(2*Pi)+ 2*int((arctan(t/ z))/(exp(2*Pi*t)- 1), t = 0..infinity)</code> || <code>Log[Gamma[z]]=(z -Divide[1,2])* Log[z]- z +Divide[1,2]*Log[2*Pi]+ 2*Integrate[Divide[ArcTan[t/ z],Exp[2*Pi*t]- 1], {t, 0, Infinity}]</code> || Failure || Failure || Skip || Error
+
| [https://dlmf.nist.gov/5.9.E10 5.9.E10] || [[Item:Q2099|<math>\Ln@@{\EulerGamma@{z}} = \left(z-\tfrac{1}{2}\right)\ln@@{z}-z+\tfrac{1}{2}\ln@{2\pi}+2\int_{0}^{\infty}\frac{\atan@{t/z}}{e^{2\pi t}-1}\diff{t}</math>]] || <code>ln(GAMMA(z))=(z -(1)/(2))* ln(z)- z +(1)/(2)*ln(2*Pi)+ 2*int((arctan(t/ z))/(exp(2*Pi*t)- 1), t = 0..infinity)</code> || <code>Log[Gamma[z]]=(z -Divide[1,2])* Log[z]- z +Divide[1,2]*Log[2*Pi]+ 2*Integrate[Divide[ArcTan[t/ z],Exp[2*Pi*t]- 1], {t, 0, Infinity}]</code> || Failure || Failure || Skip || Skip
 
|-
 
|-
| [https://dlmf.nist.gov/5.9.E12 5.9.E12] || [[Item:Q2101|<math>\digamma@{z} = \int_{0}^{\infty}\left(\frac{e^{-t}}{t}-\frac{e^{-zt}}{1-e^{-t}}\right)\diff{t}</math>]] || <code>Psi(z)= int((exp(- t))/(t)-(exp(- z*t))/(1 - exp(- t)), t = 0..infinity)</code> || <code>PolyGamma[z]= Integrate[Divide[Exp[- t],t]-Divide[Exp[- z*t],1 - Exp[- t]], {t, 0, Infinity}]</code> || Failure || Failure || Skip || Error
+
| [https://dlmf.nist.gov/5.9.E12 5.9.E12] || [[Item:Q2101|<math>\digamma@{z} = \int_{0}^{\infty}\left(\frac{e^{-t}}{t}-\frac{e^{-zt}}{1-e^{-t}}\right)\diff{t}</math>]] || <code>Psi(z)= int((exp(- t))/(t)-(exp(- z*t))/(1 - exp(- t)), t = 0..infinity)</code> || <code>PolyGamma[z]= Integrate[Divide[Exp[- t],t]-Divide[Exp[- z*t],1 - Exp[- t]], {t, 0, Infinity}]</code> || Failure || Failure || Skip || Successful
 
|-
 
|-
 
| [https://dlmf.nist.gov/5.9.E13 5.9.E13] || [[Item:Q2102|<math>\digamma@{z} = \ln@@{z}+\int_{0}^{\infty}\left(\frac{1}{t}-\frac{1}{1-e^{-t}}\right)e^{-tz}\diff{t}</math>]] || <code>Psi(z)= ln(z)+ int(((1)/(t)-(1)/(1 - exp(- t)))* exp(- t*z), t = 0..infinity)</code> || <code>PolyGamma[z]= Log[z]+ Integrate[(Divide[1,t]-Divide[1,1 - Exp[- t]])* Exp[- t*z], {t, 0, Infinity}]</code> || Failure || Failure || Skip || Error  
 
| [https://dlmf.nist.gov/5.9.E13 5.9.E13] || [[Item:Q2102|<math>\digamma@{z} = \ln@@{z}+\int_{0}^{\infty}\left(\frac{1}{t}-\frac{1}{1-e^{-t}}\right)e^{-tz}\diff{t}</math>]] || <code>Psi(z)= ln(z)+ int(((1)/(t)-(1)/(1 - exp(- t)))* exp(- t*z), t = 0..infinity)</code> || <code>PolyGamma[z]= Log[z]+ Integrate[(Divide[1,t]-Divide[1,1 - Exp[- t]])* Exp[- t*z], {t, 0, Infinity}]</code> || Failure || Failure || Skip || Error  
 
|-
 
|-
| [https://dlmf.nist.gov/5.9.E14 5.9.E14] || [[Item:Q2103|<math>\digamma@{z} = \int_{0}^{\infty}\left(e^{-t}-\frac{1}{(1+t)^{z}}\right)\frac{\diff{t}}{t}</math>]] || <code>Psi(z)= int((exp(- t)-(1)/((1 + t)^(z)))*(1)/(t), t = 0..infinity)</code> || <code>PolyGamma[z]= Integrate[(Exp[- t]-Divide[1,(1 + t)^(z)])*Divide[1,t], {t, 0, Infinity}]</code> || Failure || Failure || Skip || Error
+
| [https://dlmf.nist.gov/5.9.E14 5.9.E14] || [[Item:Q2103|<math>\digamma@{z} = \int_{0}^{\infty}\left(e^{-t}-\frac{1}{(1+t)^{z}}\right)\frac{\diff{t}}{t}</math>]] || <code>Psi(z)= int((exp(- t)-(1)/((1 + t)^(z)))*(1)/(t), t = 0..infinity)</code> || <code>PolyGamma[z]= Integrate[(Exp[- t]-Divide[1,(1 + t)^(z)])*Divide[1,t], {t, 0, Infinity}]</code> || Failure || Failure || Skip || Successful
 
|-
 
|-
| [https://dlmf.nist.gov/5.9.E15 5.9.E15] || [[Item:Q2104|<math>\digamma@{z} = \ln@@{z}-\frac{1}{2z}-2\int_{0}^{\infty}\frac{t\diff{t}}{(t^{2}+z^{2})(e^{2\pi t}-1)}</math>]] || <code>Psi(z)= ln(z)-(1)/(2*z)- 2*int((t)/(((t)^(2)+ (z)^(2))*(exp(2*Pi*t)- 1)), t = 0..infinity)</code> || <code>PolyGamma[z]= Log[z]-Divide[1,2*z]- 2*Integrate[Divide[t,((t)^(2)+ (z)^(2))*(Exp[2*Pi*t]- 1)], {t, 0, Infinity}]</code> || Failure || Failure || Skip || Error
+
| [https://dlmf.nist.gov/5.9.E15 5.9.E15] || [[Item:Q2104|<math>\digamma@{z} = \ln@@{z}-\frac{1}{2z}-2\int_{0}^{\infty}\frac{t\diff{t}}{(t^{2}+z^{2})(e^{2\pi t}-1)}</math>]] || <code>Psi(z)= ln(z)-(1)/(2*z)- 2*int((t)/(((t)^(2)+ (z)^(2))*(exp(2*Pi*t)- 1)), t = 0..infinity)</code> || <code>PolyGamma[z]= Log[z]-Divide[1,2*z]- 2*Integrate[Divide[t,((t)^(2)+ (z)^(2))*(Exp[2*Pi*t]- 1)], {t, 0, Infinity}]</code> || Failure || Failure || Skip || Skip
 
|-
 
|-
| [https://dlmf.nist.gov/5.9.E16 5.9.E16] || [[Item:Q2105|<math>\digamma@{z}+\EulerConstant = \int_{0}^{\infty}\frac{e^{-t}-e^{-zt}}{1-e^{-t}}\diff{t}</math>]] || <code>Psi(z)+ gamma = int((exp(- t)- exp(- z*t))/(1 - exp(- t)), t = 0..infinity)</code> || <code>PolyGamma[z]+ EulerGamma = Integrate[Divide[Exp[- t]- Exp[- z*t],1 - Exp[- t]], {t, 0, Infinity}]</code> || Failure || Failure || Skip || Error
+
| [https://dlmf.nist.gov/5.9.E16 5.9.E16] || [[Item:Q2105|<math>\digamma@{z}+\EulerConstant = \int_{0}^{\infty}\frac{e^{-t}-e^{-zt}}{1-e^{-t}}\diff{t}</math>]] || <code>Psi(z)+ gamma = int((exp(- t)- exp(- z*t))/(1 - exp(- t)), t = 0..infinity)</code> || <code>PolyGamma[z]+ EulerGamma = Integrate[Divide[Exp[- t]- Exp[- z*t],1 - Exp[- t]], {t, 0, Infinity}]</code> || Failure || Failure || Skip || Successful
 
|-
 
|-
| [https://dlmf.nist.gov/5.9.E16 5.9.E16] || [[Item:Q2105|<math>\int_{0}^{\infty}\frac{e^{-t}-e^{-zt}}{1-e^{-t}}\diff{t} = \int_{0}^{1}\frac{1-t^{z-1}}{1-t}\diff{t}</math>]] || <code>int((exp(- t)- exp(- z*t))/(1 - exp(- t)), t = 0..infinity)= int((1 - (t)^(z - 1))/(1 - t), t = 0..1)</code> || <code>Integrate[Divide[Exp[- t]- Exp[- z*t],1 - Exp[- t]], {t, 0, Infinity}]= Integrate[Divide[1 - (t)^(z - 1),1 - t], {t, 0, 1}]</code> || Failure || Failure || Skip || Error
+
| [https://dlmf.nist.gov/5.9.E16 5.9.E16] || [[Item:Q2105|<math>\int_{0}^{\infty}\frac{e^{-t}-e^{-zt}}{1-e^{-t}}\diff{t} = \int_{0}^{1}\frac{1-t^{z-1}}{1-t}\diff{t}</math>]] || <code>int((exp(- t)- exp(- z*t))/(1 - exp(- t)), t = 0..infinity)= int((1 - (t)^(z - 1))/(1 - t), t = 0..1)</code> || <code>Integrate[Divide[Exp[- t]- Exp[- z*t],1 - Exp[- t]], {t, 0, Infinity}]= Integrate[Divide[1 - (t)^(z - 1),1 - t], {t, 0, 1}]</code> || Failure || Failure || Skip || Skip
 
|-
 
|-
 
| [https://dlmf.nist.gov/5.9.E17 5.9.E17] || [[Item:Q2106|<math>\digamma@{z+1} = -\EulerConstant+\frac{1}{2\pi i}\int_{-c-\infty i}^{-c+\infty i}\frac{\pi z^{-s-1}}{\sin@{\pi s}}\Riemannzeta@{-s}\diff{s}</math>]] || <code>Psi(z + 1)= - gamma +(1)/(2*Pi*I)*int((Pi*(z)^(- s - 1))/(sin(Pi*s))*Zeta(- s), s = - c - infinity*I..- c + infinity*I)</code> || <code>PolyGamma[z + 1]= - EulerGamma +Divide[1,2*Pi*I]*Integrate[Divide[Pi*(z)^(- s - 1),Sin[Pi*s]]*Zeta[- s], {s, - c - Infinity*I, - c + Infinity*I}]</code> || Failure || Failure || Skip || Skip  
 
| [https://dlmf.nist.gov/5.9.E17 5.9.E17] || [[Item:Q2106|<math>\digamma@{z+1} = -\EulerConstant+\frac{1}{2\pi i}\int_{-c-\infty i}^{-c+\infty i}\frac{\pi z^{-s-1}}{\sin@{\pi s}}\Riemannzeta@{-s}\diff{s}</math>]] || <code>Psi(z + 1)= - gamma +(1)/(2*Pi*I)*int((Pi*(z)^(- s - 1))/(sin(Pi*s))*Zeta(- s), s = - c - infinity*I..- c + infinity*I)</code> || <code>PolyGamma[z + 1]= - EulerGamma +Divide[1,2*Pi*I]*Integrate[Divide[Pi*(z)^(- s - 1),Sin[Pi*s]]*Zeta[- s], {s, - c - Infinity*I, - c + Infinity*I}]</code> || Failure || Failure || Skip || Skip  
Line 155: Line 155:
 
| [https://dlmf.nist.gov/5.9.E19 5.9.E19] || [[Item:Q2108|<math>\EulerGamma^{(n)}@{z} = \int_{0}^{\infty}(\ln@@{t})^{n}e^{-t}t^{z-1}\diff{t}</math>]] || <code>subs( temp=z, diff( GAMMA(temp), temp$(n) ) )= int((ln(t))^(n)* exp(- t)*(t)^(z - 1), t = 0..infinity)</code> || <code>(D[Gamma[temp], {temp, n}]/.temp-> z)= Integrate[(Log[t])^(n)* Exp[- t]*(t)^(z - 1), {t, 0, Infinity}]</code> || Successful || Failure || - || Error  
 
| [https://dlmf.nist.gov/5.9.E19 5.9.E19] || [[Item:Q2108|<math>\EulerGamma^{(n)}@{z} = \int_{0}^{\infty}(\ln@@{t})^{n}e^{-t}t^{z-1}\diff{t}</math>]] || <code>subs( temp=z, diff( GAMMA(temp), temp$(n) ) )= int((ln(t))^(n)* exp(- t)*(t)^(z - 1), t = 0..infinity)</code> || <code>(D[Gamma[temp], {temp, n}]/.temp-> z)= Integrate[(Log[t])^(n)* Exp[- t]*(t)^(z - 1), {t, 0, Infinity}]</code> || Successful || Failure || - || Error  
 
|-
 
|-
| [https://dlmf.nist.gov/5.11.E5 5.11.E5] || [[Item:Q2127|<math>g_{k} = \sqrt{2}\Pochhammersym{\tfrac{1}{2}}{k}a_{2k}</math>]] || <code>g[k]=sqrt(2)*pochhammer((1)/(2), k)*a[2*k]</code> || <code>Subscript[g, k]=Sqrt[2]*Pochhammer[Divide[1,2], k]*Subscript[a, 2*k]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>.4142135625+.4142135625*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)+I*2^(1/2), k = 1}</code><br><code>-.8578643792e-1-.8578643792e-1*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)+I*2^(1/2), k = 2}</code><br><code>-2.335786436-2.335786436*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)+I*2^(1/2), k = 3}</code><br><code>.4142135625-2.414213561*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)-I*2^(1/2), k = 1}</code><br><code>-.8578643792e-1-2.914213562*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)-I*2^(1/2), k = 2}</code><br><code>-2.335786436-5.164213560*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)-I*2^(1/2), k = 3}</code><br><code>-2.414213561-2.414213561*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = -2^(1/2)-I*2^(1/2), k = 1}</code><br><code>-2.914213562-2.914213562*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = -2^(1/2)-I*2^(1/2), k = 2}</code><br><code>-5.164213560-5.164213560*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = -2^(1/2)-I*2^(1/2), k = 3}</code><br><code>-2.414213561+.4142135625*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = -2^(1/2)+I*2^(1/2), k = 1}</code><br><code>-2.914213562-.8578643792e-1*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = -2^(1/2)+I*2^(1/2), k = 2}</code><br><code>-5.164213560-2.335786436*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = -2^(1/2)+I*2^(1/2), k = 3}</code><br><code>.4142135625+2.414213561*I <- {a[2*k] = 2^(1/2)-I*2^(1/2), g[k] = 2^(1/2)+I*2^(1/2), k = 1}</code><br><code>-.8578643792e-1+2.914213562*I <- {a[2*k] = 2^(1/2)-I*2^(1/2), g[k] = 2^(1/2)+I*2^(1/2), k = 2}</code><br><code>-2.335786436+5.164213560*I <- {a[2*k] = 2^(1/2)-I*2^(1/2), g[k] = 2^(1/2)+I*2^(1/2), k = 3}</code><br><code>.4142135625-.4142135625*I <- {a[2*k] = 2^(1/2)-I*2^(1/2), g[k] = 2^(1/2)-I*2^(1/2), k = 1}</code><br><code>-.8578643792e-1+.8578643792e-1*I <- {a[2*k] = 2^(1/2)-I*2^(1/2), g[k] = 2^(1/2)-I*2^(1/2), k = 2}</code><br><code>-2.335786436+2.335786436*I <- {a[2*k] = 2^(1/2)-I*2^(1/2), g[k] = 2^(1/2)-I*2^(1/2), k = 3}</code><br><code>-2.414213561-.4142135625*I <- {a[2*k] = 2^(1/2)-I*2^(1/2), g[k] = -2^(1/2)-I*2^(1/2), k = 1}</code><br><code>-2.914213562+.8578643792e-1*I <- {a[2*k] = 2^(1/2)-I*2^(1/2), g[k] = -2^(1/2)-I*2^(1/2), k = 2}</code><br><code>-5.164213560+2.335786436*I <- {a[2*k] = 2^(1/2)-I*2^(1/2), g[k] = -2^(1/2)-I*2^(1/2), k = 3}</code><br><code>-2.414213561+2.414213561*I <- {a[2*k] = 2^(1/2)-I*2^(1/2), g[k] = -2^(1/2)+I*2^(1/2), k = 1}</code><br><code>-2.914213562+2.914213562*I <- {a[2*k] = 2^(1/2)-I*2^(1/2), g[k] = -2^(1/2)+I*2^(1/2), k = 2}</code><br><code>-5.164213560+5.164213560*I <- {a[2*k] = 2^(1/2)-I*2^(1/2), g[k] = -2^(1/2)+I*2^(1/2), k = 3}</code><br><code>2.414213561+2.414213561*I <- {a[2*k] = -2^(1/2)-I*2^(1/2), g[k] = 2^(1/2)+I*2^(1/2), k = 1}</code><br><code>2.914213562+2.914213562*I <- {a[2*k] = -2^(1/2)-I*2^(1/2), g[k] = 2^(1/2)+I*2^(1/2), k = 2}</code><br><code>5.164213560+5.164213560*I <- {a[2*k] = -2^(1/2)-I*2^(1/2), g[k] = 2^(1/2)+I*2^(1/2), k = 3}</code><br><code>2.414213561-.4142135625*I <- {a[2*k] = -2^(1/2)-I*2^(1/2), g[k] = 2^(1/2)-I*2^(1/2), k = 1}</code><br><code>2.914213562+.8578643792e-1*I <- {a[2*k] = -2^(1/2)-I*2^(1/2), g[k] = 2^(1/2)-I*2^(1/2), k = 2}</code><br><code>5.164213560+2.335786436*I <- {a[2*k] = -2^(1/2)-I*2^(1/2), g[k] = 2^(1/2)-I*2^(1/2), k = 3}</code><br><code>-.4142135625-.4142135625*I <- {a[2*k] = -2^(1/2)-I*2^(1/2), g[k] = -2^(1/2)-I*2^(1/2), k = 1}</code><br><code>.8578643792e-1+.8578643792e-1*I <- {a[2*k] = -2^(1/2)-I*2^(1/2), g[k] = -2^(1/2)-I*2^(1/2), k = 2}</code><br><code>2.335786436+2.335786436*I <- {a[2*k] = -2^(1/2)-I*2^(1/2), g[k] = -2^(1/2)-I*2^(1/2), k = 3}</code><br><code>-.4142135625+2.414213561*I <- {a[2*k] = -2^(1/2)-I*2^(1/2), g[k] = -2^(1/2)+I*2^(1/2), k = 1}</code><br><code>.8578643792e-1+2.914213562*I <- {a[2*k] = -2^(1/2)-I*2^(1/2), g[k] = -2^(1/2)+I*2^(1/2), k = 2}</code><br><code>2.335786436+5.164213560*I <- {a[2*k] = -2^(1/2)-I*2^(1/2), g[k] = -2^(1/2)+I*2^(1/2), k = 3}</code><br><code>2.414213561+.4142135625*I <- {a[2*k] = -2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)+I*2^(1/2), k = 1}</code><br><code>2.914213562-.8578643792e-1*I <- {a[2*k] = -2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)+I*2^(1/2), k = 2}</code><br><code>5.164213560-2.335786436*I <- {a[2*k] = -2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)+I*2^(1/2), k = 3}</code><br><code>2.414213561-2.414213561*I <- {a[2*k] = -2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)-I*2^(1/2), k = 1}</code><br><code>2.914213562-2.914213562*I <- {a[2*k] = -2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)-I*2^(1/2), k = 2}</code><br><code>5.164213560-5.164213560*I <- {a[2*k] = -2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)-I*2^(1/2), k = 3}</code><br><code>-.4142135625-2.414213561*I <- {a[2*k] = -2^(1/2)+I*2^(1/2), g[k] = -2^(1/2)-I*2^(1/2), k = 1}</code><br><code>.8578643792e-1-2.914213562*I <- {a[2*k] = -2^(1/2)+I*2^(1/2), g[k] = -2^(1/2)-I*2^(1/2), k = 2}</code><br><code>2.335786436-5.164213560*I <- {a[2*k] = -2^(1/2)+I*2^(1/2), g[k] = -2^(1/2)-I*2^(1/2), k = 3}</code><br><code>-.4142135625+.4142135625*I <- {a[2*k] = -2^(1/2)+I*2^(1/2), g[k] = -2^(1/2)+I*2^(1/2), k = 1}</code><br><code>.8578643792e-1-.8578643792e-1*I <- {a[2*k] = -2^(1/2)+I*2^(1/2), g[k] = -2^(1/2)+I*2^(1/2), k = 2}</code><br><code>2.335786436-2.335786436*I <- {a[2*k] = -2^(1/2)+I*2^(1/2), g[k] = -2^(1/2)+I*2^(1/2), k = 3}</code><br></div></div> || Successful  
+
| [https://dlmf.nist.gov/5.11.E5 5.11.E5] || [[Item:Q2127|<math>g_{k} = \sqrt{2}\Pochhammersym{\tfrac{1}{2}}{k}a_{2k}</math>]] || <code>g[k]=sqrt(2)*pochhammer((1)/(2), k)*a[2*k]</code> || <code>Subscript[g, k]=Sqrt[2]*Pochhammer[Divide[1,2], k]*Subscript[a, 2*k]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>.4142135625+.4142135625*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)+I*2^(1/2), k = 1}</code><br><code>-.8578643792e-1-.8578643792e-1*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)+I*2^(1/2), k = 2}</code><br><code>-2.335786436-2.335786436*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)+I*2^(1/2), k = 3}</code><br><code>.4142135625-2.414213561*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)-I*2^(1/2), k = 1}</code><br>... skip entries to safe data<br></div></div> || Successful  
 
|-
 
|-
 
| [https://dlmf.nist.gov/5.11.E10 5.11.E10] || [[Item:Q2132|<math>\EulerGamma@{z} = e^{-z}z^{z}\left(\frac{2\pi}{z}\right)^{1/2}\left(\sum_{k=0}^{K-1}\frac{g_{k}}{z^{k}}+R_{K}(z)\right)</math>]] || <code>GAMMA(z)= exp(- z)*(z)^(z)*((2*Pi)/(z))^(1/ 2)*(sum((g[k])/((z)^(k)), k = 0..K - 1)+ R[K]*(z))</code> || <code>Gamma[z]= Exp[- z]*(z)^(z)*(Divide[2*Pi,z])^(1/ 2)*(Sum[Divide[Subscript[g, k],(z)^(k)], {k, 0, K - 1}]+ Subscript[R, K]*(z))</code> || Failure || Failure || Skip || Skip  
 
| [https://dlmf.nist.gov/5.11.E10 5.11.E10] || [[Item:Q2132|<math>\EulerGamma@{z} = e^{-z}z^{z}\left(\frac{2\pi}{z}\right)^{1/2}\left(\sum_{k=0}^{K-1}\frac{g_{k}}{z^{k}}+R_{K}(z)\right)</math>]] || <code>GAMMA(z)= exp(- z)*(z)^(z)*((2*Pi)/(z))^(1/ 2)*(sum((g[k])/((z)^(k)), k = 0..K - 1)+ R[K]*(z))</code> || <code>Gamma[z]= Exp[- z]*(z)^(z)*(Divide[2*Pi,z])^(1/ 2)*(Sum[Divide[Subscript[g, k],(z)^(k)], {k, 0, K - 1}]+ Subscript[R, K]*(z))</code> || Failure || Failure || Skip || Skip  
Line 165: Line 165:
 
| [https://dlmf.nist.gov/5.11#Ex13 5.11#Ex13] || [[Item:Q2142|<math>H_{2}(a,b) = \frac{1}{240}\binom{a-b}{4}(2(a-b+1)+5(a-b+1)^{2})</math>]] || <code>H[2]*(a , b)=(1)/(240)*binomial(a - b,4)*(2*(a - b + 1)+ 5*(a - b + 1)^(2))</code> || <code>Subscript[H, 2]*(a , b)=Divide[1,240]*Binomial[a - b,4]*(2*(a - b + 1)+ 5*(a - b + 1)^(2))</code> || Failure || Failure || Error || Error  
 
| [https://dlmf.nist.gov/5.11#Ex13 5.11#Ex13] || [[Item:Q2142|<math>H_{2}(a,b) = \frac{1}{240}\binom{a-b}{4}(2(a-b+1)+5(a-b+1)^{2})</math>]] || <code>H[2]*(a , b)=(1)/(240)*binomial(a - b,4)*(2*(a - b + 1)+ 5*(a - b + 1)^(2))</code> || <code>Subscript[H, 2]*(a , b)=Divide[1,240]*Binomial[a - b,4]*(2*(a - b + 1)+ 5*(a - b + 1)^(2))</code> || Failure || Failure || Error || Error  
 
|-
 
|-
| [https://dlmf.nist.gov/5.12.E1 5.12.E1] || [[Item:Q2146|<math>\EulerBeta@{a}{b} = \int_{0}^{1}t^{a-1}(1-t)^{b-1}\diff{t}</math>]] || <code>Beta(a, b)= int((t)^(a - 1)*(1 - t)^(b - 1), t = 0..1)</code> || <code>Beta[a, b]= Integrate[(t)^(a - 1)*(1 - t)^(b - 1), {t, 0, 1}]</code> || Failure || Failure || Skip || Skip
+
| [https://dlmf.nist.gov/5.12.E1 5.12.E1] || [[Item:Q2146|<math>\EulerBeta@{a}{b} = \int_{0}^{1}t^{a-1}(1-t)^{b-1}\diff{t}</math>]] || <code>Beta(a, b)= int((t)^(a - 1)*(1 - t)^(b - 1), t = 0..1)</code> || <code>Beta[a, b]= Integrate[(t)^(a - 1)*(1 - t)^(b - 1), {t, 0, 1}]</code> || Failure || Failure || Skip || Successful
 
|-
 
|-
| [https://dlmf.nist.gov/5.12.E1 5.12.E1] || [[Item:Q2146|<math>\int_{0}^{1}t^{a-1}(1-t)^{b-1}\diff{t} = \frac{\EulerGamma@{a}\EulerGamma@{b}}{\EulerGamma@{a+b}}</math>]] || <code>int((t)^(a - 1)*(1 - t)^(b - 1), t = 0..1)=(GAMMA(a)*GAMMA(b))/(GAMMA(a + b))</code> || <code>Integrate[(t)^(a - 1)*(1 - t)^(b - 1), {t, 0, 1}]=Divide[Gamma[a]*Gamma[b],Gamma[a + b]]</code> || Successful || Failure || - || Skip
+
| [https://dlmf.nist.gov/5.12.E1 5.12.E1] || [[Item:Q2146|<math>\int_{0}^{1}t^{a-1}(1-t)^{b-1}\diff{t} = \frac{\EulerGamma@{a}\EulerGamma@{b}}{\EulerGamma@{a+b}}</math>]] || <code>int((t)^(a - 1)*(1 - t)^(b - 1), t = 0..1)=(GAMMA(a)*GAMMA(b))/(GAMMA(a + b))</code> || <code>Integrate[(t)^(a - 1)*(1 - t)^(b - 1), {t, 0, 1}]=Divide[Gamma[a]*Gamma[b],Gamma[a + b]]</code> || Successful || Failure || - || Successful
 
|-
 
|-
| [https://dlmf.nist.gov/5.12.E2 5.12.E2] || [[Item:Q2147|<math>\int_{0}^{\pi/2}\sin^{2a-1}@@{\theta}\cos^{2b-1}@@{\theta}\diff{\theta} = \tfrac{1}{2}\EulerBeta@{a}{b}</math>]] || <code>int((sin(theta))^(2*a - 1)* (cos(theta))^(2*b - 1), theta = 0..Pi/ 2)=(1)/(2)*Beta(a, b)</code> || <code>Integrate[(Sin[\[Theta]])^(2*a - 1)* (Cos[\[Theta]])^(2*b - 1), {\[Theta], 0, Pi/ 2}]=Divide[1,2]*Beta[a, b]</code> || Failure || Failure || Skip || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[ConditionalExpression[Plus[Times[Rational[-1, 2], Beta[a, b]], Times[Rational[1, 2], Gamma[a], Gamma[b], Power[Gamma[Plus[a, b]], -1]]], And[Greater[Re[a], 0], Greater[Re[b], 0]]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[ConditionalExpression[Plus[Times[Rational[-1, 2], Beta[a, b]], Times[Rational[1, 2], Gamma[a], Gamma[b], Power[Gamma[Plus[a, b]], -1]]], And[Greater[Re[a], 0], Greater[Re[b], 0]]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[ConditionalExpression[Plus[Times[Rational[-1, 2], Beta[a, b]], Times[Rational[1, 2], Gamma[a], Gamma[b], Power[Gamma[Plus[a, b]], -1]]], And[Greater[Re[a], 0], Greater[Re[b], 0]]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[ConditionalExpression[Plus[Times[Rational[-1, 2], Beta[a, b]], Times[Rational[1, 2], Gamma[a], Gamma[b], Power[Gamma[Plus[a, b]], -1]]], And[Greater[Re[a], 0], Greater[Re[b], 0]]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br></div></div>
+
| [https://dlmf.nist.gov/5.12.E2 5.12.E2] || [[Item:Q2147|<math>\int_{0}^{\pi/2}\sin^{2a-1}@@{\theta}\cos^{2b-1}@@{\theta}\diff{\theta} = \tfrac{1}{2}\EulerBeta@{a}{b}</math>]] || <code>int((sin(theta))^(2*a - 1)* (cos(theta))^(2*b - 1), theta = 0..Pi/ 2)=(1)/(2)*Beta(a, b)</code> || <code>Integrate[(Sin[\[Theta]])^(2*a - 1)* (Cos[\[Theta]])^(2*b - 1), {\[Theta], 0, Pi/ 2}]=Divide[1,2]*Beta[a, b]</code> || Failure || Failure || Skip || Successful
 
|-
 
|-
| [https://dlmf.nist.gov/5.12.E3 5.12.E3] || [[Item:Q2148|<math>\int_{0}^{\infty}\frac{t^{a-1}\diff{t}}{(1+t)^{a+b}} = \EulerBeta@{a}{b}</math>]] || <code>int(((t)^(a - 1))/((1 + t)^(a + b)), t = 0..infinity)= Beta(a, b)</code> || <code>Integrate[Divide[(t)^(a - 1),(1 + t)^(a + b)], {t, 0, Infinity}]= Beta[a, b]</code> || Failure || Failure || Skip || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[ConditionalExpression[Plus[Times[-1, Beta[a, b]], Times[Gamma[a], Gamma[b], Power[Gamma[Plus[a, b]], -1]]], And[Greater[Re[a], 0], Greater[Re[b], 0]]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[ConditionalExpression[Plus[Times[-1, Beta[a, b]], Times[Gamma[a], Gamma[b], Power[Gamma[Plus[a, b]], -1]]], And[Greater[Re[a], 0], Greater[Re[b], 0]]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[ConditionalExpression[Plus[Times[-1, Beta[a, b]], Times[Gamma[a], Gamma[b], Power[Gamma[Plus[a, b]], -1]]], And[Greater[Re[a], 0], Greater[Re[b], 0]]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[ConditionalExpression[Plus[Times[-1, Beta[a, b]], Times[Gamma[a], Gamma[b], Power[Gamma[Plus[a, b]], -1]]], And[Greater[Re[a], 0], Greater[Re[b], 0]]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br></div></div>
+
| [https://dlmf.nist.gov/5.12.E3 5.12.E3] || [[Item:Q2148|<math>\int_{0}^{\infty}\frac{t^{a-1}\diff{t}}{(1+t)^{a+b}} = \EulerBeta@{a}{b}</math>]] || <code>int(((t)^(a - 1))/((1 + t)^(a + b)), t = 0..infinity)= Beta(a, b)</code> || <code>Integrate[Divide[(t)^(a - 1),(1 + t)^(a + b)], {t, 0, Infinity}]= Beta[a, b]</code> || Failure || Failure || Skip || Successful
 
|-
 
|-
| [https://dlmf.nist.gov/5.12.E4 5.12.E4] || [[Item:Q2149|<math>\int_{0}^{1}\frac{t^{a-1}(1-t)^{b-1}}{(t+z)^{a+b}}\diff{t} = \EulerBeta@{a}{b}(1+z)^{-a}z^{-b}</math>]] || <code>int(((t)^(a - 1)*(1 - t)^(b - 1))/((t + z)^(a + b)), t = 0..1)= Beta(a, b)*(1 + z)^(- a)* (z)^(- b)</code> || <code>Integrate[Divide[(t)^(a - 1)*(1 - t)^(b - 1),(t + z)^(a + b)], {t, 0, 1}]= Beta[a, b]*(1 + z)^(- a)* (z)^(- b)</code> || Failure || Failure || Skip || Error
+
| [https://dlmf.nist.gov/5.12.E4 5.12.E4] || [[Item:Q2149|<math>\int_{0}^{1}\frac{t^{a-1}(1-t)^{b-1}}{(t+z)^{a+b}}\diff{t} = \EulerBeta@{a}{b}(1+z)^{-a}z^{-b}</math>]] || <code>int(((t)^(a - 1)*(1 - t)^(b - 1))/((t + z)^(a + b)), t = 0..1)= Beta(a, b)*(1 + z)^(- a)* (z)^(- b)</code> || <code>Integrate[Divide[(t)^(a - 1)*(1 - t)^(b - 1),(t + z)^(a + b)], {t, 0, 1}]= Beta[a, b]*(1 + z)^(- a)* (z)^(- b)</code> || Failure || Failure || Skip || Successful
 
|-
 
|-
 
| [https://dlmf.nist.gov/5.12.E5 5.12.E5] || [[Item:Q2150|<math>\int_{0}^{\pi/2}(\cos@@{t})^{a-1}\cos@{bt}\diff{t} = \frac{\pi}{2^{a}}\frac{1}{a\EulerBeta@{\frac{1}{2}(a+b+1)}{\frac{1}{2}(a-b+1)}}</math>]] || <code>int((cos(t))^(a - 1)* cos(b*t), t = 0..Pi/ 2)=(Pi)/((2)^(a))*(1)/(a*Beta((1)/(2)*(a + b + 1), (1)/(2)*(a - b + 1)))</code> || <code>Integrate[(Cos[t])^(a - 1)* Cos[b*t], {t, 0, Pi/ 2}]=Divide[Pi,(2)^(a)]*Divide[1,a*Beta[Divide[1,2]*(a + b + 1), Divide[1,2]*(a - b + 1)]]</code> || Failure || Failure || Skip || Error  
 
| [https://dlmf.nist.gov/5.12.E5 5.12.E5] || [[Item:Q2150|<math>\int_{0}^{\pi/2}(\cos@@{t})^{a-1}\cos@{bt}\diff{t} = \frac{\pi}{2^{a}}\frac{1}{a\EulerBeta@{\frac{1}{2}(a+b+1)}{\frac{1}{2}(a-b+1)}}</math>]] || <code>int((cos(t))^(a - 1)* cos(b*t), t = 0..Pi/ 2)=(Pi)/((2)^(a))*(1)/(a*Beta((1)/(2)*(a + b + 1), (1)/(2)*(a - b + 1)))</code> || <code>Integrate[(Cos[t])^(a - 1)* Cos[b*t], {t, 0, Pi/ 2}]=Divide[Pi,(2)^(a)]*Divide[1,a*Beta[Divide[1,2]*(a + b + 1), Divide[1,2]*(a - b + 1)]]</code> || Failure || Failure || Skip || Error  
 
|-
 
|-
| [https://dlmf.nist.gov/5.12.E6 5.12.E6] || [[Item:Q2151|<math>\int_{0}^{\pi}(\sin@@{t})^{a-1}e^{ibt}\diff{t} = \frac{\pi}{2^{a-1}}\frac{e^{i\pi b/2}}{a\EulerBeta@{\frac{1}{2}(a+b+1)}{\frac{1}{2}(a-b+1)}}</math>]] || <code>int((sin(t))^(a - 1)* exp(I*b*t), t = 0..Pi)=(Pi)/((2)^(a - 1))*(exp(I*Pi*b/ 2))/(a*Beta((1)/(2)*(a + b + 1), (1)/(2)*(a - b + 1)))</code> || <code>Integrate[(Sin[t])^(a - 1)* Exp[I*b*t], {t, 0, Pi}]=Divide[Pi,(2)^(a - 1)]*Divide[Exp[I*Pi*b/ 2],a*Beta[Divide[1,2]*(a + b + 1), Divide[1,2]*(a - b + 1)]]</code> || Failure || Failure || Skip || Error
+
| [https://dlmf.nist.gov/5.12.E6 5.12.E6] || [[Item:Q2151|<math>\int_{0}^{\pi}(\sin@@{t})^{a-1}e^{ibt}\diff{t} = \frac{\pi}{2^{a-1}}\frac{e^{i\pi b/2}}{a\EulerBeta@{\frac{1}{2}(a+b+1)}{\frac{1}{2}(a-b+1)}}</math>]] || <code>int((sin(t))^(a - 1)* exp(I*b*t), t = 0..Pi)=(Pi)/((2)^(a - 1))*(exp(I*Pi*b/ 2))/(a*Beta((1)/(2)*(a + b + 1), (1)/(2)*(a - b + 1)))</code> || <code>Integrate[(Sin[t])^(a - 1)* Exp[I*b*t], {t, 0, Pi}]=Divide[Pi,(2)^(a - 1)]*Divide[Exp[I*Pi*b/ 2],a*Beta[Divide[1,2]*(a + b + 1), Divide[1,2]*(a - b + 1)]]</code> || Failure || Failure || Skip || Skip
 
|-
 
|-
| [https://dlmf.nist.gov/5.12.E7 5.12.E7] || [[Item:Q2152|<math>\int_{0}^{\infty}\frac{\cosh@{2bt}}{(\cosh@@{t})^{2a}}\diff{t} = 4^{a-1}\EulerBeta@{a+b}{a-b}</math>]] || <code>int((cosh(2*b*t))/((cosh(t))^(2*a)), t = 0..infinity)= (4)^(a - 1)* Beta(a + b, a - b)</code> || <code>Integrate[Divide[Cosh[2*b*t],(Cosh[t])^(2*a)], {t, 0, Infinity}]= (4)^(a - 1)* Beta[a + b, a - b]</code> || Failure || Failure || Skip || Error
+
| [https://dlmf.nist.gov/5.12.E7 5.12.E7] || [[Item:Q2152|<math>\int_{0}^{\infty}\frac{\cosh@{2bt}}{(\cosh@@{t})^{2a}}\diff{t} = 4^{a-1}\EulerBeta@{a+b}{a-b}</math>]] || <code>int((cosh(2*b*t))/((cosh(t))^(2*a)), t = 0..infinity)= (4)^(a - 1)* Beta(a + b, a - b)</code> || <code>Integrate[Divide[Cosh[2*b*t],(Cosh[t])^(2*a)], {t, 0, Infinity}]= (4)^(a - 1)* Beta[a + b, a - b]</code> || Failure || Failure || Skip || Skip
 
|-
 
|-
 
| [https://dlmf.nist.gov/5.12.E11 5.12.E11] || [[Item:Q2156|<math>\frac{1}{e^{2\pi ia}-1}\int_{\infty}^{(0+)}t^{a-1}(1+t)^{-a-b}\diff{t} = \EulerBeta@{a}{b}</math>]] || <code>(1)/(exp(2*Pi*I*a)- 1)*int((t)^(a - 1)*(1 + t)^(- a - b), t = infinity..(0 +))= Beta(a, b)</code> || <code>Divide[1,Exp[2*Pi*I*a]- 1]*Integrate[(t)^(a - 1)*(1 + t)^(- a - b), {t, Infinity, (0 +)}]= Beta[a, b]</code> || Error || Failure || - || Error  
 
| [https://dlmf.nist.gov/5.12.E11 5.12.E11] || [[Item:Q2156|<math>\frac{1}{e^{2\pi ia}-1}\int_{\infty}^{(0+)}t^{a-1}(1+t)^{-a-b}\diff{t} = \EulerBeta@{a}{b}</math>]] || <code>(1)/(exp(2*Pi*I*a)- 1)*int((t)^(a - 1)*(1 + t)^(- a - b), t = infinity..(0 +))= Beta(a, b)</code> || <code>Divide[1,Exp[2*Pi*I*a]- 1]*Integrate[(t)^(a - 1)*(1 + t)^(- a - b), {t, Infinity, (0 +)}]= Beta[a, b]</code> || Error || Failure || - || Error  
Line 185: Line 185:
 
| [https://dlmf.nist.gov/5.12.E12 5.12.E12] || [[Item:Q2157|<math>\int_{P}^{(1+,0+,1-,0-)}t^{a-1}(1-t)^{b-1}\diff{t} = -4e^{\pi i(a+b)}\sin@{\pi a}\sin@{\pi b}\EulerBeta@{a}{b}</math>]] || <code>int((t)^(a - 1)*(1 - t)^(b - 1), t = P..(1 + , 0 + , 1 - , 0 -))= - 4*exp(Pi*I*(a + b))*sin(Pi*a)*sin(Pi*b)*Beta(a, b)</code> || <code>Integrate[(t)^(a - 1)*(1 - t)^(b - 1), {t, P, (1 + , 0 + , 1 - , 0 -)}]= - 4*Exp[Pi*I*(a + b)]*Sin[Pi*a]*Sin[Pi*b]*Beta[a, b]</code> || Error || Failure || - || Error  
 
| [https://dlmf.nist.gov/5.12.E12 5.12.E12] || [[Item:Q2157|<math>\int_{P}^{(1+,0+,1-,0-)}t^{a-1}(1-t)^{b-1}\diff{t} = -4e^{\pi i(a+b)}\sin@{\pi a}\sin@{\pi b}\EulerBeta@{a}{b}</math>]] || <code>int((t)^(a - 1)*(1 - t)^(b - 1), t = P..(1 + , 0 + , 1 - , 0 -))= - 4*exp(Pi*I*(a + b))*sin(Pi*a)*sin(Pi*b)*Beta(a, b)</code> || <code>Integrate[(t)^(a - 1)*(1 - t)^(b - 1), {t, P, (1 + , 0 + , 1 - , 0 -)}]= - 4*Exp[Pi*I*(a + b)]*Sin[Pi*a]*Sin[Pi*b]*Beta[a, b]</code> || Error || Failure || - || Error  
 
|-
 
|-
| [https://dlmf.nist.gov/5.13.E3 5.13.E3] || [[Item:Q2160|<math>\frac{1}{2\pi}\int_{-\infty}^{\infty}\EulerGamma@{a+it}\EulerGamma@{b+it}\EulerGamma@{c-it}\EulerGamma@{d-it}\diff{t} = \frac{\EulerGamma@{a+c}\EulerGamma@{a+d}\EulerGamma@{b+c}\EulerGamma@{b+d}}{\EulerGamma@{a+b+c+d}}</math>]] || <code>(1)/(2*Pi)*int(GAMMA(a + I*t)*GAMMA(b + I*t)*GAMMA(c - I*t)*GAMMA(d - I*t), t = - infinity..infinity)=(GAMMA(a + c)*GAMMA(a + d)*GAMMA(b + c)*GAMMA(b + d))/(GAMMA(a + b + c + d))</code> || <code>Divide[1,2*Pi]*Integrate[Gamma[a + I*t]*Gamma[b + I*t]*Gamma[c - I*t]*Gamma[d - I*t], {t, - Infinity, Infinity}]=Divide[Gamma[a + c]*Gamma[a + d]*Gamma[b + c]*Gamma[b + d],Gamma[a + b + c + d]]</code> || Error || Failure || - || Error
+
| [https://dlmf.nist.gov/5.13.E3 5.13.E3] || [[Item:Q2160|<math>\frac{1}{2\pi}\int_{-\infty}^{\infty}\EulerGamma@{a+it}\EulerGamma@{b+it}\EulerGamma@{c-it}\EulerGamma@{d-it}\diff{t} = \frac{\EulerGamma@{a+c}\EulerGamma@{a+d}\EulerGamma@{b+c}\EulerGamma@{b+d}}{\EulerGamma@{a+b+c+d}}</math>]] || <code>(1)/(2*Pi)*int(GAMMA(a + I*t)*GAMMA(b + I*t)*GAMMA(c - I*t)*GAMMA(d - I*t), t = - infinity..infinity)=(GAMMA(a + c)*GAMMA(a + d)*GAMMA(b + c)*GAMMA(b + d))/(GAMMA(a + b + c + d))</code> || <code>Divide[1,2*Pi]*Integrate[Gamma[a + I*t]*Gamma[b + I*t]*Gamma[c - I*t]*Gamma[d - I*t], {t, - Infinity, Infinity}]=Divide[Gamma[a + c]*Gamma[a + d]*Gamma[b + c]*Gamma[b + d],Gamma[a + b + c + d]]</code> || Error || Failure || - || Successful
 
|-
 
|-
 
| [https://dlmf.nist.gov/5.13.E4 5.13.E4] || [[Item:Q2161|<math>\int_{-\infty}^{\infty}\frac{\diff{t}}{\EulerGamma@{a+t}\EulerGamma@{b+t}\EulerGamma@{c-t}\EulerGamma@{d-t}} = \frac{\EulerGamma@{a+b+c+d-3}}{\EulerGamma@{a+c-1}\EulerGamma@{a+d-1}\EulerGamma@{b+c-1}\EulerGamma@{b+d-1}}</math>]] || <code>int((1)/(GAMMA(a + t)*GAMMA(b + t)*GAMMA(c - t)*GAMMA(d - t)), t = - infinity..infinity)=(GAMMA(a + b + c + d - 3))/(GAMMA(a + c - 1)*GAMMA(a + d - 1)*GAMMA(b + c - 1)*GAMMA(b + d - 1))</code> || <code>Integrate[Divide[1,Gamma[a + t]*Gamma[b + t]*Gamma[c - t]*Gamma[d - t]], {t, - Infinity, Infinity}]=Divide[Gamma[a + b + c + d - 3],Gamma[a + c - 1]*Gamma[a + d - 1]*Gamma[b + c - 1]*Gamma[b + d - 1]]</code> || Failure || Failure || Skip || Error  
 
| [https://dlmf.nist.gov/5.13.E4 5.13.E4] || [[Item:Q2161|<math>\int_{-\infty}^{\infty}\frac{\diff{t}}{\EulerGamma@{a+t}\EulerGamma@{b+t}\EulerGamma@{c-t}\EulerGamma@{d-t}} = \frac{\EulerGamma@{a+b+c+d-3}}{\EulerGamma@{a+c-1}\EulerGamma@{a+d-1}\EulerGamma@{b+c-1}\EulerGamma@{b+d-1}}</math>]] || <code>int((1)/(GAMMA(a + t)*GAMMA(b + t)*GAMMA(c - t)*GAMMA(d - t)), t = - infinity..infinity)=(GAMMA(a + b + c + d - 3))/(GAMMA(a + c - 1)*GAMMA(a + d - 1)*GAMMA(b + c - 1)*GAMMA(b + d - 1))</code> || <code>Integrate[Divide[1,Gamma[a + t]*Gamma[b + t]*Gamma[c - t]*Gamma[d - t]], {t, - Infinity, Infinity}]=Divide[Gamma[a + b + c + d - 3],Gamma[a + c - 1]*Gamma[a + d - 1]*Gamma[b + c - 1]*Gamma[b + d - 1]]</code> || Failure || Failure || Skip || Error  
Line 217: Line 217:
 
| [https://dlmf.nist.gov/5.17.E3 5.17.E3] || [[Item:Q2183|<math>\BarnesG@{z+1} = (2\pi)^{z/2}\exp@{-\tfrac{1}{2}z(z+1)-\tfrac{1}{2}\EulerConstant z^{2}}\*\prod_{k=1}^{\infty}\left(\left(1+\frac{z}{k}\right)^{k}\exp@{-z+\frac{z^{2}}{2k}}\right)</math>]] || <code>Error</code> || <code>BarnesG[z + 1]=(2*Pi)^(z/ 2)* Exp[-Divide[1,2]*z*(z + 1)-Divide[1,2]*EulerGamma*(z)^(2)]* Product[(1 +Divide[z,k])^(k)* Exp[- z +Divide[(z)^(2),2*k]], {k, 1, Infinity}]</code> || Error || Successful || - || -  
 
| [https://dlmf.nist.gov/5.17.E3 5.17.E3] || [[Item:Q2183|<math>\BarnesG@{z+1} = (2\pi)^{z/2}\exp@{-\tfrac{1}{2}z(z+1)-\tfrac{1}{2}\EulerConstant z^{2}}\*\prod_{k=1}^{\infty}\left(\left(1+\frac{z}{k}\right)^{k}\exp@{-z+\frac{z^{2}}{2k}}\right)</math>]] || <code>Error</code> || <code>BarnesG[z + 1]=(2*Pi)^(z/ 2)* Exp[-Divide[1,2]*z*(z + 1)-Divide[1,2]*EulerGamma*(z)^(2)]* Product[(1 +Divide[z,k])^(k)* Exp[- z +Divide[(z)^(2),2*k]], {k, 1, Infinity}]</code> || Error || Successful || - || -  
 
|-
 
|-
| [https://dlmf.nist.gov/5.17.E4 5.17.E4] || [[Item:Q2184|<math>\Ln@@{\BarnesG@{z+1}} = \tfrac{1}{2}z\ln@{2\pi}-\tfrac{1}{2}z(z+1)+z\Ln@@{\EulerGamma@{z+1}}-\int_{0}^{z}\Ln@@{\EulerGamma@{t+1}}\diff{t}</math>]] || <code>Error</code> || <code>Log[BarnesG[z + 1]]=Divide[1,2]*z*Log[2*Pi]-Divide[1,2]*z*(z + 1)+ z*Log[Gamma[z + 1]]- Integrate[Log[Gamma[t + 1]], {t, 0, z}]</code> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[ConditionalExpression[Plus[Times[Rational[1, 2], z, Plus[1, z]], Times[Rational[-1, 2], Log[Times[2, Pi]]], Times[Rational[-1, 2], z, Log[Times[2, Pi]]], Log[BarnesG[Plus[1, z]]], Times[-1, z, Log[Gamma[Plus[1, z]]]], Times[Plus[1, z], Log[Gamma[Plus[1, z]]]], Times[-1, Plus[1, z], LogGamma[Plus[1, z]]], PolyGamma[-2, Plus[1, z]]], GreaterEqual[Gamma[Plus[1, z]], 0]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[ConditionalExpression[Plus[Times[Rational[1, 2], z, Plus[1, z]], Times[Rational[-1, 2], Log[Times[2, Pi]]], Times[Rational[-1, 2], z, Log[Times[2, Pi]]], Log[BarnesG[Plus[1, z]]], Times[-1, z, Log[Gamma[Plus[1, z]]]], Times[Plus[1, z], Log[Gamma[Plus[1, z]]]], Times[-1, Plus[1, z], LogGamma[Plus[1, z]]], PolyGamma[-2, Plus[1, z]]], GreaterEqual[Gamma[Plus[1, z]], 0]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[ConditionalExpression[Plus[Times[Rational[1, 2], z, Plus[1, z]], Times[Rational[-1, 2], Log[Times[2, Pi]]], Times[Rational[-1, 2], z, Log[Times[2, Pi]]], Log[BarnesG[Plus[1, z]]], Times[-1, z, Log[Gamma[Plus[1, z]]]], Times[Plus[1, z], Log[Gamma[Plus[1, z]]]], Times[-1, Plus[1, z], LogGamma[Plus[1, z]]], PolyGamma[-2, Plus[1, z]]], GreaterEqual[Gamma[Plus[1, z]], 0]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[ConditionalExpression[Plus[Times[Rational[1, 2], z, Plus[1, z]], Times[Rational[-1, 2], Log[Times[2, Pi]]], Times[Rational[-1, 2], z, Log[Times[2, Pi]]], Log[BarnesG[Plus[1, z]]], Times[-1, z, Log[Gamma[Plus[1, z]]]], Times[Plus[1, z], Log[Gamma[Plus[1, z]]]], Times[-1, Plus[1, z], LogGamma[Plus[1, z]]], PolyGamma[-2, Plus[1, z]]], GreaterEqual[Gamma[Plus[1, z]], 0]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br></div></div>
+
| [https://dlmf.nist.gov/5.17.E4 5.17.E4] || [[Item:Q2184|<math>\Ln@@{\BarnesG@{z+1}} = \tfrac{1}{2}z\ln@{2\pi}-\tfrac{1}{2}z(z+1)+z\Ln@@{\EulerGamma@{z+1}}-\int_{0}^{z}\Ln@@{\EulerGamma@{t+1}}\diff{t}</math>]] || <code>Error</code> || <code>Log[BarnesG[z + 1]]=Divide[1,2]*z*Log[2*Pi]-Divide[1,2]*z*(z + 1)+ z*Log[Gamma[z + 1]]- Integrate[Log[Gamma[t + 1]], {t, 0, z}]</code> || Error || Failure || - || Successful
 
|-
 
|-
 
| [https://dlmf.nist.gov/5.17.E7 5.17.E7] || [[Item:Q2187|<math>C = \lim_{n\to\infty}\left(\sum_{k=1}^{n}k\ln@@{k}-\left(\tfrac{1}{2}n^{2}+\tfrac{1}{2}n+\tfrac{1}{12}\right)\ln@@{n}+\tfrac{1}{4}n^{2}\right)</math>]] || <code>C = limit(sum(k*ln(k), k = 1..n)-((1)/(2)*(n)^(2)+(1)/(2)*n +(1)/(12))* ln(n)+(1)/(4)*(n)^(2), n = infinity)</code> || <code>C = Limit[Sum[k*Log[k], {k, 1, n}]-(Divide[1,2]*(n)^(2)+Divide[1,2]*n +Divide[1,12])* Log[n]+Divide[1,4]*(n)^(2), n -> Infinity]</code> || Failure || Failure || Skip || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>Complex[1.165459085339311, 1.4142135623730951] <- {Rule[C, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.165459085339311, -1.4142135623730951] <- {Rule[C, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.6629680394068793, -1.4142135623730951] <- {Rule[C, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.6629680394068793, 1.4142135623730951] <- {Rule[C, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br></div></div>  
 
| [https://dlmf.nist.gov/5.17.E7 5.17.E7] || [[Item:Q2187|<math>C = \lim_{n\to\infty}\left(\sum_{k=1}^{n}k\ln@@{k}-\left(\tfrac{1}{2}n^{2}+\tfrac{1}{2}n+\tfrac{1}{12}\right)\ln@@{n}+\tfrac{1}{4}n^{2}\right)</math>]] || <code>C = limit(sum(k*ln(k), k = 1..n)-((1)/(2)*(n)^(2)+(1)/(2)*n +(1)/(12))* ln(n)+(1)/(4)*(n)^(2), n = infinity)</code> || <code>C = Limit[Sum[k*Log[k], {k, 1, n}]-(Divide[1,2]*(n)^(2)+Divide[1,2]*n +Divide[1,12])* Log[n]+Divide[1,4]*(n)^(2), n -> Infinity]</code> || Failure || Failure || Skip || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>Complex[1.165459085339311, 1.4142135623730951] <- {Rule[C, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.165459085339311, -1.4142135623730951] <- {Rule[C, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.6629680394068793, -1.4142135623730951] <- {Rule[C, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.6629680394068793, 1.4142135623730951] <- {Rule[C, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br></div></div>  
Line 229: Line 229:
 
| [https://dlmf.nist.gov/5.19.E3 5.19.E3] || [[Item:Q2203|<math>\digamma@{\tfrac{1}{2}}-2\digamma@{\tfrac{2}{3}}-\EulerConstant = 3\ln@@{3}-2\ln@@{2}-\tfrac{1}{3}\pi\sqrt{3}</math>]] || <code>Psi((1)/(2))- 2*Psi((2)/(3))- gamma = 3*ln(3)- 2*ln(2)-(1)/(3)*Pi*sqrt(3)</code> || <code>PolyGamma[Divide[1,2]]- 2*PolyGamma[Divide[2,3]]- EulerGamma = 3*Log[3]- 2*Log[2]-Divide[1,3]*Pi*Sqrt[3]</code> || Successful || Successful || - || -  
 
| [https://dlmf.nist.gov/5.19.E3 5.19.E3] || [[Item:Q2203|<math>\digamma@{\tfrac{1}{2}}-2\digamma@{\tfrac{2}{3}}-\EulerConstant = 3\ln@@{3}-2\ln@@{2}-\tfrac{1}{3}\pi\sqrt{3}</math>]] || <code>Psi((1)/(2))- 2*Psi((2)/(3))- gamma = 3*ln(3)- 2*ln(2)-(1)/(3)*Pi*sqrt(3)</code> || <code>PolyGamma[Divide[1,2]]- 2*PolyGamma[Divide[2,3]]- EulerGamma = 3*Log[3]- 2*Log[2]-Divide[1,3]*Pi*Sqrt[3]</code> || Successful || Successful || - || -  
 
|-
 
|-
| [https://dlmf.nist.gov/5.19#Ex3 5.19#Ex3] || [[Item:Q2204|<math>V = \frac{\pi^{\frac{1}{2}n}r^{n}}{\EulerGamma@{\frac{1}{2}n+1}}</math>]] || <code>V =((Pi)^((1)/(2)*n)* (r)^(n))/(GAMMA((1)/(2)*n + 1))</code> || <code>V =Divide[(Pi)^(Divide[1,2]*n)* (r)^(n),Gamma[Divide[1,2]*n + 1]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>-1.414213562-1.414213562*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 1}</code><br><code>1.414213562-11.15215705*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 2}</code><br><code>25.10958922-22.28116210*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 3}</code><br><code>-1.414213562+4.242640686*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 1}</code><br><code>1.414213562+13.98058417*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 2}</code><br><code>25.10958922+25.10958922*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 3}</code><br><code>4.242640686+4.242640686*I <- {V = 2^(1/2)+I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 1}</code><br><code>1.414213562-11.15215705*I <- {V = 2^(1/2)+I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 2}</code><br><code>-22.28116210+25.10958922*I <- {V = 2^(1/2)+I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 3}</code><br><code>4.242640686-1.414213562*I <- {V = 2^(1/2)+I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 1}</code><br><code>1.414213562+13.98058417*I <- {V = 2^(1/2)+I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 2}</code><br><code>-22.28116210-22.28116210*I <- {V = 2^(1/2)+I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 3}</code><br><code>-1.414213562-4.242640686*I <- {V = 2^(1/2)-I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 1}</code><br><code>1.414213562-13.98058417*I <- {V = 2^(1/2)-I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 2}</code><br><code>25.10958922-25.10958922*I <- {V = 2^(1/2)-I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 3}</code><br><code>-1.414213562+1.414213562*I <- {V = 2^(1/2)-I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 1}</code><br><code>1.414213562+11.15215705*I <- {V = 2^(1/2)-I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 2}</code><br><code>25.10958922+22.28116210*I <- {V = 2^(1/2)-I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 3}</code><br><code>4.242640686+1.414213562*I <- {V = 2^(1/2)-I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 1}</code><br><code>1.414213562-13.98058417*I <- {V = 2^(1/2)-I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 2}</code><br><code>-22.28116210+22.28116210*I <- {V = 2^(1/2)-I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 3}</code><br><code>4.242640686-4.242640686*I <- {V = 2^(1/2)-I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 1}</code><br><code>1.414213562+11.15215705*I <- {V = 2^(1/2)-I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 2}</code><br><code>-22.28116210-25.10958922*I <- {V = 2^(1/2)-I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 3}</code><br><code>-4.242640686-4.242640686*I <- {V = -2^(1/2)-I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 1}</code><br><code>-1.414213562-13.98058417*I <- {V = -2^(1/2)-I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 2}</code><br><code>22.28116210-25.10958922*I <- {V = -2^(1/2)-I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 3}</code><br><code>-4.242640686+1.414213562*I <- {V = -2^(1/2)-I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 1}</code><br><code>-1.414213562+11.15215705*I <- {V = -2^(1/2)-I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 2}</code><br><code>22.28116210+22.28116210*I <- {V = -2^(1/2)-I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 3}</code><br><code>1.414213562+1.414213562*I <- {V = -2^(1/2)-I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 1}</code><br><code>-1.414213562-13.98058417*I <- {V = -2^(1/2)-I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 2}</code><br><code>-25.10958922+22.28116210*I <- {V = -2^(1/2)-I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 3}</code><br><code>1.414213562-4.242640686*I <- {V = -2^(1/2)-I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 1}</code><br><code>-1.414213562+11.15215705*I <- {V = -2^(1/2)-I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 2}</code><br><code>-25.10958922-25.10958922*I <- {V = -2^(1/2)-I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 3}</code><br><code>-4.242640686-1.414213562*I <- {V = -2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 1}</code><br><code>-1.414213562-11.15215705*I <- {V = -2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 2}</code><br><code>22.28116210-22.28116210*I <- {V = -2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 3}</code><br><code>-4.242640686+4.242640686*I <- {V = -2^(1/2)+I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 1}</code><br><code>-1.414213562+13.98058417*I <- {V = -2^(1/2)+I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 2}</code><br><code>22.28116210+25.10958922*I <- {V = -2^(1/2)+I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 3}</code><br><code>1.414213562+4.242640686*I <- {V = -2^(1/2)+I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 1}</code><br><code>-1.414213562-11.15215705*I <- {V = -2^(1/2)+I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 2}</code><br><code>-25.10958922+25.10958922*I <- {V = -2^(1/2)+I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 3}</code><br><code>1.414213562-1.414213562*I <- {V = -2^(1/2)+I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 1}</code><br><code>-1.414213562+13.98058417*I <- {V = -2^(1/2)+I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 2}</code><br><code>-25.10958922-22.28116210*I <- {V = -2^(1/2)+I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 3}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, -11.152157051986077] <- {Rule[n, 2], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[25.10958923255105, -22.281162107804857] <- {Rule[n, 3], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, -4.242640687119286] <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, -13.980584176732268] <- {Rule[n, 2], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[25.10958923255105, -25.10958923255105] <- {Rule[n, 3], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-4.242640687119286, -4.242640687119286] <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, -13.980584176732268] <- {Rule[n, 2], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[22.281162107804857, -25.10958923255105] <- {Rule[n, 3], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-4.242640687119286, -1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, -11.152157051986077] <- {Rule[n, 2], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[22.281162107804857, -22.281162107804857] <- {Rule[n, 3], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, 4.242640687119286] <- {Rule[n, 1], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, 13.980584176732268] <- {Rule[n, 2], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[25.10958923255105, 25.10958923255105] <- {Rule[n, 3], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, 11.152157051986077] <- {Rule[n, 2], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[25.10958923255105, 22.281162107804857] <- {Rule[n, 3], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-4.242640687119286, 1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, 11.152157051986077] <- {Rule[n, 2], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[22.281162107804857, 22.281162107804857] <- {Rule[n, 3], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-4.242640687119286, 4.242640687119286] <- {Rule[n, 1], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, 13.980584176732268] <- {Rule[n, 2], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[22.281162107804857, 25.10958923255105] <- {Rule[n, 3], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[4.242640687119286, 4.242640687119286] <- {Rule[n, 1], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, -11.152157051986077] <- {Rule[n, 2], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-22.281162107804857, 25.10958923255105] <- {Rule[n, 3], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[4.242640687119286, 1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, -13.980584176732268] <- {Rule[n, 2], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-22.281162107804857, 22.281162107804857] <- {Rule[n, 3], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, -13.980584176732268] <- {Rule[n, 2], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-25.10958923255105, 22.281162107804857] <- {Rule[n, 3], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, 4.242640687119286] <- {Rule[n, 1], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, -11.152157051986077] <- {Rule[n, 2], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-25.10958923255105, 25.10958923255105] <- {Rule[n, 3], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[4.242640687119286, -1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, 13.980584176732268] <- {Rule[n, 2], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-22.281162107804857, -22.281162107804857] <- {Rule[n, 3], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[4.242640687119286, -4.242640687119286] <- {Rule[n, 1], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, 11.152157051986077] <- {Rule[n, 2], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-22.281162107804857, -25.10958923255105] <- {Rule[n, 3], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, -4.242640687119286] <- {Rule[n, 1], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, 11.152157051986077] <- {Rule[n, 2], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-25.10958923255105, -25.10958923255105] <- {Rule[n, 3], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, 13.980584176732268] <- {Rule[n, 2], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-25.10958923255105, -22.281162107804857] <- {Rule[n, 3], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br></div></div>  
+
| [https://dlmf.nist.gov/5.19#Ex3 5.19#Ex3] || [[Item:Q2204|<math>V = \frac{\pi^{\frac{1}{2}n}r^{n}}{\EulerGamma@{\frac{1}{2}n+1}}</math>]] || <code>V =((Pi)^((1)/(2)*n)* (r)^(n))/(GAMMA((1)/(2)*n + 1))</code> || <code>V =Divide[(Pi)^(Divide[1,2]*n)* (r)^(n),Gamma[Divide[1,2]*n + 1]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>-1.414213562-1.414213562*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 1}</code><br><code>1.414213562-11.15215705*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 2}</code><br><code>25.10958922-22.28116210*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 3}</code><br><code>-1.414213562+4.242640686*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 1}</code><br>... skip entries to safe data<br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, -11.152157051986077] <- {Rule[n, 2], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[25.10958923255105, -22.281162107804857] <- {Rule[n, 3], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, -4.242640687119286] <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  
 
|-
 
|-
| [https://dlmf.nist.gov/5.19#Ex4 5.19#Ex4] || [[Item:Q2205|<math>S = \frac{2\pi^{\frac{1}{2}n}r^{n-1}}{\EulerGamma@{\frac{1}{2}n}}</math>]] || <code>S =(2*(Pi)^((1)/(2)*n)* (r)^(n - 1))/(GAMMA((1)/(2)*n))</code> || <code>S =Divide[2*(Pi)^(Divide[1,2]*n)* (r)^(n - 1),Gamma[Divide[1,2]*n]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>-.585786438+1.414213562*I <- {S = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 1}</code><br><code>-7.471552313-7.471552313*I <- {S = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 2}</code><br><code>1.414213562-48.85126889*I <- {S = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 3}</code><br><code>-.585786438+1.414213562*I <- {S = 2^(1/2)+I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 1}</code><br><code>-7.471552313+10.29997944*I <- {S = 2^(1/2)+I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 2}</code><br><code>1.414213562+51.67969601*I <- {S = 2^(1/2)+I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 3}</code><br><code>-.585786438+1.414213562*I <- {S = 2^(1/2)+I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 1}</code><br><code>10.29997944+10.29997944*I <- {S = 2^(1/2)+I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 2}</code><br><code>1.414213562-48.85126889*I <- {S = 2^(1/2)+I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 3}</code><br><code>-.585786438+1.414213562*I <- {S = 2^(1/2)+I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 1}</code><br><code>10.29997944-7.471552313*I <- {S = 2^(1/2)+I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 2}</code><br><code>1.414213562+51.67969601*I <- {S = 2^(1/2)+I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 3}</code><br><code>-.585786438-1.414213562*I <- {S = 2^(1/2)-I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 1}</code><br><code>-7.471552313-10.29997944*I <- {S = 2^(1/2)-I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 2}</code><br><code>1.414213562-51.67969601*I <- {S = 2^(1/2)-I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 3}</code><br><code>-.585786438-1.414213562*I <- {S = 2^(1/2)-I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 1}</code><br><code>-7.471552313+7.471552313*I <- {S = 2^(1/2)-I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 2}</code><br><code>1.414213562+48.85126889*I <- {S = 2^(1/2)-I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 3}</code><br><code>-.585786438-1.414213562*I <- {S = 2^(1/2)-I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 1}</code><br><code>10.29997944+7.471552313*I <- {S = 2^(1/2)-I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 2}</code><br><code>1.414213562-51.67969601*I <- {S = 2^(1/2)-I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 3}</code><br><code>-.585786438-1.414213562*I <- {S = 2^(1/2)-I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 1}</code><br><code>10.29997944-10.29997944*I <- {S = 2^(1/2)-I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 2}</code><br><code>1.414213562+48.85126889*I <- {S = 2^(1/2)-I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 3}</code><br><code>-3.414213562-1.414213562*I <- {S = -2^(1/2)-I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 1}</code><br><code>-10.29997944-10.29997944*I <- {S = -2^(1/2)-I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 2}</code><br><code>-1.414213562-51.67969601*I <- {S = -2^(1/2)-I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 3}</code><br><code>-3.414213562-1.414213562*I <- {S = -2^(1/2)-I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 1}</code><br><code>-10.29997944+7.471552313*I <- {S = -2^(1/2)-I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 2}</code><br><code>-1.414213562+48.85126889*I <- {S = -2^(1/2)-I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 3}</code><br><code>-3.414213562-1.414213562*I <- {S = -2^(1/2)-I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 1}</code><br><code>7.471552313+7.471552313*I <- {S = -2^(1/2)-I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 2}</code><br><code>-1.414213562-51.67969601*I <- {S = -2^(1/2)-I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 3}</code><br><code>-3.414213562-1.414213562*I <- {S = -2^(1/2)-I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 1}</code><br><code>7.471552313-10.29997944*I <- {S = -2^(1/2)-I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 2}</code><br><code>-1.414213562+48.85126889*I <- {S = -2^(1/2)-I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 3}</code><br><code>-3.414213562+1.414213562*I <- {S = -2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 1}</code><br><code>-10.29997944-7.471552313*I <- {S = -2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 2}</code><br><code>-1.414213562-48.85126889*I <- {S = -2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 3}</code><br><code>-3.414213562+1.414213562*I <- {S = -2^(1/2)+I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 1}</code><br><code>-10.29997944+10.29997944*I <- {S = -2^(1/2)+I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 2}</code><br><code>-1.414213562+51.67969601*I <- {S = -2^(1/2)+I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 3}</code><br><code>-3.414213562+1.414213562*I <- {S = -2^(1/2)+I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 1}</code><br><code>7.471552313+10.29997944*I <- {S = -2^(1/2)+I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 2}</code><br><code>-1.414213562-48.85126889*I <- {S = -2^(1/2)+I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 3}</code><br><code>-3.414213562+1.414213562*I <- {S = -2^(1/2)+I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 1}</code><br><code>7.471552313-7.471552313*I <- {S = -2^(1/2)+I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 2}</code><br><code>-1.414213562+51.67969601*I <- {S = -2^(1/2)+I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 3}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>Complex[-0.5857864376269049, 1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-7.471552313943637, -7.471552313943637] <- {Rule[n, 2], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, -48.8512688950636] <- {Rule[n, 3], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-0.5857864376269049, -1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-7.471552313943637, -10.299979438689828] <- {Rule[n, 2], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, -51.67969601980978] <- {Rule[n, 3], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-3.414213562373095, -1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-10.299979438689828, -10.299979438689828] <- {Rule[n, 2], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, -51.67969601980978] <- {Rule[n, 3], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-3.414213562373095, 1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-10.299979438689828, -7.471552313943637] <- {Rule[n, 2], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, -48.8512688950636] <- {Rule[n, 3], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-0.5857864376269049, 1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-7.471552313943637, 10.299979438689828] <- {Rule[n, 2], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, 51.67969601980978] <- {Rule[n, 3], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-0.5857864376269049, -1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-7.471552313943637, 7.471552313943637] <- {Rule[n, 2], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, 48.8512688950636] <- {Rule[n, 3], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-3.414213562373095, -1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-10.299979438689828, 7.471552313943637] <- {Rule[n, 2], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, 48.8512688950636] <- {Rule[n, 3], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-3.414213562373095, 1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-10.299979438689828, 10.299979438689828] <- {Rule[n, 2], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, 51.67969601980978] <- {Rule[n, 3], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-0.5857864376269049, 1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[10.299979438689828, 10.299979438689828] <- {Rule[n, 2], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, -48.8512688950636] <- {Rule[n, 3], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-0.5857864376269049, -1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[10.299979438689828, 7.471552313943637] <- {Rule[n, 2], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, -51.67969601980978] <- {Rule[n, 3], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-3.414213562373095, -1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[7.471552313943637, 7.471552313943637] <- {Rule[n, 2], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, -51.67969601980978] <- {Rule[n, 3], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-3.414213562373095, 1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[7.471552313943637, 10.299979438689828] <- {Rule[n, 2], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, -48.8512688950636] <- {Rule[n, 3], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-0.5857864376269049, 1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[10.299979438689828, -7.471552313943637] <- {Rule[n, 2], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, 51.67969601980978] <- {Rule[n, 3], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-0.5857864376269049, -1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[10.299979438689828, -10.299979438689828] <- {Rule[n, 2], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, 48.8512688950636] <- {Rule[n, 3], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-3.414213562373095, -1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[7.471552313943637, -10.299979438689828] <- {Rule[n, 2], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, 48.8512688950636] <- {Rule[n, 3], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-3.414213562373095, 1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[7.471552313943637, -7.471552313943637] <- {Rule[n, 2], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, 51.67969601980978] <- {Rule[n, 3], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br></div></div>  
+
| [https://dlmf.nist.gov/5.19#Ex4 5.19#Ex4] || [[Item:Q2205|<math>S = \frac{2\pi^{\frac{1}{2}n}r^{n-1}}{\EulerGamma@{\frac{1}{2}n}}</math>]] || <code>S =(2*(Pi)^((1)/(2)*n)* (r)^(n - 1))/(GAMMA((1)/(2)*n))</code> || <code>S =Divide[2*(Pi)^(Divide[1,2]*n)* (r)^(n - 1),Gamma[Divide[1,2]*n]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>-.585786438+1.414213562*I <- {S = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 1}</code><br><code>-7.471552313-7.471552313*I <- {S = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 2}</code><br><code>1.414213562-48.85126889*I <- {S = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 3}</code><br><code>-.585786438+1.414213562*I <- {S = 2^(1/2)+I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 1}</code><br>... skip entries to safe data<br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>Complex[-0.5857864376269049, 1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-7.471552313943637, -7.471552313943637] <- {Rule[n, 2], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, -48.8512688950636] <- {Rule[n, 3], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-0.5857864376269049, -1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  
 
|-
 
|-
| [https://dlmf.nist.gov/5.19#Ex4 5.19#Ex4] || [[Item:Q2205|<math>\frac{2\pi^{\frac{1}{2}n}r^{n-1}}{\EulerGamma@{\frac{1}{2}n}} = \frac{n}{r}V</math>]] || <code>(2*(Pi)^((1)/(2)*n)* (r)^(n - 1))/(GAMMA((1)/(2)*n))=(n)/(r)*V</code> || <code>Divide[2*(Pi)^(Divide[1,2]*n)* (r)^(n - 1),Gamma[Divide[1,2]*n]]=Divide[n,r]*V</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>1.000000000 <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 1}</code><br><code>6.885765875+8.885765875*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 2}</code><br><code>-3.+50.26548245*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 3}</code><br><code>2.000000000-1.000000000*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 1}</code><br><code>8.885765875-10.88576588*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 2}</code><br><code>0.-53.26548245*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 3}</code><br><code>3.000000000-0.*I <- {V = 2^(1/2)+I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 1}</code><br><code>-6.885765875-8.885765875*I <- {V = 2^(1/2)+I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 2}</code><br><code>3.000000000+50.26548245*I <- {V = 2^(1/2)+I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 3}</code><br><code>2.000000000+1.000000000*I <- {V = 2^(1/2)+I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 1}</code><br><code>-8.885765875+10.88576588*I <- {V = 2^(1/2)+I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 2}</code><br><code>0.-47.26548245*I <- {V = 2^(1/2)+I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 3}</code><br><code>2.000000000+1.000000000*I <- {V = 2^(1/2)-I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 1}</code><br><code>8.885765875+10.88576588*I <- {V = 2^(1/2)-I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 2}</code><br><code>0.+53.26548245*I <- {V = 2^(1/2)-I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 3}</code><br><code>1.000000000 <- {V = 2^(1/2)-I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 1}</code><br><code>6.885765875-8.885765875*I <- {V = 2^(1/2)-I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 2}</code><br><code>-3.-50.26548245*I <- {V = 2^(1/2)-I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 3}</code><br><code>2.000000000-1.000000000*I <- {V = 2^(1/2)-I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 1}</code><br><code>-8.885765875-10.88576588*I <- {V = 2^(1/2)-I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 2}</code><br><code>0.+47.26548245*I <- {V = 2^(1/2)-I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 3}</code><br><code>3.000000000-0.*I <- {V = 2^(1/2)-I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 1}</code><br><code>-6.885765875+8.885765875*I <- {V = 2^(1/2)-I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 2}</code><br><code>3.000000000-50.26548245*I <- {V = 2^(1/2)-I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 3}</code><br><code>3.000000000-0.*I <- {V = -2^(1/2)-I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 1}</code><br><code>10.88576588+8.885765875*I <- {V = -2^(1/2)-I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 2}</code><br><code>3.000000000+50.26548245*I <- {V = -2^(1/2)-I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 3}</code><br><code>2.000000000+1.000000000*I <- {V = -2^(1/2)-I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 1}</code><br><code>8.885765875-6.885765875*I <- {V = -2^(1/2)-I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 2}</code><br><code>0.-47.26548245*I <- {V = -2^(1/2)-I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 3}</code><br><code>1.000000000 <- {V = -2^(1/2)-I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 1}</code><br><code>-10.88576588-8.885765875*I <- {V = -2^(1/2)-I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 2}</code><br><code>-3.+50.26548245*I <- {V = -2^(1/2)-I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 3}</code><br><code>2.000000000-1.000000000*I <- {V = -2^(1/2)-I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 1}</code><br><code>-8.885765875+6.885765875*I <- {V = -2^(1/2)-I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 2}</code><br><code>0.-53.26548245*I <- {V = -2^(1/2)-I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 3}</code><br><code>2.000000000-1.000000000*I <- {V = -2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 1}</code><br><code>8.885765875+6.885765875*I <- {V = -2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 2}</code><br><code>0.+47.26548245*I <- {V = -2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 3}</code><br><code>3.000000000-0.*I <- {V = -2^(1/2)+I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 1}</code><br><code>10.88576588-8.885765875*I <- {V = -2^(1/2)+I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 2}</code><br><code>3.000000000-50.26548245*I <- {V = -2^(1/2)+I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 3}</code><br><code>2.000000000+1.000000000*I <- {V = -2^(1/2)+I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 1}</code><br><code>-8.885765875-6.885765875*I <- {V = -2^(1/2)+I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 2}</code><br><code>0.+53.26548245*I <- {V = -2^(1/2)+I*2^(1/2), r = -2^(1/2)-I*2^(1/2), n = 3}</code><br><code>1.000000000 <- {V = -2^(1/2)+I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 1}</code><br><code>-10.88576588+8.885765875*I <- {V = -2^(1/2)+I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 2}</code><br><code>-3.-50.26548245*I <- {V = -2^(1/2)+I*2^(1/2), r = -2^(1/2)+I*2^(1/2), n = 3}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>1.0 <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[6.885765876316732, 8.885765876316732] <- {Rule[n, 2], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-3.0, 50.26548245743669] <- {Rule[n, 3], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[2.0, 1.0] <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[8.885765876316732, 10.885765876316732] <- {Rule[n, 2], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[0.0, 53.26548245743669] <- {Rule[n, 3], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>3.0 <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[10.885765876316732, 8.885765876316732] <- {Rule[n, 2], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[3.0, 50.26548245743669] <- {Rule[n, 3], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[2.0, -1.0] <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[8.885765876316732, 6.885765876316732] <- {Rule[n, 2], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[0.0, 47.26548245743669] <- {Rule[n, 3], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[2.0, -1.0] <- {Rule[n, 1], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[8.885765876316732, -10.885765876316732] <- {Rule[n, 2], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[0.0, -53.26548245743669] <- {Rule[n, 3], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>1.0 <- {Rule[n, 1], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[6.885765876316732, -8.885765876316732] <- {Rule[n, 2], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-3.0, -50.26548245743669] <- {Rule[n, 3], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[2.0, 1.0] <- {Rule[n, 1], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[8.885765876316732, -6.885765876316732] <- {Rule[n, 2], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[0.0, -47.26548245743669] <- {Rule[n, 3], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>3.0 <- {Rule[n, 1], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[10.885765876316732, -8.885765876316732] <- {Rule[n, 2], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[3.0, -50.26548245743669] <- {Rule[n, 3], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>3.0 <- {Rule[n, 1], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-6.885765876316732, -8.885765876316732] <- {Rule[n, 2], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[3.0, 50.26548245743669] <- {Rule[n, 3], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[2.0, -1.0] <- {Rule[n, 1], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-8.885765876316732, -10.885765876316732] <- {Rule[n, 2], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[0.0, 47.26548245743669] <- {Rule[n, 3], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>1.0 <- {Rule[n, 1], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-10.885765876316732, -8.885765876316732] <- {Rule[n, 2], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-3.0, 50.26548245743669] <- {Rule[n, 3], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[2.0, 1.0] <- {Rule[n, 1], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-8.885765876316732, -6.885765876316732] <- {Rule[n, 2], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[0.0, 53.26548245743669] <- {Rule[n, 3], Rule[r, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[2.0, 1.0] <- {Rule[n, 1], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-8.885765876316732, 10.885765876316732] <- {Rule[n, 2], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[0.0, -47.26548245743669] <- {Rule[n, 3], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>3.0 <- {Rule[n, 1], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-6.885765876316732, 8.885765876316732] <- {Rule[n, 2], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[3.0, -50.26548245743669] <- {Rule[n, 3], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[2.0, -1.0] <- {Rule[n, 1], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-8.885765876316732, 6.885765876316732] <- {Rule[n, 2], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[0.0, -53.26548245743669] <- {Rule[n, 3], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>1.0 <- {Rule[n, 1], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-10.885765876316732, 8.885765876316732] <- {Rule[n, 2], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-3.0, -50.26548245743669] <- {Rule[n, 3], Rule[r, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br></div></div>  
+
| [https://dlmf.nist.gov/5.19#Ex4 5.19#Ex4] || [[Item:Q2205|<math>\frac{2\pi^{\frac{1}{2}n}r^{n-1}}{\EulerGamma@{\frac{1}{2}n}} = \frac{n}{r}V</math>]] || <code>(2*(Pi)^((1)/(2)*n)* (r)^(n - 1))/(GAMMA((1)/(2)*n))=(n)/(r)*V</code> || <code>Divide[2*(Pi)^(Divide[1,2]*n)* (r)^(n - 1),Gamma[Divide[1,2]*n]]=Divide[n,r]*V</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>1.000000000 <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 1}</code><br><code>6.885765875+8.885765875*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 2}</code><br><code>-3.+50.26548245*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 3}</code><br><code>2.000000000-1.000000000*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 1}</code><br>... skip entries to safe data<br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>1.0 <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[6.885765876316732, 8.885765876316732] <- {Rule[n, 2], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-3.0, 50.26548245743669] <- {Rule[n, 3], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[2.0, 1.0] <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  
 
|-
 
|-
 
| [https://dlmf.nist.gov/5.20.E1 5.20.E1] || [[Item:Q2206|<math>W = \frac{1}{2}\sum_{\ell=1}^{n}x_{\ell}^{2}-\sum_{1\leq\ell<j\leq n}\ln@@{|x_{\ell}-x_{j}|}</math>]] || <code>W =sum(x(x[ell])^(2), ell = 1..n)- sum(sum(ln(abs(x[ell]- x[j])), j = ell + 1..n), ell = 1..j - 1)</code> || <code>W =Sum[x(Subscript[x, \[ScriptL]])^(2), {\[ScriptL], 1, n}]- Sum[Sum[Log[Abs[Subscript[x, \[ScriptL]]- Subscript[x, j]]], {j, [ScriptL] + 1, n}], {\[ScriptL], 1, j - 1}]</code> || Failure || Failure || Skip || Error  
 
| [https://dlmf.nist.gov/5.20.E1 5.20.E1] || [[Item:Q2206|<math>W = \frac{1}{2}\sum_{\ell=1}^{n}x_{\ell}^{2}-\sum_{1\leq\ell<j\leq n}\ln@@{|x_{\ell}-x_{j}|}</math>]] || <code>W =sum(x(x[ell])^(2), ell = 1..n)- sum(sum(ln(abs(x[ell]- x[j])), j = ell + 1..n), ell = 1..j - 1)</code> || <code>W =Sum[x(Subscript[x, \[ScriptL]])^(2), {\[ScriptL], 1, n}]- Sum[Sum[Log[Abs[Subscript[x, \[ScriptL]]- Subscript[x, j]]], {j, [ScriptL] + 1, n}], {\[ScriptL], 1, j - 1}]</code> || Failure || Failure || Skip || Error  

Latest revision as of 12:43, 19 January 2020

DLMF Formula Maple Mathematica Symbolic
Maple
Symbolic
Mathematica
Numeric
Maple
Numeric
Mathematica
5.2.E1 GAMMA(z)= int(exp(- t)*(t)^(z - 1), t = 0..infinity) Gamma[z]= Integrate[Exp[- t]*(t)^(z - 1), {t, 0, Infinity}] Successful Successful - -
5.2.E2 Psi(z)= subs( temp=z, diff( GAMMA(temp), temp$(1) ) )/ GAMMA(z) PolyGamma[z]= (D[Gamma[temp], {temp, 1}]/.temp-> z)/ Gamma[z] Successful Successful - -
5.2#Ex1 pochhammer(a, 0)= 1 Pochhammer[a, 0]= 1 Successful Successful - -
5.2.E5 pochhammer(a, n)= GAMMA(a + n)/ GAMMA(a) Pochhammer[a, n]= Gamma[a + n]/ Gamma[a] Successful Successful - -
5.2.E6 pochhammer(- a, n)=(- 1)^(n)* pochhammer(a - n + 1, n) Pochhammer[- a, n]=(- 1)^(n)* Pochhammer[a - n + 1, n] Failure Failure Successful Successful
5.2.E7 pochhammer(- m, n)= Pochhammer[- m, n]= Error Failure - -
5.2#Ex3 pochhammer(a, 2*n)= (2)^(2*n)* pochhammer((a)/(2), n)*pochhammer((a + 1)/(2), n) Pochhammer[a, 2*n]= (2)^(2*n)* Pochhammer[Divide[a,2], n]*Pochhammer[Divide[a + 1,2], n] Successful Successful - -
5.2#Ex4 pochhammer(a, 2*n + 1)= (2)^(2*n + 1)* pochhammer((a)/(2), n + 1)*pochhammer((a + 1)/(2), n) Pochhammer[a, 2*n + 1]= (2)^(2*n + 1)* Pochhammer[Divide[a,2], n + 1]*Pochhammer[Divide[a + 1,2], n] Successful Successful - -
5.4#Ex1 GAMMA(1)= 1 Gamma[1]= 1 Successful Successful - -
5.4#Ex2 factorial(n)= GAMMA(n + 1) (n)!= Gamma[n + 1] Successful Successful - -
5.4.E3 abs(GAMMA(I*y))=((Pi)/(y*sinh(Pi*y)))^(1/ 2) Abs[Gamma[I*y]]=(Divide[Pi,y*Sinh[Pi*y]])^(1/ 2) Failure Failure Successful Successful
5.4.E4 GAMMA((1)/(2)+ I*y)*GAMMA((1)/(2)- I*y)=(abs(GAMMA((1)/(2)+ I*y)))^(2) Gamma[Divide[1,2]+ I*y]*Gamma[Divide[1,2]- I*y]=(Abs[Gamma[Divide[1,2]+ I*y]])^(2) Failure Failure Successful Successful
5.4.E4 (abs(GAMMA((1)/(2)+ I*y)))^(2)=(Pi)/(cosh(Pi*y)) (Abs[Gamma[Divide[1,2]+ I*y]])^(2)=Divide[Pi,Cosh[Pi*y]] Failure Failure Successful Successful
5.4.E5 GAMMA((1)/(4)+ I*y)*GAMMA((3)/(4)- I*y)=(Pi*sqrt(2))/(cosh(Pi*y)+ I*sinh(Pi*y)) Gamma[Divide[1,4]+ I*y]*Gamma[Divide[3,4]- I*y]=Divide[Pi*Sqrt[2],Cosh[Pi*y]+ I*Sinh[Pi*y]] Failure Successful Successful -
5.4.E11 subs( temp=1, diff( GAMMA(temp), temp$(1) ) )= - gamma (D[Gamma[temp], {temp, 1}]/.temp-> 1)= - EulerGamma Successful Successful - -
5.4#Ex3 Psi(1)= - gamma PolyGamma[1]= - EulerGamma Successful Successful - -
5.4#Ex4 subs( temp=1, diff( Psi(temp), temp$(1) ) )=(1)/(6)*(Pi)^(2) (D[PolyGamma[temp], {temp, 1}]/.temp-> 1)=Divide[1,6]*(Pi)^(2) Successful Successful - -
5.4#Ex5 Psi((1)/(2))= - gamma - 2*ln(2) PolyGamma[Divide[1,2]]= - EulerGamma - 2*Log[2] Successful Successful - -
5.4#Ex6 subs( temp=(1)/(2), diff( Psi(temp), temp$(1) ) )=(1)/(2)*(Pi)^(2) (D[PolyGamma[temp], {temp, 1}]/.temp-> Divide[1,2])=Divide[1,2]*(Pi)^(2) Successful Successful - -
5.4.E14 Psi(n + 1)= sum((1)/(k), k = 1..n)- gamma PolyGamma[n + 1]= Sum[Divide[1,k], {k, 1, n}]- EulerGamma Successful Successful - -
5.4.E16 Im(Psi(I*y))=(1)/(2*y)+(Pi)/(2)*coth(Pi*y) Im[PolyGamma[I*y]]=Divide[1,2*y]+Divide[Pi,2]*Coth[Pi*y] Failure Failure Successful Successful
5.4.E17 Im(Psi((1)/(2)+ I*y))=(Pi)/(2)*tanh(Pi*y) Im[PolyGamma[Divide[1,2]+ I*y]]=Divide[Pi,2]*Tanh[Pi*y] Failure Failure Successful Successful
5.4.E18 Im(Psi(1 + I*y))= -(1)/(2*y)+(Pi)/(2)*coth(Pi*y) Im[PolyGamma[1 + I*y]]= -Divide[1,2*y]+Divide[Pi,2]*Coth[Pi*y] Failure Failure Successful Successful
5.4.E19 Psi((p)/(q))= - gamma - ln(q)-(Pi)/(2)*cot((Pi*p)/(q))+(1)/(2)*sum(cos((2*Pi*k*p)/(q))*ln(2 - 2*cos((2*Pi*k)/(q))), k = 1..q - 1) PolyGamma[Divide[p,q]]= - EulerGamma - Log[q]-Divide[Pi,2]*Cot[Divide[Pi*p,q]]+Divide[1,2]*Sum[Cos[Divide[2*Pi*k*p,q]]*Log[2 - 2*Cos[Divide[2*Pi*k,q]]], {k, 1, q - 1}] Failure Failure Skip
Fail
DirectedInfinity[] <- {Rule[k, 1], Rule[p, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
DirectedInfinity[] <- {Rule[k, 2], Rule[p, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
DirectedInfinity[] <- {Rule[k, 3], Rule[p, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
DirectedInfinity[] <- {Rule[k, 1], Rule[p, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
5.5.E1 GAMMA(z + 1)= z*GAMMA(z) Gamma[z + 1]= z*Gamma[z] Successful Successful - -
5.5.E2 Psi(z + 1)= Psi(z)+(1)/(z) PolyGamma[z + 1]= PolyGamma[z]+Divide[1,z] Successful Successful - -
5.5.E3 GAMMA(z)*GAMMA(1 - z)= Pi/ sin(Pi*z) Gamma[z]*Gamma[1 - z]= Pi/ Sin[Pi*z] Successful Successful - -
5.5.E4 Psi(z)- Psi(1 - z)= - Pi/ tan(Pi*z) PolyGamma[z]- PolyGamma[1 - z]= - Pi/ Tan[Pi*z] Successful Successful - -
5.5.E5 GAMMA(2*z)= (Pi)^(- 1/ 2)* (2)^(2*z - 1)* GAMMA(z)*GAMMA(z +(1)/(2)) Gamma[2*z]= (Pi)^(- 1/ 2)* (2)^(2*z - 1)* Gamma[z]*Gamma[z +Divide[1,2]] Successful Successful - -
5.5.E6 GAMMA(n*z)=(2*Pi)^((1 - n)/ 2)* (n)^(n*z -(1/ 2))* product(GAMMA(z +(k)/(n)), k = 0..n - 1) Gamma[n*z]=(2*Pi)^((1 - n)/ 2)* (n)^(n*z -(1/ 2))* Product[Gamma[z +Divide[k,n]], {k, 0, n - 1}] Failure Successful Skip -
5.5.E7 product(GAMMA((k)/(n)), k = 1..n - 1)=(2*Pi)^((n - 1)/ 2)* (n)^(- 1/ 2) Product[Gamma[Divide[k,n]], {k, 1, n - 1}]=(2*Pi)^((n - 1)/ 2)* (n)^(- 1/ 2) Failure Failure Skip
Fail
Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[Indeterminate, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[Indeterminate, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[Indeterminate, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[Indeterminate, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
5.5.E8 Psi(2*z)=(1)/(2)*(Psi(z)+ Psi(z +(1)/(2)))+ ln(2) PolyGamma[2*z]=Divide[1,2]*(PolyGamma[z]+ PolyGamma[z +Divide[1,2]])+ Log[2] Successful Successful - -
5.5.E9 Psi(n*z)=(1)/(n)*sum(Psi(z +(k)/(n)), k = 0..n - 1)+ ln(n) PolyGamma[n*z]=Divide[1,n]*Sum[PolyGamma[z +Divide[k,n]], {k, 0, n - 1}]+ Log[n] Failure Successful Skip -
5.6.E1 1 <(2*Pi)^(- 1/ 2)* (x)^((1/ 2)- x)* exp(x)*GAMMA(x) 1 <(2*Pi)^(- 1/ 2)* (x)^((1/ 2)- x)* Exp[x]*Gamma[x] Failure Failure Successful Successful
5.6.E1 (2*Pi)^(- 1/ 2)* (x)^((1/ 2)- x)* exp(x)*GAMMA(x)< exp(1/(12*x)) (2*Pi)^(- 1/ 2)* (x)^((1/ 2)- x)* Exp[x]*Gamma[x]< Exp[1/(12*x)] Failure Failure Successful Successful
5.6.E2 (1)/(GAMMA(x))+(1)/(GAMMA(1/ x))< = 2 Divide[1,Gamma[x]]+Divide[1,Gamma[1/ x]]< = 2 Failure Failure Successful Successful
5.6.E3 (1)/((GAMMA(x))^(2))+(1)/((GAMMA(1/ x))^(2))< = 2 Divide[1,(Gamma[x])^(2)]+Divide[1,(Gamma[1/ x])^(2)]< = 2 Failure Failure Successful Successful
5.6.E4 (x)^(1 - s)<(GAMMA(x + 1))/(GAMMA(x + s)) (x)^(1 - s)<Divide[Gamma[x + 1],Gamma[x + s]] Failure Failure Successful Successful
5.6.E4 (GAMMA(x + 1))/(GAMMA(x + s))<(x + 1)^(1 - s) Divide[Gamma[x + 1],Gamma[x + s]]<(x + 1)^(1 - s) Failure Failure Successful Successful
5.6.E5 exp((1 - s)* Psi(x + (s)^(1/ 2)))< =(GAMMA(x + 1))/(GAMMA(x + s)) Exp[(1 - s)* PolyGamma[x + (s)^(1/ 2)]]< =Divide[Gamma[x + 1],Gamma[x + s]] Failure Failure Successful Successful
5.6.E5 (GAMMA(x + 1))/(GAMMA(x + s))< = exp((1 - s)* Psi(x +(1)/(2)*(s + 1))) Divide[Gamma[x + 1],Gamma[x + s]]< = Exp[(1 - s)* PolyGamma[x +Divide[1,2]*(s + 1)]] Failure Failure Successful Successful
5.6.E6 abs(GAMMA(x + I*y))< =abs(GAMMA(x)) Abs[Gamma[x + I*y]]< =Abs[Gamma[x]] Failure Failure Successful Successful
5.6.E7 abs(GAMMA(x + I*y))> =(sech(Pi*y))^(1/ 2)* GAMMA(x) Abs[Gamma[x + I*y]]> =(Sech[Pi*y])^(1/ 2)* Gamma[x] Failure Failure Skip Successful
5.6.E8 abs((GAMMA(z + a))/(GAMMA(z + b)))< =(1)/((abs(z))^(b - a)) Abs[Divide[Gamma[z + a],Gamma[z + b]]]< =Divide[1,(Abs[z])^(b - a)] Failure Failure Error Successful
5.6.E9 abs(GAMMA(z))< =(2*Pi)^(1/ 2)*(abs(z))^(x -(1/ 2))* exp(- Pi*abs(y)/ 2)*exp((1)/(6)*(abs(z))^(- 1)) Abs[Gamma[z]]< =(2*Pi)^(1/ 2)*(Abs[z])^(x -(1/ 2))* Exp[- Pi*Abs[y]/ 2]*Exp[Divide[1,6]*(Abs[z])^(- 1)] Failure Failure
Fail
.3896047846 <= .1665021267 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 2}
.3896047846 <= .3461239156e-1 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 3}
.3896047846 <= .3330042534 <- {z = 2^(1/2)+I*2^(1/2), x = 2, y = 2}
.3896047846 <= .6922478312e-1 <- {z = 2^(1/2)+I*2^(1/2), x = 2, y = 3}
... skip entries to safe data
Successful
5.7.E1 (1)/(GAMMA(z))= sum(c[k]*(z)^(k), k = 1..infinity) Divide[1,Gamma[z]]= Sum[Subscript[c, k]*(z)^(k), {k, 1, Infinity}] Failure Failure Skip Skip
5.7.E3 ln(GAMMA(1 + z))= - ln(1 + z)+ z*(1 - gamma)+ sum((- 1)^(k)*(Zeta(k)- 1)*((z)^(k))/(k), k = 2..infinity) Log[Gamma[1 + z]]= - Log[1 + z]+ z*(1 - EulerGamma)+ Sum[(- 1)^(k)*(Zeta[k]- 1)*Divide[(z)^(k),k], {k, 2, Infinity}] Failure Successful Skip -
5.7.E4 Psi(1 + z)= - gamma + sum((- 1)^(k)* Zeta(k)*(z)^(k - 1), k = 2..infinity) PolyGamma[1 + z]= - EulerGamma + Sum[(- 1)^(k)* Zeta[k]*(z)^(k - 1), {k, 2, Infinity}] Failure Successful Skip -
5.7.E5 Psi(1 + z)=(1)/(2*z)-(Pi)/(2)*cot(Pi*z)+(1)/((z)^(2)- 1)+ 1 - gamma - sum((Zeta(2*k + 1)- 1)* (z)^(2*k), k = 1..infinity) PolyGamma[1 + z]=Divide[1,2*z]-Divide[Pi,2]*Cot[Pi*z]+Divide[1,(z)^(2)- 1]+ 1 - EulerGamma - Sum[(Zeta[2*k + 1]- 1)* (z)^(2*k), {k, 1, Infinity}] Failure Successful Skip -
5.7.E6 Psi(z)= - gamma -(1)/(z)+ sum((z)/(k*(k + z)), k = 1..infinity) PolyGamma[z]= - EulerGamma -Divide[1,z]+ Sum[Divide[z,k*(k + z)], {k, 1, Infinity}] Successful Successful - -
5.7.E6 - gamma -(1)/(z)+ sum((z)/(k*(k + z)), k = 1..infinity)= - gamma + sum((1)/(k + 1)-(1)/(k + z), k = 0..infinity) - EulerGamma -Divide[1,z]+ Sum[Divide[z,k*(k + z)], {k, 1, Infinity}]= - EulerGamma + Sum[Divide[1,k + 1]-Divide[1,k + z], {k, 0, Infinity}] Successful Successful - -
5.7.E7 Psi((z + 1)/(2))- Psi((z)/(2))= 2*sum(((- 1)^(k))/(k + z), k = 0..infinity) PolyGamma[Divide[z + 1,2]]- PolyGamma[Divide[z,2]]= 2*Sum[Divide[(- 1)^(k),k + z], {k, 0, Infinity}] Successful Successful - -
5.7.E8 Im(Psi(1 + I*y))= sum((y)/((k)^(2)+ (y)^(2)), k = 1..infinity) Im[PolyGamma[1 + I*y]]= Sum[Divide[y,(k)^(2)+ (y)^(2)], {k, 1, Infinity}] Failure Failure Skip Successful
5.8.E2 (1)/(GAMMA(z))= z*exp(gamma*z)*product((1 +(z)/(k))* exp(- z/ k), k = 1..infinity) Divide[1,Gamma[z]]= z*Exp[EulerGamma*z]*Product[(1 +Divide[z,k])* Exp[- z/ k], {k, 1, Infinity}] Successful Failure - Successful
5.8.E3 (abs((GAMMA(x))/(GAMMA(x + I*y))))^(2)= product(1 +((y)^(2))/((x + k)^(2)), k = 0..infinity) (Abs[Divide[Gamma[x],Gamma[x + I*y]]])^(2)= Product[1 +Divide[(y)^(2),(x + k)^(2)], {k, 0, Infinity}] Failure Failure Skip Successful
5.9.E1 (1)/(mu)*GAMMA((nu)/(mu))*(1)/((z)^(nu/ mu))= int(exp(- z*(t)^(mu))*(t)^(nu - 1), t = 0..infinity) Divide[1,\[Mu]]*Gamma[Divide[\[Nu],\[Mu]]]*Divide[1,(z)^(\[Nu]/ \[Mu])]= Integrate[Exp[- z*(t)^(\[Mu])]*(t)^(\[Nu]- 1), {t, 0, Infinity}] Failure Failure Skip Successful
5.9.E2 (1)/(GAMMA(z))=(1)/(2*Pi*I)*int(exp(t)*(t)^(- z), t = - infinity..(0 +)) Divide[1,Gamma[z]]=Divide[1,2*Pi*I]*Integrate[Exp[t]*(t)^(- z), {t, - Infinity, (0 +)}] Error Failure - Error
5.9.E4 GAMMA(z)= int((t)^(z - 1)* exp(- t), t = 1..infinity)+ sum(((- 1)^(k))/((z + k)* factorial(k)), k = 0..infinity) Gamma[z]= Integrate[(t)^(z - 1)* Exp[- t], {t, 1, Infinity}]+ Sum[Divide[(- 1)^(k),(z + k)* (k)!], {k, 0, Infinity}] Failure Successful Skip -
5.9.E5 GAMMA(z)= int((t)^(z - 1)*(exp(- t)- sum(((- 1)^(k)* (t)^(k))/(factorial(k)), k = 0..n)), t = 0..infinity) Gamma[z]= Integrate[(t)^(z - 1)*(Exp[- t]- Sum[Divide[(- 1)^(k)* (t)^(k),(k)!], {k, 0, n}]), {t, 0, Infinity}] Failure Failure Skip Skip
5.9.E6 GAMMA(z)*cos((1)/(2)*Pi*z)= int((t)^(z - 1)* cos(t), t = 0..infinity) Gamma[z]*Cos[Divide[1,2]*Pi*z]= Integrate[(t)^(z - 1)* Cos[t], {t, 0, Infinity}] Successful Failure - Successful
5.9.E7 GAMMA(z)*sin((1)/(2)*Pi*z)= int((t)^(z - 1)* sin(t), t = 0..infinity) Gamma[z]*Sin[Divide[1,2]*Pi*z]= Integrate[(t)^(z - 1)* Sin[t], {t, 0, Infinity}] Successful Failure - Successful
5.9.E8 GAMMA(1 +(1)/(n))*cos((Pi)/(2*n))= int(cos((t)^(n)), t = 0..infinity) Gamma[1 +Divide[1,n]]*Cos[Divide[Pi,2*n]]= Integrate[Cos[(t)^(n)], {t, 0, Infinity}] Successful Failure - Successful
5.9.E9 GAMMA(1 +(1)/(n))*sin((Pi)/(2*n))= int(sin((t)^(n)), t = 0..infinity) Gamma[1 +Divide[1,n]]*Sin[Divide[Pi,2*n]]= Integrate[Sin[(t)^(n)], {t, 0, Infinity}] Successful Failure - Successful
5.9.E10 ln(GAMMA(z))=(z -(1)/(2))* ln(z)- z +(1)/(2)*ln(2*Pi)+ 2*int((arctan(t/ z))/(exp(2*Pi*t)- 1), t = 0..infinity) Log[Gamma[z]]=(z -Divide[1,2])* Log[z]- z +Divide[1,2]*Log[2*Pi]+ 2*Integrate[Divide[ArcTan[t/ z],Exp[2*Pi*t]- 1], {t, 0, Infinity}] Failure Failure Skip Skip
5.9.E12 Psi(z)= int((exp(- t))/(t)-(exp(- z*t))/(1 - exp(- t)), t = 0..infinity) PolyGamma[z]= Integrate[Divide[Exp[- t],t]-Divide[Exp[- z*t],1 - Exp[- t]], {t, 0, Infinity}] Failure Failure Skip Successful
5.9.E13 Psi(z)= ln(z)+ int(((1)/(t)-(1)/(1 - exp(- t)))* exp(- t*z), t = 0..infinity) PolyGamma[z]= Log[z]+ Integrate[(Divide[1,t]-Divide[1,1 - Exp[- t]])* Exp[- t*z], {t, 0, Infinity}] Failure Failure Skip Error
5.9.E14 Psi(z)= int((exp(- t)-(1)/((1 + t)^(z)))*(1)/(t), t = 0..infinity) PolyGamma[z]= Integrate[(Exp[- t]-Divide[1,(1 + t)^(z)])*Divide[1,t], {t, 0, Infinity}] Failure Failure Skip Successful
5.9.E15 Psi(z)= ln(z)-(1)/(2*z)- 2*int((t)/(((t)^(2)+ (z)^(2))*(exp(2*Pi*t)- 1)), t = 0..infinity) PolyGamma[z]= Log[z]-Divide[1,2*z]- 2*Integrate[Divide[t,((t)^(2)+ (z)^(2))*(Exp[2*Pi*t]- 1)], {t, 0, Infinity}] Failure Failure Skip Skip
5.9.E16 Psi(z)+ gamma = int((exp(- t)- exp(- z*t))/(1 - exp(- t)), t = 0..infinity) PolyGamma[z]+ EulerGamma = Integrate[Divide[Exp[- t]- Exp[- z*t],1 - Exp[- t]], {t, 0, Infinity}] Failure Failure Skip Successful
5.9.E16 int((exp(- t)- exp(- z*t))/(1 - exp(- t)), t = 0..infinity)= int((1 - (t)^(z - 1))/(1 - t), t = 0..1) Integrate[Divide[Exp[- t]- Exp[- z*t],1 - Exp[- t]], {t, 0, Infinity}]= Integrate[Divide[1 - (t)^(z - 1),1 - t], {t, 0, 1}] Failure Failure Skip Skip
5.9.E17 Psi(z + 1)= - gamma +(1)/(2*Pi*I)*int((Pi*(z)^(- s - 1))/(sin(Pi*s))*Zeta(- s), s = - c - infinity*I..- c + infinity*I) PolyGamma[z + 1]= - EulerGamma +Divide[1,2*Pi*I]*Integrate[Divide[Pi*(z)^(- s - 1),Sin[Pi*s]]*Zeta[- s], {s, - c - Infinity*I, - c + Infinity*I}] Failure Failure Skip Skip
5.9.E18 gamma = - int(exp(- t)*ln(t), t = 0..infinity) EulerGamma = - Integrate[Exp[- t]*Log[t], {t, 0, Infinity}] Successful Successful - -
5.9.E18 - int(exp(- t)*ln(t), t = 0..infinity)= int(((1)/(1 + t)- exp(- t))*(1)/(t), t = 0..infinity) - Integrate[Exp[- t]*Log[t], {t, 0, Infinity}]= Integrate[(Divide[1,1 + t]- Exp[- t])*Divide[1,t], {t, 0, Infinity}] Successful Successful - -
5.9.E18 int(((1)/(1 + t)- exp(- t))*(1)/(t), t = 0..infinity)= int((1 - exp(- t))*(1)/(t), t = 0..1)- int(exp(- t)*(1)/(t), t = 1..infinity) Integrate[(Divide[1,1 + t]- Exp[- t])*Divide[1,t], {t, 0, Infinity}]= Integrate[(1 - Exp[- t])*Divide[1,t], {t, 0, 1}]- Integrate[Exp[- t]*Divide[1,t], {t, 1, Infinity}] Successful Successful - -
5.9.E18 int((1 - exp(- t))*(1)/(t), t = 0..1)- int(exp(- t)*(1)/(t), t = 1..infinity)= int((exp(- t))/(1 - exp(- t))-(exp(- t))/(t), t = 0..infinity) Integrate[(1 - Exp[- t])*Divide[1,t], {t, 0, 1}]- Integrate[Exp[- t]*Divide[1,t], {t, 1, Infinity}]= Integrate[Divide[Exp[- t],1 - Exp[- t]]-Divide[Exp[- t],t], {t, 0, Infinity}] Successful Successful - -
5.9.E19 subs( temp=z, diff( GAMMA(temp), temp$(n) ) )= int((ln(t))^(n)* exp(- t)*(t)^(z - 1), t = 0..infinity) (D[Gamma[temp], {temp, n}]/.temp-> z)= Integrate[(Log[t])^(n)* Exp[- t]*(t)^(z - 1), {t, 0, Infinity}] Successful Failure - Error
5.11.E5 g[k]=sqrt(2)*pochhammer((1)/(2), k)*a[2*k] Subscript[g, k]=Sqrt[2]*Pochhammer[Divide[1,2], k]*Subscript[a, 2*k] Failure Failure
Fail
.4142135625+.4142135625*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)+I*2^(1/2), k = 1}
-.8578643792e-1-.8578643792e-1*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)+I*2^(1/2), k = 2}
-2.335786436-2.335786436*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)+I*2^(1/2), k = 3}
.4142135625-2.414213561*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)-I*2^(1/2), k = 1}
... skip entries to safe data
Successful
5.11.E10 GAMMA(z)= exp(- z)*(z)^(z)*((2*Pi)/(z))^(1/ 2)*(sum((g[k])/((z)^(k)), k = 0..K - 1)+ R[K]*(z)) Gamma[z]= Exp[- z]*(z)^(z)*(Divide[2*Pi,z])^(1/ 2)*(Sum[Divide[Subscript[g, k],(z)^(k)], {k, 0, K - 1}]+ Subscript[R, K]*(z)) Failure Failure Skip Skip
5.11#Ex10 G[2]*(a , b)=(1)/(12)*binomial(a - b,2)*(3*(a + b - 1)^(2)-(a - b + 1)) Subscript[G, 2]*(a , b)=Divide[1,12]*Binomial[a - b,2]*(3*(a + b - 1)^(2)-(a - b + 1)) Failure Failure Error Error
5.11#Ex12 H[1]*(a , b)= -(1)/(12)*binomial(a - b,2)*(a - b + 1) Subscript[H, 1]*(a , b)= -Divide[1,12]*Binomial[a - b,2]*(a - b + 1) Failure Failure Error Error
5.11#Ex13 H[2]*(a , b)=(1)/(240)*binomial(a - b,4)*(2*(a - b + 1)+ 5*(a - b + 1)^(2)) Subscript[H, 2]*(a , b)=Divide[1,240]*Binomial[a - b,4]*(2*(a - b + 1)+ 5*(a - b + 1)^(2)) Failure Failure Error Error
5.12.E1 Beta(a, b)= int((t)^(a - 1)*(1 - t)^(b - 1), t = 0..1) Beta[a, b]= Integrate[(t)^(a - 1)*(1 - t)^(b - 1), {t, 0, 1}] Failure Failure Skip Successful
5.12.E1 int((t)^(a - 1)*(1 - t)^(b - 1), t = 0..1)=(GAMMA(a)*GAMMA(b))/(GAMMA(a + b)) Integrate[(t)^(a - 1)*(1 - t)^(b - 1), {t, 0, 1}]=Divide[Gamma[a]*Gamma[b],Gamma[a + b]] Successful Failure - Successful
5.12.E2 int((sin(theta))^(2*a - 1)* (cos(theta))^(2*b - 1), theta = 0..Pi/ 2)=(1)/(2)*Beta(a, b) Integrate[(Sin[\[Theta]])^(2*a - 1)* (Cos[\[Theta]])^(2*b - 1), {\[Theta], 0, Pi/ 2}]=Divide[1,2]*Beta[a, b] Failure Failure Skip Successful
5.12.E3 int(((t)^(a - 1))/((1 + t)^(a + b)), t = 0..infinity)= Beta(a, b) Integrate[Divide[(t)^(a - 1),(1 + t)^(a + b)], {t, 0, Infinity}]= Beta[a, b] Failure Failure Skip Successful
5.12.E4 int(((t)^(a - 1)*(1 - t)^(b - 1))/((t + z)^(a + b)), t = 0..1)= Beta(a, b)*(1 + z)^(- a)* (z)^(- b) Integrate[Divide[(t)^(a - 1)*(1 - t)^(b - 1),(t + z)^(a + b)], {t, 0, 1}]= Beta[a, b]*(1 + z)^(- a)* (z)^(- b) Failure Failure Skip Successful
5.12.E5 int((cos(t))^(a - 1)* cos(b*t), t = 0..Pi/ 2)=(Pi)/((2)^(a))*(1)/(a*Beta((1)/(2)*(a + b + 1), (1)/(2)*(a - b + 1))) Integrate[(Cos[t])^(a - 1)* Cos[b*t], {t, 0, Pi/ 2}]=Divide[Pi,(2)^(a)]*Divide[1,a*Beta[Divide[1,2]*(a + b + 1), Divide[1,2]*(a - b + 1)]] Failure Failure Skip Error
5.12.E6 int((sin(t))^(a - 1)* exp(I*b*t), t = 0..Pi)=(Pi)/((2)^(a - 1))*(exp(I*Pi*b/ 2))/(a*Beta((1)/(2)*(a + b + 1), (1)/(2)*(a - b + 1))) Integrate[(Sin[t])^(a - 1)* Exp[I*b*t], {t, 0, Pi}]=Divide[Pi,(2)^(a - 1)]*Divide[Exp[I*Pi*b/ 2],a*Beta[Divide[1,2]*(a + b + 1), Divide[1,2]*(a - b + 1)]] Failure Failure Skip Skip
5.12.E7 int((cosh(2*b*t))/((cosh(t))^(2*a)), t = 0..infinity)= (4)^(a - 1)* Beta(a + b, a - b) Integrate[Divide[Cosh[2*b*t],(Cosh[t])^(2*a)], {t, 0, Infinity}]= (4)^(a - 1)* Beta[a + b, a - b] Failure Failure Skip Skip
5.12.E11 (1)/(exp(2*Pi*I*a)- 1)*int((t)^(a - 1)*(1 + t)^(- a - b), t = infinity..(0 +))= Beta(a, b) Divide[1,Exp[2*Pi*I*a]- 1]*Integrate[(t)^(a - 1)*(1 + t)^(- a - b), {t, Infinity, (0 +)}]= Beta[a, b] Error Failure - Error
5.12.E12 int((t)^(a - 1)*(1 - t)^(b - 1), t = P..(1 + , 0 + , 1 - , 0 -))= - 4*exp(Pi*I*(a + b))*sin(Pi*a)*sin(Pi*b)*Beta(a, b) Integrate[(t)^(a - 1)*(1 - t)^(b - 1), {t, P, (1 + , 0 + , 1 - , 0 -)}]= - 4*Exp[Pi*I*(a + b)]*Sin[Pi*a]*Sin[Pi*b]*Beta[a, b] Error Failure - Error
5.13.E3 (1)/(2*Pi)*int(GAMMA(a + I*t)*GAMMA(b + I*t)*GAMMA(c - I*t)*GAMMA(d - I*t), t = - infinity..infinity)=(GAMMA(a + c)*GAMMA(a + d)*GAMMA(b + c)*GAMMA(b + d))/(GAMMA(a + b + c + d)) Divide[1,2*Pi]*Integrate[Gamma[a + I*t]*Gamma[b + I*t]*Gamma[c - I*t]*Gamma[d - I*t], {t, - Infinity, Infinity}]=Divide[Gamma[a + c]*Gamma[a + d]*Gamma[b + c]*Gamma[b + d],Gamma[a + b + c + d]] Error Failure - Successful
5.13.E4