# Difference between revisions of "Results of Error Functions, Dawson’s and Fresnel Integrals"

DLMF Formula Maple Mathematica Symbolic
Maple
Symbolic
Mathematica
Numeric
Maple
Numeric
Mathematica
7.2.E1 ${\displaystyle{\displaystyle\operatorname{erf}z=\frac{2}{\sqrt{\pi}}\int_{0}^{% z}e^{-t^{2}}\mathrm{d}t}}$ erf(z)=(2)/(sqrt(Pi))*int(exp(- (t)^(2)), t = 0..z) Erf[z]=Divide[2,Sqrt[Pi]]*Integrate[Exp[- (t)^(2)], {t, 0, z}] Successful Successful - -
7.2.E2 ${\displaystyle{\displaystyle\operatorname{erfc}z=\frac{2}{\sqrt{\pi}}\int_{z}^% {\infty}e^{-t^{2}}\mathrm{d}t}}$ erfc(z)=(2)/(sqrt(Pi))*int(exp(- (t)^(2)), t = z..infinity) Erfc[z]=Divide[2,Sqrt[Pi]]*Integrate[Exp[- (t)^(2)], {t, z, Infinity}] Successful Successful - -
7.2.E2 ${\displaystyle{\displaystyle\frac{2}{\sqrt{\pi}}\int_{z}^{\infty}e^{-t^{2}}% \mathrm{d}t=1-\operatorname{erf}z}}$ (2)/(sqrt(Pi))*int(exp(- (t)^(2)), t = z..infinity)= 1 - erf(z) Divide[2,Sqrt[Pi]]*Integrate[Exp[- (t)^(2)], {t, z, Infinity}]= 1 - Erf[z] Successful Successful - -
7.2.E3 ${\displaystyle{\displaystyle e^{-z^{2}}\left(1+\frac{2i}{\sqrt{\pi}}\int_{0}^{% z}e^{t^{2}}\mathrm{d}t\right)=e^{-z^{2}}\operatorname{erfc}\left(-iz\right)}}$ exp(- (z)^(2))*(1 +(2*I)/(sqrt(Pi))*int(exp((t)^(2)), t = 0..z))= exp(- (z)^(2))*erfc(- I*z) Exp[- (z)^(2)]*(1 +Divide[2*I,Sqrt[Pi]]*Integrate[Exp[(t)^(2)], {t, 0, z}])= Exp[- (z)^(2)]*Erfc[- I*z] Successful Successful - -
7.2#Ex1 ${\displaystyle{\displaystyle\lim_{z\to\infty}\operatorname{erf}z=1}}$ limit(erf(z), z = infinity)= 1 Limit[Erf[z], z -> Infinity]= 1 Successful Successful - -
7.2#Ex2 ${\displaystyle{\displaystyle\lim_{z\to\infty}\operatorname{erfc}z=0}}$ limit(erfc(z), z = infinity)= 0 Limit[Erfc[z], z -> Infinity]= 0 Successful Successful - -
7.2.E5 ${\displaystyle{\displaystyle F\left(z\right)=e^{-z^{2}}\int_{0}^{z}e^{t^{2}}% \mathrm{d}t}}$ dawson(z)= exp(- (z)^(2))*int(exp((t)^(2)), t = 0..z) DawsonF[z]= Exp[- (z)^(2)]*Integrate[Exp[(t)^(2)], {t, 0, z}] Successful Successful - -
7.2.E7 ${\displaystyle{\displaystyle C\left(z\right)=\int_{0}^{z}\cos\left(\tfrac{1}{2% }\pi t^{2}\right)\mathrm{d}t}}$ FresnelC(z)= int(cos((1)/(2)*Pi*(t)^(2)), t = 0..z) FresnelC[z]= Integrate[Cos[Divide[1,2]*Pi*(t)^(2)], {t, 0, z}] Successful Successful - -
7.2.E8 ${\displaystyle{\displaystyle S\left(z\right)=\int_{0}^{z}\sin\left(\tfrac{1}{2% }\pi t^{2}\right)\mathrm{d}t}}$ FresnelS(z)= int(sin((1)/(2)*Pi*(t)^(2)), t = 0..z) FresnelS[z]= Integrate[Sin[Divide[1,2]*Pi*(t)^(2)], {t, 0, z}] Successful Successful - -
7.2#Ex3 ${\displaystyle{\displaystyle\lim_{x\to\infty}C\left(x\right)=\tfrac{1}{2}}}$ limit(FresnelC(x), x = infinity)=(1)/(2) Limit[FresnelC[x], x -> Infinity]=Divide[1,2] Successful Successful - -
7.2#Ex4 ${\displaystyle{\displaystyle\lim_{x\to\infty}S\left(x\right)=\tfrac{1}{2}}}$ limit(FresnelS(x), x = infinity)=(1)/(2) Limit[FresnelS[x], x -> Infinity]=Divide[1,2] Successful Successful - -
7.2.E10 ${\displaystyle{\displaystyle\mathrm{f}\left(z\right)=\left(\tfrac{1}{2}-S\left% (z\right)\right)\cos\left(\tfrac{1}{2}\pi z^{2}\right)-\left(\tfrac{1}{2}-C% \left(z\right)\right)\sin\left(\tfrac{1}{2}\pi z^{2}\right)}}$ Fresnelf(z)=((1)/(2)- FresnelS(z))* cos((1)/(2)*Pi*(z)^(2))-((1)/(2)- FresnelC(z))* sin((1)/(2)*Pi*(z)^(2)) FresnelF[z]=(Divide[1,2]- FresnelS[z])* Cos[Divide[1,2]*Pi*(z)^(2)]-(Divide[1,2]- FresnelC[z])* Sin[Divide[1,2]*Pi*(z)^(2)] Successful Successful - -
7.2.E11 ${\displaystyle{\displaystyle\mathrm{g}\left(z\right)=\left(\tfrac{1}{2}-C\left% (z\right)\right)\cos\left(\tfrac{1}{2}\pi z^{2}\right)+\left(\tfrac{1}{2}-S% \left(z\right)\right)\sin\left(\tfrac{1}{2}\pi z^{2}\right)}}$ Fresnelg(z)=((1)/(2)- FresnelC(z))* cos((1)/(2)*Pi*(z)^(2))+((1)/(2)- FresnelS(z))* sin((1)/(2)*Pi*(z)^(2)) FresnelG[z]=(Divide[1,2]- FresnelC[z])* Cos[Divide[1,2]*Pi*(z)^(2)]+(Divide[1,2]- FresnelS[z])* Sin[Divide[1,2]*Pi*(z)^(2)] Successful Successful - -
7.4.E1 ${\displaystyle{\displaystyle\operatorname{erf}\left(-z\right)=-\operatorname{% erf}\left(z\right)}}$ erf(- z)= - erf(z) Erf[- z]= - Erf[z] Successful Successful - -
7.4.E2 ${\displaystyle{\displaystyle\operatorname{erfc}\left(-z\right)=2-\operatorname% {erfc}\left(z\right)}}$ erfc(- z)= 2 - erfc(z) Erfc[- z]= 2 - Erfc[z] Successful Successful - -
7.4.E4 ${\displaystyle{\displaystyle F\left(-z\right)=-F\left(z\right)}}$ dawson(- z)= - dawson(z) DawsonF[- z]= - DawsonF[z] Successful Successful - -
7.4#Ex1 ${\displaystyle{\displaystyle C\left(-z\right)=-C\left(z\right)}}$ FresnelC(- z)= - FresnelC(z) FresnelC[- z]= - FresnelC[z] Successful Successful - -
7.4#Ex2 ${\displaystyle{\displaystyle S\left(-z\right)=-S\left(z\right)}}$ FresnelS(- z)= - FresnelS(z) FresnelS[- z]= - FresnelS[z] Successful Successful - -
7.4#Ex3 ${\displaystyle{\displaystyle C\left(iz\right)=iC\left(z\right)}}$ FresnelC(I*z)= I*FresnelC(z) FresnelC[I*z]= I*FresnelC[z] Successful Successful - -
7.4#Ex4 ${\displaystyle{\displaystyle S\left(iz\right)=-iS\left(z\right)}}$ FresnelS(I*z)= - I*FresnelS(z) FresnelS[I*z]= - I*FresnelS[z] Successful Successful - -
7.4#Ex5 ${\displaystyle{\displaystyle\mathrm{f}\left(iz\right)=(1/\sqrt{2})e^{\frac{1}{% 4}\pi i-\frac{1}{2}\pi iz^{2}}-i\mathrm{f}\left(z\right)}}$ Fresnelf(I*z)=(1/sqrt(2))* exp((1)/(4)*Pi*I -(1)/(2)*Pi*I*(z)^(2))- I*Fresnelf(z) FresnelF[I*z]=(1/Sqrt[2])* Exp[Divide[1,4]*Pi*I -Divide[1,2]*Pi*I*(z)^(2)]- I*FresnelF[z] Failure Failure Successful Successful
7.4#Ex6 ${\displaystyle{\displaystyle\mathrm{g}\left(iz\right)=(1/\sqrt{2})e^{-\frac{1}% {4}\pi i-\frac{1}{2}\pi iz^{2}}+i\mathrm{g}\left(z\right)}}$ Fresnelg(I*z)=(1/sqrt(2))* exp(-(1)/(4)*Pi*I -(1)/(2)*Pi*I*(z)^(2))+ I*Fresnelg(z) FresnelG[I*z]=(1/Sqrt[2])* Exp[-Divide[1,4]*Pi*I -Divide[1,2]*Pi*I*(z)^(2)]+ I*FresnelG[z] Failure Failure Successful Successful
7.4#Ex7 ${\displaystyle{\displaystyle\mathrm{f}\left(-z\right)=\sqrt{2}\cos\left(\tfrac% {1}{4}\pi+\tfrac{1}{2}\pi z^{2}\right)-\mathrm{f}\left(z\right)}}$ Fresnelf(- z)=sqrt(2)*cos((1)/(4)*Pi +(1)/(2)*Pi*(z)^(2))- Fresnelf(z) FresnelF[- z]=Sqrt[2]*Cos[Divide[1,4]*Pi +Divide[1,2]*Pi*(z)^(2)]- FresnelF[z] Failure Successful Successful -
7.4#Ex8 ${\displaystyle{\displaystyle\mathrm{g}\left(-z\right)=\sqrt{2}\sin\left(\tfrac% {1}{4}\pi+\tfrac{1}{2}\pi z^{2}\right)-\mathrm{g}\left(z\right)}}$ Fresnelg(- z)=sqrt(2)*sin((1)/(4)*Pi +(1)/(2)*Pi*(z)^(2))- Fresnelg(z) FresnelG[- z]=Sqrt[2]*Sin[Divide[1,4]*Pi +Divide[1,2]*Pi*(z)^(2)]- FresnelG[z] Failure Failure Successful Successful
7.5.E3 ${\displaystyle{\displaystyle C\left(z\right)=\tfrac{1}{2}+\mathrm{f}\left(z% \right)\sin\left(\tfrac{1}{2}\pi z^{2}\right)-\mathrm{g}\left(z\right)\cos% \left(\tfrac{1}{2}\pi z^{2}\right)}}$ FresnelC(z)=(1)/(2)+ Fresnelf(z)*sin((1)/(2)*Pi*(z)^(2))- Fresnelg(z)*cos((1)/(2)*Pi*(z)^(2)) FresnelC[z]=Divide[1,2]+ FresnelF[z]*Sin[Divide[1,2]*Pi*(z)^(2)]- FresnelG[z]*Cos[Divide[1,2]*Pi*(z)^(2)] Successful Failure - Successful
7.5.E4 ${\displaystyle{\displaystyle S\left(z\right)=\tfrac{1}{2}-\mathrm{f}\left(z% \right)\cos\left(\tfrac{1}{2}\pi z^{2}\right)-\mathrm{g}\left(z\right)\sin% \left(\tfrac{1}{2}\pi z^{2}\right)}}$ FresnelS(z)=(1)/(2)- Fresnelf(z)*cos((1)/(2)*Pi*(z)^(2))- Fresnelg(z)*sin((1)/(2)*Pi*(z)^(2)) FresnelS[z]=Divide[1,2]- FresnelF[z]*Cos[Divide[1,2]*Pi*(z)^(2)]- FresnelG[z]*Sin[Divide[1,2]*Pi*(z)^(2)] Successful Failure - Successful
7.5.E6 ${\displaystyle{\displaystyle e^{+\frac{1}{2}\pi iz^{2}}(\mathrm{g}\left(z% \right)+i\mathrm{f}\left(z\right))=\tfrac{1}{2}(1+i)-(C\left(z\right)+iS\left(% z\right))}}$ exp(+(1)/(2)*Pi*I*(z)^(2))*(Fresnelg(z)+ I*Fresnelf(z))=(1)/(2)*(1 + I)-(FresnelC(z)+ I*FresnelS(z)) Exp[+Divide[1,2]*Pi*I*(z)^(2)]*(FresnelG[z]+ I*FresnelF[z])=Divide[1,2]*(1 + I)-(FresnelC[z]+ I*FresnelS[z]) Failure Failure
Fail
.149314e-2-.173022e-2*I <- {z = 2^(1/2)-I*2^(1/2)}
-.119473e-2+.149314e-2*I <- {z = -2^(1/2)+I*2^(1/2)}
Successful
7.5.E6 ${\displaystyle{\displaystyle e^{-\frac{1}{2}\pi iz^{2}}(\mathrm{g}\left(z% \right)-i\mathrm{f}\left(z\right))=\tfrac{1}{2}(1-i)-(C\left(z\right)-iS\left(% z\right))}}$ exp(-(1)/(2)*Pi*I*(z)^(2))*(Fresnelg(z)- I*Fresnelf(z))=(1)/(2)*(1 - I)-(FresnelC(z)- I*FresnelS(z)) Exp[-Divide[1,2]*Pi*I*(z)^(2)]*(FresnelG[z]- I*FresnelF[z])=Divide[1,2]*(1 - I)-(FresnelC[z]- I*FresnelS[z]) Failure Failure
Fail
.149314e-2+.173022e-2*I <- {z = 2^(1/2)+I*2^(1/2)}
-.119473e-2-.149314e-2*I <- {z = -2^(1/2)-I*2^(1/2)}
Successful
7.5.E8 ${\displaystyle{\displaystyle C\left(z\right)+iS\left(z\right)=\tfrac{1}{2}(1+i% )\operatorname{erf}\zeta}}$ FresnelC(z)+ I*FresnelS(z)=(1)/(2)*(1 + I)* erf(zeta) FresnelC[z]+ I*FresnelS[z]=Divide[1,2]*(1 + I)* Erf[\[zeta]] Failure Failure
Fail
-.1423151062+.1316106532*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2)}
.1316106532-.1423151062*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)-I*2^(1/2)}
1.141922366+.8679966068*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)-I*2^(1/2)}
.8679966068+1.141922366*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Error
7.5.E8 ${\displaystyle{\displaystyle C\left(z\right)-iS\left(z\right)=\tfrac{1}{2}(1-i% )\operatorname{erf}\zeta}}$ FresnelC(z)- I*FresnelS(z)=(1)/(2)*(1 - I)* erf(zeta) FresnelC[z]- I*FresnelS[z]=Divide[1,2]*(1 - I)* Erf[\[zeta]] Failure Failure
Fail
66.79933367+67.80964539*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2)}
66.52540791+67.53571963*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)-I*2^(1/2)}
67.53571963+66.52540791*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)-I*2^(1/2)}
67.80964539+66.79933367*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Error
7.5.E10 ${\displaystyle{\displaystyle\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)% =\tfrac{1}{2}(1+i)e^{\zeta^{2}}\operatorname{erfc}\zeta}}$ Fresnelg(z)+ I*Fresnelf(z)=(1)/(2)*(1 + I)* exp((zeta)^(2))*erfc(zeta) FresnelG[z]+ I*FresnelF[z]=Divide[1,2]*(1 + I)* Exp[(\[zeta])^(2)]*Erfc[\[zeta]] Failure Failure
Fail
-.874918896e-1+.8375300635e-1*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2)}
.8375400635e-1-.874928896e-1*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)-I*2^(1/2)}
.1946430180+1.537001110*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)-I*2^(1/2)}
1.537002110+.1946420180*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Error
7.5.E10 ${\displaystyle{\displaystyle\mathrm{g}\left(z\right)-i\mathrm{f}\left(z\right)% =\tfrac{1}{2}(1-i)e^{\zeta^{2}}\operatorname{erfc}\zeta}}$ Fresnelg(z)- I*Fresnelf(z)=(1)/(2)*(1 - I)* exp((zeta)^(2))*erfc(zeta) FresnelG[z]- I*FresnelF[z]=Divide[1,2]*(1 - I)* Exp[(\[zeta])^(2)]*Erfc[\[zeta]] Failure Failure
Fail
-.1458959936+.662848896e-1*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2)}
-.3171418896-.1049610064*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)-I*2^(1/2)}
1.307352110-.2158500180*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)-I*2^(1/2)}
-.3500698200e-1-1.558209110*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Error
7.6.E1 ${\displaystyle{\displaystyle\operatorname{erf}z=\frac{2}{\sqrt{\pi}}\sum_{n=0}% ^{\infty}\frac{(-1)^{n}z^{2n+1}}{n!(2n+1)}}}$ erf(z)=(2)/(sqrt(Pi))*sum(((- 1)^(n)* (z)^(2*n + 1))/(factorial(n)*(2*n + 1)), n = 0..infinity) Erf[z]=Divide[2,Sqrt[Pi]]*Sum[Divide[(- 1)^(n)* (z)^(2*n + 1),(n)!*(2*n + 1)], {n, 0, Infinity}] Successful Successful - -
7.6.E4 ${\displaystyle{\displaystyle C\left(z\right)=\sum_{n=0}^{\infty}\frac{(-1)^{n}% (\frac{1}{2}\pi)^{2n}}{(2n)!(4n+1)}z^{4n+1}}}$ FresnelC(z)= sum(((- 1)^(n)*((1)/(2)*Pi)^(2*n))/(factorial(2*n)*(4*n + 1))*(z)^(4*n + 1), n = 0..infinity) FresnelC[z]= Sum[Divide[(- 1)^(n)*(Divide[1,2]*Pi)^(2*n),(2*n)!*(4*n + 1)]*(z)^(4*n + 1), {n, 0, Infinity}] Successful Successful - -
7.6.E6 ${\displaystyle{\displaystyle S\left(z\right)=\sum_{n=0}^{\infty}\frac{(-1)^{n}% (\frac{1}{2}\pi)^{2n+1}}{(2n+1)!(4n+3)}z^{4n+3}}}$ FresnelS(z)= sum(((- 1)^(n)*((1)/(2)*Pi)^(2*n + 1))/(factorial(2*n + 1)*(4*n + 3))*(z)^(4*n + 3), n = 0..infinity) FresnelS[z]= Sum[Divide[(- 1)^(n)*(Divide[1,2]*Pi)^(2*n + 1),(2*n + 1)!*(4*n + 3)]*(z)^(4*n + 3), {n, 0, Infinity}] Successful Successful - -
7.6.E8 ${\displaystyle{\displaystyle\operatorname{erf}z=\frac{2z}{\sqrt{\pi}}\sum_{n=0% }^{\infty}(-1)^{n}\left({\mathsf{i}^{(1)}_{2n}}\left(z^{2}\right)-{\mathsf{i}^% {(1)}_{2n+1}}\left(z^{2}\right)\right)}}$ Error \|Sqrt[1/2 Pi /$2] BesselI[-2*n - 1/2, 2*n]*(z)^(2)- Sqrt[1/2 Pi /$2] BesselI[(-1)^(1-1)*2*n + 1 + 1/2, 2*n + 1]\|\|Sqrt[1/2 Pi /$2] BesselI[-2*n + 1 - 1/2, 2*n + 1]*(z)^(2)), {n, 0, Infinity}] Error Error - - 7.6.E9 ${\displaystyle{\displaystyle\operatorname{erf}\left(az\right)=\frac{2z}{\sqrt{% \pi}}e^{(\frac{1}{2}-a^{2})z^{2}}\sum_{n=0}^{\infty}T_{2n+1}\left(a\right){% \mathsf{i}^{(1)}_{n}}\left(\tfrac{1}{2}z^{2}\right)}}$ Error \|Sqrt[1/2 Pi /$2] BesselI[-n - 1/2, n]*Divide[1,2]*(z)^(2), {n, 0, Infinity}] Error Error - -
7.6.E10 ${\displaystyle{\displaystyle C\left(z\right)=z\sum_{n=0}^{\infty}\mathsf{j}_{2% n}\left(\tfrac{1}{2}\pi z^{2}\right)}}$ Error FresnelC[z]= z*Sum[SphericalBesselJ[2*n, Divide[1,2]*Pi*(z)^(2)], {n, 0, Infinity}] Error Failure - Skip
7.6.E11 ${\displaystyle{\displaystyle S\left(z\right)=z\sum_{n=0}^{\infty}\mathsf{j}_{2% n+1}\left(\tfrac{1}{2}\pi z^{2}\right)}}$ Error FresnelS[z]= z*Sum[SphericalBesselJ[2*n + 1, Divide[1,2]*Pi*(z)^(2)], {n, 0, Infinity}] Error Failure - Skip
7.7.E1 ${\displaystyle{\displaystyle\operatorname{erfc}z=\frac{2}{\pi}e^{-z^{2}}\int_{% 0}^{\infty}\frac{e^{-z^{2}t^{2}}}{t^{2}+1}\mathrm{d}t}}$ erfc(z)=(2)/(Pi)*exp(- (z)^(2))*int((exp(- (z)^(2)* (t)^(2)))/((t)^(2)+ 1), t = 0..infinity) Erfc[z]=Divide[2,Pi]*Exp[- (z)^(2)]*Integrate[Divide[Exp[- (z)^(2)* (t)^(2)],(t)^(2)+ 1], {t, 0, Infinity}] Successful Failure - Error
7.7.E2 ${\displaystyle{\displaystyle\frac{1}{\pi i}\int_{-\infty}^{\infty}\frac{e^{-t^% {2}}\mathrm{d}t}{t-z}=\frac{2z}{\pi i}\int_{0}^{\infty}\frac{e^{-t^{2}}\mathrm% {d}t}{t^{2}-z^{2}}}}$ (1)/(Pi*I)*int((exp(- (t)^(2)))/(t - z), t = - infinity..infinity)=(2*z)/(Pi*I)*int((exp(- (t)^(2)))/((t)^(2)- (z)^(2)), t = 0..infinity) Divide[1,Pi*I]*Integrate[Divide[Exp[- (t)^(2)],t - z], {t, - Infinity, Infinity}]=Divide[2*z,Pi*I]*Integrate[Divide[Exp[- (t)^(2)],(t)^(2)- (z)^(2)], {t, 0, Infinity}] Failure Failure Skip Successful
7.7.E3 ${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at^{2}+2izt}\mathrm{d}t=\frac% {1}{2}\sqrt{\frac{\pi}{a}}e^{-z^{2}/a}+\frac{i}{\sqrt{a}}F\left(\frac{z}{\sqrt% {a}}\right)}}$ int(exp(- a*(t)^(2)+ 2*I*z*t), t = 0..infinity)=(1)/(2)*sqrt((Pi)/(a))*exp(- (z)^(2)/ a)+(I)/(sqrt(a))*dawson((z)/(sqrt(a))) Integrate[Exp[- a*(t)^(2)+ 2*I*z*t], {t, 0, Infinity}]=Divide[1,2]*Sqrt[Divide[Pi,a]]*Exp[- (z)^(2)/ a]+Divide[I,Sqrt[a]]*DawsonF[Divide[z,Sqrt[a]]] Failure Successful Skip -
7.7.E4 ${\displaystyle{\displaystyle\int_{0}^{\infty}\frac{e^{-at}}{\sqrt{t+z^{2}}}% \mathrm{d}t=\sqrt{\frac{\pi}{a}}e^{az^{2}}\operatorname{erfc}\left(\sqrt{a}z% \right)}}$ int((exp(- a*t))/(sqrt(t + (z)^(2))), t = 0..infinity)=sqrt((Pi)/(a))*exp(a*(z)^(2))*erfc(sqrt(a)*z) Integrate[Divide[Exp[- a*t],Sqrt[t + (z)^(2)]], {t, 0, Infinity}]=Sqrt[Divide[Pi,a]]*Exp[a*(z)^(2)]*Erfc[Sqrt[a]*z] Successful Failure - Successful
7.7.E6 ${\displaystyle{\displaystyle\int_{x}^{\infty}e^{-(at^{2}+2bt+c)}\mathrm{d}t=% \frac{1}{2}\sqrt{\frac{\pi}{a}}e^{(b^{2}-ac)/a}\operatorname{erfc}\left(\sqrt{% a}x+\frac{b}{\sqrt{a}}\right)}}$ int(exp(-(a*(t)^(2)+ 2*b*t + c)), t = x..infinity)=(1)/(2)*sqrt((Pi)/(a))*exp(((b)^(2)- a*c)/ a)*erfc(sqrt(a)*x +(b)/(sqrt(a))) Integrate[Exp[-(a*(t)^(2)+ 2*b*t + c)], {t, x, Infinity}]=Divide[1,2]*Sqrt[Divide[Pi,a]]*Exp[((b)^(2)- a*c)/ a]*Erfc[Sqrt[a]*x +Divide[b,Sqrt[a]]] Failure Failure Skip Successful
7.7.E7 ${\displaystyle{\displaystyle\int_{x}^{\infty}e^{-a^{2}t^{2}-(b^{2}/t^{2})}% \mathrm{d}t=\frac{\sqrt{\pi}}{4a}\left(e^{2ab}\operatorname{erfc}\left(ax+(b/x% )\right)+e^{-2ab}\operatorname{erfc}\left(ax-(b/x)\right)\right)}}$ int(exp(- (a)^(2)* (t)^(2)-((b)^(2)/ (t)^(2))), t = x..infinity)=(sqrt(Pi))/(4*a)*(exp(2*a*b)*erfc(a*x +(b/ x))+ exp(- 2*a*b)*erfc(a*x -(b/ x))) Integrate[Exp[- (a)^(2)* (t)^(2)-((b)^(2)/ (t)^(2))], {t, x, Infinity}]=Divide[Sqrt[Pi],4*a]*(Exp[2*a*b]*Erfc[a*x +(b/ x)]+ Exp[- 2*a*b]*Erfc[a*x -(b/ x)]) Failure Failure Skip Error
7.7.E8 ${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-a^{2}t^{2}-(b^{2}/t^{2})}% \mathrm{d}t=\frac{\sqrt{\pi}}{2a}e^{-2ab}}}$ int(exp(- (a)^(2)* (t)^(2)-((b)^(2)/ (t)^(2))), t = 0..infinity)=(sqrt(Pi))/(2*a)*exp(- 2*a*b) Integrate[Exp[- (a)^(2)* (t)^(2)-((b)^(2)/ (t)^(2))], {t, 0, Infinity}]=Divide[Sqrt[Pi],2*a]*Exp[- 2*a*b] Successful Failure - Successful
7.7.E9 ${\displaystyle{\displaystyle\int_{0}^{x}\operatorname{erf}t\mathrm{d}t=x% \operatorname{erf}x+\frac{1}{\sqrt{\pi}}\left(e^{-x^{2}}-1\right)}}$ int(erf(t), t = 0..x)= x*erf(x)+(1)/(sqrt(Pi))*(exp(- (x)^(2))- 1) Integrate[Erf[t], {t, 0, x}]= x*Erf[x]+Divide[1,Sqrt[Pi]]*(Exp[- (x)^(2)]- 1) Successful Successful - -
7.7.E10 ${\displaystyle{\displaystyle\mathrm{f}\left(z\right)=\frac{1}{\pi\sqrt{2}}\int% _{0}^{\infty}\frac{e^{-\pi z^{2}t/2}}{\sqrt{t}(t^{2}+1)}\mathrm{d}t}}$ Fresnelf(z)=(1)/(Pi*sqrt(2))*int((exp(- Pi*(z)^(2)* t/ 2))/(sqrt(t)*((t)^(2)+ 1)), t = 0..infinity) FresnelF[z]=Divide[1,Pi*Sqrt[2]]*Integrate[Divide[Exp[- Pi*(z)^(2)* t/ 2],Sqrt[t]*((t)^(2)+ 1)], {t, 0, Infinity}] Failure Failure Skip Error
7.7.E11 ${\displaystyle{\displaystyle\mathrm{g}\left(z\right)=\frac{1}{\pi\sqrt{2}}\int% _{0}^{\infty}\frac{\sqrt{t}e^{-\pi z^{2}t/2}}{t^{2}+1}\mathrm{d}t}}$ Fresnelg(z)=(1)/(Pi*sqrt(2))*int((sqrt(t)*exp(- Pi*(z)^(2)* t/ 2))/((t)^(2)+ 1), t = 0..infinity) FresnelG[z]=Divide[1,Pi*Sqrt[2]]*Integrate[Divide[Sqrt[t]*Exp[- Pi*(z)^(2)* t/ 2],(t)^(2)+ 1], {t, 0, Infinity}] Failure Failure Skip Error
7.7.E12 ${\displaystyle{\displaystyle\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)% =e^{-\pi iz^{2}/2}\int_{z}^{\infty}e^{\pi it^{2}/2}\mathrm{d}t}}$ Fresnelg(z)+ I*Fresnelf(z)= exp(- Pi*I*(z)^(2)/ 2)*int(exp(Pi*I*(t)^(2)/ 2), t = z..infinity) FresnelG[z]+ I*FresnelF[z]= Exp[- Pi*I*(z)^(2)/ 2]*Integrate[Exp[Pi*I*(t)^(2)/ 2], {t, z, Infinity}] Failure Failure Skip
Fail
Complex[-0.12449815517713354, 0.12449815517716199] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.12449815517710515, -0.12449815517713354] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
7.7.E13 ${\displaystyle{\displaystyle\mathrm{f}\left(z\right)=\frac{(2\pi)^{-3/2}}{2\pi i% }\int_{c-i\infty}^{c+i\infty}\zeta^{-s}\Gamma\left(s\right)\Gamma\left(s+% \tfrac{1}{2}\right)\*\Gamma\left(s+\tfrac{3}{4}\right)\Gamma\left(\tfrac{1}{4}% -s\right)\mathrm{d}s}}$ Fresnelf(z)=((2*Pi)^(- 3/ 2))/(2*Pi*I)*int((zeta)^(- s)* GAMMA(s)*GAMMA(s +(1)/(2))* GAMMA(s +(3)/(4))*GAMMA((1)/(4)- s), s = c - I*infinity..c + I*infinity) FresnelF[z]=Divide[(2*Pi)^(- 3/ 2),2*Pi*I]*Integrate[(\[zeta])^(- s)* Gamma[s]*Gamma[s +Divide[1,2]]* Gamma[s +Divide[3,4]]*Gamma[Divide[1,4]- s], {s, c - I*Infinity, c + I*Infinity}] Failure Failure Skip Error
7.7.E14 ${\displaystyle{\displaystyle\mathrm{g}\left(z\right)=\frac{(2\pi)^{-3/2}}{2\pi i% }\int_{c-i\infty}^{c+i\infty}\zeta^{-s}\Gamma\left(s\right)\Gamma\left(s+% \tfrac{1}{2}\right)\*\Gamma\left(s+\tfrac{1}{4}\right)\Gamma\left(\tfrac{3}{4}% -s\right)\mathrm{d}s}}$ Fresnelg(z)=((2*Pi)^(- 3/ 2))/(2*Pi*I)*int((zeta)^(- s)* GAMMA(s)*GAMMA(s +(1)/(2))* GAMMA(s +(1)/(4))*GAMMA((3)/(4)- s), s = c - I*infinity..c + I*infinity) FresnelG[z]=Divide[(2*Pi)^(- 3/ 2),2*Pi*I]*Integrate[(\[zeta])^(- s)* Gamma[s]*Gamma[s +Divide[1,2]]* Gamma[s +Divide[1,4]]*Gamma[Divide[3,4]- s], {s, c - I*Infinity, c + I*Infinity}] Failure Failure Skip Error
7.7.E15 ${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}\cos\left(t^{2}\right)% \mathrm{d}t=\sqrt{\frac{\pi}{2}}\mathrm{f}\left(\frac{a}{\sqrt{2\pi}}\right)}}$ int(exp(- a*t)*cos((t)^(2)), t = 0..infinity)=sqrt((Pi)/(2))*Fresnelf((a)/(sqrt(2*Pi))) Integrate[Exp[- a*t]*Cos[(t)^(2)], {t, 0, Infinity}]=Sqrt[Divide[Pi,2]]*FresnelF[Divide[a,Sqrt[2*Pi]]] Successful Failure - Error
7.7.E16 ${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}\sin\left(t^{2}\right)% \mathrm{d}t=\sqrt{\frac{\pi}{2}}\mathrm{g}\left(\frac{a}{\sqrt{2\pi}}\right)}}$ int(exp(- a*t)*sin((t)^(2)), t = 0..infinity)=sqrt((Pi)/(2))*Fresnelg((a)/(sqrt(2*Pi))) Integrate[Exp[- a*t]*Sin[(t)^(2)], {t, 0, Infinity}]=Sqrt[Divide[Pi,2]]*FresnelG[Divide[a,Sqrt[2*Pi]]] Successful Failure - Error
7.8.E1 ${\displaystyle{\displaystyle\frac{\int_{x}^{\infty}e^{-t^{2}}\mathrm{d}t}{e^{-% x^{2}}}=e^{x^{2}}\int_{x}^{\infty}e^{-t^{2}}\mathrm{d}t}}$ (int(exp(- (t)^(2)), t = x..infinity))/(exp(- (x)^(2)))= exp((x)^(2))*int(exp(- (t)^(2)), t = x..infinity) Divide[Integrate[Exp[- (t)^(2)], {t, x, Infinity}],Exp[- (x)^(2)]]= Exp[(x)^(2)]*Integrate[Exp[- (t)^(2)], {t, x, Infinity}] Successful Successful - -
7.8.E5 ${\displaystyle{\displaystyle\frac{x^{2}}{2x^{2}+1}<=\frac{x^{2}(2x^{2}+5)}{4x^% {4}+12x^{2}+3}}}$ ((x)^(2))/(2*(x)^(2)+ 1)< =((x)^(2)*(2*(x)^(2)+ 5))/(4*(x)^(4)+ 12*(x)^(2)+ 3) Divide[(x)^(2),2*(x)^(2)+ 1]< =Divide[(x)^(2)*(2*(x)^(2)+ 5),4*(x)^(4)+ 12*(x)^(2)+ 3] Failure Failure Successful Successful
7.8.E5 ${\displaystyle{\displaystyle\frac{2x^{4}+9x^{2}+4}{4x^{4}+20x^{2}+15}<\frac{x^% {2}+1}{2x^{2}+3}}}$ (2*(x)^(4)+ 9*(x)^(2)+ 4)/(4*(x)^(4)+ 20*(x)^(2)+ 15)<((x)^(2)+ 1)/(2*(x)^(2)+ 3) Divide[2*(x)^(4)+ 9*(x)^(2)+ 4,4*(x)^(4)+ 20*(x)^(2)+ 15]<Divide[(x)^(2)+ 1,2*(x)^(2)+ 3] Failure Failure Skip Successful
7.8.E6 ${\displaystyle{\displaystyle\int_{0}^{x}e^{at^{2}}\mathrm{d}t<\frac{1}{3ax}% \left(2e^{ax^{2}}+ax^{2}-2\right)}}$ int(exp(a*(t)^(2)), t = 0..x)<(1)/(3*a*x)*(2*exp(a*(x)^(2))+ a*(x)^(2)- 2) Integrate[Exp[a*(t)^(2)], {t, 0, x}]<Divide[1,3*a*x]*(2*Exp[a*(x)^(2)]+ a*(x)^(2)- 2) Error Failure - Successful
7.8.E7 ${\displaystyle{\displaystyle\int_{0}^{x}e^{t^{2}}\mathrm{d}t<\frac{e^{x^{2}}-1% }{x}}}$ int(exp((t)^(2)), t = 0..x)<(exp((x)^(2))- 1)/(x) Integrate[Exp[(t)^(2)], {t, 0, x}]<Divide[Exp[(x)^(2)]- 1,x] Failure Failure Skip Successful
7.8.E8 ${\displaystyle{\displaystyle\operatorname{erf}x<\sqrt{1-{\mathrm{e}^{-4x^{2}/% \pi}}}}}$ erf(x)<sqrt(1 - exp(- 4*(x)^(2)/ Pi)) Erf[x]<Sqrt[1 - Exp[- 4*(x)^(2)/ Pi]] Failure Failure Skip Successful
7.10.E1 ${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n+1}\operatorname{erf}z}{{% \mathrm{d}z}^{n+1}}=(-1)^{n}\frac{2}{\sqrt{\pi}}H_{n}\left(z\right)e^{-z^{2}}}}$ diff(erf(z), [z\$(n + 1)])=(- 1)^(n)*(2)/(sqrt(Pi))*HermiteH(n, z)*exp(- (z)^(2)) D[Erf[z], {z, n + 1}]=(- 1)^(n)*Divide[2,Sqrt[Pi]]*HermiteH[n, z]*Exp[- (z)^(2)] Failure Failure Skip Successful
7.10#Ex1 ${\displaystyle{\displaystyle\frac{\mathrm{d}\mathrm{f}\left(z\right)}{\mathrm{% d}z}=-\pi z\mathrm{g}\left(z\right)}}$ diff(Fresnelf(z), z)= - Pi*z*Fresnelg(z) D[FresnelF[z], z]= - Pi*z*FresnelG[z] Successful Successful - -
7.10#Ex2 ${\displaystyle{\displaystyle\frac{\mathrm{d}\mathrm{g}\left(z\right)}{\mathrm{% d}z}=\pi z\mathrm{f}\left(z\right)-1}}$ diff(Fresnelg(z), z)= Pi*z*Fresnelf(z)- 1 D[FresnelG[z], z]= Pi*z*FresnelF[z]- 1 Successful Successful - -
7.11.E1 ${\displaystyle{\displaystyle\operatorname{erf}z=\frac{1}{\sqrt{\pi}}\gamma% \left(\tfrac{1}{2},z^{2}\right)}}$ erf(z)=(1)/(sqrt(Pi))*GAMMA((1)/(2))-GAMMA((1)/(2), (z)^(2)) Erf[z]=Divide[1,Sqrt[Pi]]*Gamma[Divide[1,2], 0, (z)^(2)] Failure Failure
Fail
-.796532174e-2+.2115950078*I <- {z = 2^(1/2)+I*2^(1/2)}
-.796532174e-2-.2115950078*I <- {z = 2^(1/2)-I*2^(1/2)}
-2.028588748+.7594465268*I <- {z = -2^(1/2)-I*2^(1/2)}
-2.028588748-.7594465268*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-2.020623424050978, 0.547851518927081] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.020623424050978, -0.547851518927081] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
7.11.E2 ${\displaystyle{\displaystyle\operatorname{erfc}z=\frac{1}{\sqrt{\pi}}\Gamma% \left(\tfrac{1}{2},z^{2}\right)}}$ erfc(z)=(1)/(sqrt(Pi))*GAMMA((1)/(2), (z)^(2)) Erfc[z]=Divide[1,Sqrt[Pi]]*Gamma[Divide[1,2], (z)^(2)] Failure Failure
Fail
2.020623426-.5478515190*I <- {z = -2^(1/2)-I*2^(1/2)}
2.020623426+.5478515190*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[2.0206234240509775, -0.547851518927081] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.0206234240509775, 0.547851518927081] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
7.11.E3 ${\displaystyle{\displaystyle\operatorname{erfc}z=\frac{z}{\sqrt{\pi}}E_{\frac{% 1}{2}}\left(z^{2}\right)}}$ erfc(z)=(z)/(sqrt(Pi))*Ei((1)/(2), (z)^(2)) Erfc[z]=Divide[z,Sqrt[Pi]]*ExpIntegralE[Divide[1,2], (z)^(2)] Failure Failure
Fail
2.000000000-.1e-9*I <- {z = -2^(1/2)-I*2^(1/2)}
2.000000000+.1e-9*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[1.9999999999999996, -1.1102230246251565*^-16] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.9999999999999996, 1.1102230246251565*^-16] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
7.11.E4 ${\displaystyle{\displaystyle\operatorname{erf}z=\frac{2z}{\sqrt{\pi}}M\left(% \tfrac{1}{2},\tfrac{3}{2},-z^{2}\right)}}$ erf(z)=(2*z)/(sqrt(Pi))*KummerM((1)/(2), (3)/(2), - (z)^(2)) Erf[z]=Divide[2*z,Sqrt[Pi]]*Hypergeometric1F1[Divide[1,2], Divide[3,2], - (z)^(2)] Successful Successful - -
7.11.E4 ${\displaystyle{\displaystyle\frac{2z}{\sqrt{\pi}}M\left(\tfrac{1}{2},\tfrac{3}% {2},-z^{2}\right)=\frac{2z}{\sqrt{\pi}}e^{-z^{2}}M\left(1,\tfrac{3}{2},z^{2}% \right)}}$ (2*z)/(sqrt(Pi))*KummerM((1)/(2), (3)/(2), - (z)^(2))=(2*z)/(sqrt(Pi))*exp(- (z)^(2))*KummerM(1, (3)/(2), (z)^(2)) Divide[2*z,Sqrt[Pi]]*Hypergeometric1F1[Divide[1,2], Divide[3,2], - (z)^(2)]=Divide[2*z,Sqrt[Pi]]*Exp[- (z)^(2)]*Hypergeometric1F1[1, Divide[3,2], (z)^(2)] Successful Successful - -
7.11.E5 ${\displaystyle{\displaystyle\operatorname{erfc}z=\frac{1}{\sqrt{\pi}}e^{-z^{2}% }U\left(\tfrac{1}{2},\tfrac{1}{2},z^{2}\right)}}$ erfc(z)=(1)/(sqrt(Pi))*exp(- (z)^(2))*KummerU((1)/(2), (1)/(2), (z)^(2)) Erfc[z]=Divide[1,Sqrt[Pi]]*Exp[- (z)^(2)]*HypergeometricU[Divide[1,2], Divide[1,2], (z)^(2)] Failure Failure
Fail
2.020623426-.5478515190*I <- {z = -2^(1/2)-I*2^(1/2)}
2.020623426+.5478515190*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[2.0206234240509775, -0.547851518927081] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.0206234240509775, 0.547851518927081] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
7.11.E5 ${\displaystyle{\displaystyle\frac{1}{\sqrt{\pi}}e^{-z^{2}}U\left(\tfrac{1}{2},% \tfrac{1}{2},z^{2}\right)=\frac{z}{\sqrt{\pi}}e^{-z^{2}}U\left(1,\tfrac{3}{2},% z^{2}\right)}}$ (1)/(sqrt(Pi))*exp(- (z)^(2))*KummerU((1)/(2), (1)/(2), (z)^(2))=(z)/(sqrt(Pi))*exp(- (z)^(2))*KummerU(1, (3)/(2), (z)^(2)) Divide[1,Sqrt[Pi]]*Exp[- (z)^(2)]*HypergeometricU[Divide[1,2], Divide[1,2], (z)^(2)]=Divide[z,Sqrt[Pi]]*Exp[- (z)^(2)]*HypergeometricU[1, Divide[3,2], (z)^(2)] Failure Failure
Fail
-.2062342514e-1+.5478515190*I <- {z = -2^(1/2)-I*2^(1/2)}
-.2062342514e-1-.5478515190*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-0.02062342405097809, 0.5478515189270807] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.020623424050978133, -0.5478515189270807] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
7.11.E6 ${\displaystyle{\displaystyle C\left(z\right)+iS\left(z\right)=zM\left(\tfrac{1% }{2},\tfrac{3}{2},\tfrac{1}{2}\pi iz^{2}\right)}}$ FresnelC(z)+ I*FresnelS(z)= z*KummerM((1)/(2), (3)/(2), (1)/(2)*Pi*I*(z)^(2)) FresnelC[z]+ I*FresnelS[z]= z*Hypergeometric1F1[Divide[1,2], Divide[3,2], Divide[1,2]*Pi*I*(z)^(2)] Failure Successful Successful -
7.11.E6 ${\displaystyle{\displaystyle zM\left(\tfrac{1}{2},\tfrac{3}{2},\tfrac{1}{2}\pi iz% ^{2}\right)=ze^{\pi iz^{2}/2}M\left(1,\tfrac{3}{2},-\tfrac{1}{2}\pi iz^{2}% \right)}}$ z*KummerM((1)/(2), (3)/(2), (1)/(2)*Pi*I*(z)^(2))= z*exp(Pi*I*(z)^(2)/ 2)*KummerM(1, (3)/(2), -(1)/(2)*Pi*I*(z)^(2)) z*Hypergeometric1F1[Divide[1,2], Divide[3,2], Divide[1,2]*Pi*I*(z)^(2)]= z*Exp[Pi*I*(z)^(2)/ 2]*Hypergeometric1F1[1, Divide[3,2], -Divide[1,2]*Pi*I*(z)^(2)] Successful Successful - -
7.11.E7 ${\displaystyle{\displaystyle C\left(z\right)=z{{}_{1}F_{2}}\left(\tfrac{1}{4};% \tfrac{5}{4},\tfrac{1}{2};-\tfrac{1}{16}\pi^{2}z^{4}\right)}}$ FresnelC(z)= z*hypergeom([(1)/(4)], [(5)/(4),(1)/(2)], -(1)/(16)*(Pi)^(2)* (z)^(4)) FresnelC[z]= z*HypergeometricPFQ[{Divide[1,4]}, {Divide[5,4],Divide[1,2]}, -Divide[1,16]*(Pi)^(2)* (z)^(4)] Successful Successful - -
7.11.E8 ${\displaystyle{\displaystyle S\left(z\right)=\tfrac{1}{6}\pi z^{3}{{}_{1}F_{2}% }\left(\tfrac{3}{4};\tfrac{7}{4},\tfrac{3}{2};-\tfrac{1}{16}\pi^{2}z^{4}\right% )}}$ FresnelS(z)=(1)/(6)*Pi*(z)^(3)* hypergeom([(3)/(4)], [(7)/(4),(3)/(2)], -(1)/(16)*(Pi)^(2)* (z)^(4)) FresnelS[z]=Divide[1,6]*Pi*(z)^(3)* HypergeometricPFQ[{Divide[3,4]}, {Divide[7,4],Divide[3,2]}, -Divide[1,16]*(Pi)^(2)* (z)^(4)] Successful Successful - -
7.13#Ex4 ${\displaystyle{\displaystyle\mu=\ln\left(\lambda\sqrt{2\pi}\right)}}$ mu = ln(lambda*sqrt(2*Pi)) \[Mu]= Log[\[Lambda]*Sqrt[2*Pi]] Failure Failure
Fail
-.197872151+.6288153986*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2)}
-.197872151-2.199611725*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)-I*2^(1/2)}
-3.026299275-2.199611725*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = -2^(1/2)-I*2^(1/2)}
-3.026299275+.6288153986*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-0.1978721513915227, 0.6288153989756469] <- {Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.1978721513915227, -2.199611725770543] <- {Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-3.026299276137713, -2.199611725770543] <- {Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-3.026299276137713, 0.6288153989756469] <- {Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
7.13#Ex8 ${\displaystyle{\displaystyle\mu=\ln\left(2\lambda\sqrt{2\pi}\right)}}$ mu = ln(2*lambda*sqrt(2*Pi)) \[Mu]= Log[2*\[Lambda]*Sqrt[2*Pi]] Failure Failure
Fail
-.891019332+.6288153986*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2)}
-.891019332-2.199611725*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)-I*2^(1/2)}
-3.719446456-2.199611725*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = -2^(1/2)-I*2^(1/2)}
-3.719446456+.6288153986*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-0.8910193319514683, 0.6288153989756469] <- {Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.8910193319514683, -2.199611725770543] <- {Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-3.719446456697659, -2.199611725770543] <- {Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-3.719446456697659, 0.6288153989756469] <- {Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
7.13#Ex12 ${\displaystyle{\displaystyle\alpha=(2/\pi)\ln\left(\pi\lambda\right)}}$ alpha =(2/ Pi)* ln(Pi*lambda) \[Alpha]=(2/ Pi)* Log[Pi*\[Lambda]] Failure Failure
Fail
.244184683+.9142135621*I <- {alpha = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)+I*2^(1/2)}
.244184683+1.914213562*I <- {alpha = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)-I*2^(1/2)}
.244184683+2.914213561*I <- {alpha = 2^(1/2)+I*2^(1/2), lambda = -2^(1/2)-I*2^(1/2)}
.244184683-.85786437e-1*I <- {alpha = 2^(1/2)+I*2^(1/2), lambda = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[0.24418468271597948, 0.9142135623730951] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.24418468271597948, 1.9142135623730951] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[λ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.24418468271597948, 2.914213562373095] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[λ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.24418468271597948, -0.08578643762690485] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[λ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
7.13#Ex14 ${\displaystyle{\displaystyle\alpha=(2/\pi)\ln\left(\pi\lambda\right)}}$ alpha =(2/ Pi)* ln(Pi*lambda) \[Alpha]=(2/ Pi)* Log[Pi*\[Lambda]] Failure Failure
Fail
.244184683+.9142135621*I <- {alpha = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)+I*2^(1/2)}
.244184683+1.914213562*I <- {alpha = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)-I*2^(1/2)}
.244184683+2.914213561*I <- {alpha = 2^(1/2)+I*2^(1/2), lambda = -2^(1/2)-I*2^(1/2)}
.244184683-.85786437e-1*I <- {alpha = 2^(1/2)+I*2^(1/2), lambda = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[0.24418468271597948, 0.9142135623730951] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.24418468271597948, 1.9142135623730951] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[λ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.24418468271597948, 2.914213562373095] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[λ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.24418468271597948, -0.08578643762690485] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[λ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
7.14.E1 ${\displaystyle{\displaystyle\int_{0}^{\infty}e^{2iat}\operatorname{erfc}\left(% bt\right)\mathrm{d}t={\frac{1}{a\sqrt{\pi}}F\left(\frac{a}{b}\right)+\frac{i}{% 2a}\left(1-e^{-(a/b)^{2}}\right)}}}$ int(exp(2*I*a*t)*erfc(b*t), t = 0..infinity)=(1)/(a*sqrt(Pi))*dawson((a)/(b))+(I)/(2*a)*(1 - exp(-(a/ b)^(2))) Integrate[Exp[2*I*a*t]*Erfc[b*t], {t, 0, Infinity}]=Divide[1,a*Sqrt[Pi]]*DawsonF[Divide[a,b]]+Divide[I,2*a]*(1 - Exp[-(a/ b)^(2)]) Failure Failure Skip Error
7.14.E2 ${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}\operatorname{erf}\left(bt% \right)\mathrm{d}t=\frac{1}{a}e^{a^{2}/(4b^{2})}\operatorname{erfc}\left(\frac% {a}{2b}\right)}}$ int(exp(- a*t)*erf(b*t), t = 0..infinity)=(1)/(a)*exp((a)^(2)/(4*(b)^(2)))*erfc((a)/(2*b)) Integrate[Exp[- a*t]*Erf[b*t], {t, 0, Infinity}]=Divide[1,a]*Exp[(a)^(2)/(4*(b)^(2))]*Erfc[Divide[a,2*b]] Successful Failure - Error
7.14.E4 ${\displaystyle{\displaystyle\int_{0}^{\infty}e^{(a-b)t}\operatorname{erfc}% \left(\sqrt{at}+\sqrt{\frac{c}{t}}\right)\mathrm{d}t=\frac{e^{-2(\sqrt{ac}+% \sqrt{bc})}}{\sqrt{b}(\sqrt{a}+\sqrt{b})}}}$ int(exp((a - b)* t)*erfc(sqrt(a*t)+sqrt((c)/(t))), t = 0..infinity)=(exp(- 2*(sqrt(a*c)+sqrt(b*c))))/(sqrt(b)*(sqrt(a)+sqrt(b))) Integrate[Exp[(a - b)* t]*Erfc[Sqrt[a*t]+Sqrt[Divide[c,t]]], {t, 0, Infinity}]=Divide[Exp[- 2*(Sqrt[a*c]+Sqrt[b*c])],Sqrt[b]*(Sqrt[a]+Sqrt[b])] Failure Failure Skip Error
7.14.E5 ${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}C\left(t\right)\mathrm{d}t% =\frac{1}{a}\mathrm{f}\left(\frac{a}{\pi}\right)}}$ int(exp(- a*t)*FresnelC(t), t = 0..infinity)=(1)/(a)*Fresnelf((a)/(Pi)) Integrate[Exp[- a*t]*FresnelC[t], {t, 0, Infinity}]=Divide[1,a]*FresnelF[Divide[a,Pi]] Failure Failure Skip Successful
7.14.E6 ${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}S\left(t\right)\mathrm{d}t% =\frac{1}{a}\mathrm{g}\left(\frac{a}{\pi}\right)}}$ int(exp(- a*t)*FresnelS(t), t = 0..infinity)=(1)/(a)*Fresnelg((a)/(Pi)) Integrate[Exp[- a*t]*FresnelS[t], {t, 0, Infinity}]=Divide[1,a]*FresnelG[Divide[a,Pi]] Failure Failure Skip Successful
7.17#Ex1 ${\displaystyle{\displaystyle y=\operatorname{inverf}x}}$ Error y = InverseErf[x] Error Failure -
Fail
DirectedInfinity[-1] <- {Rule[x, 1], Rule[y, 1]}
DirectedInfinity[-1] <- {Rule[x, 1], Rule[y, 2]}
DirectedInfinity[-1] <- {Rule[x, 1], Rule[y, 3]}
7.17#Ex2 ${\displaystyle{\displaystyle y=\operatorname{inverfc}x}}$ Error y = InverseErfc[x] Error Failure -
Fail
1.0 <- {Rule[x, 1], Rule[y, 1]}
2.0 <- {Rule[x, 1], Rule[y, 2]}
3.0 <- {Rule[x, 1], Rule[y, 3]}
DirectedInfinity[1] <- {Rule[x, 2], Rule[y, 1]}
... skip entries to safe data
7.18#Ex1 ${\displaystyle{\displaystyle\mathop{\mathrm{i}^{-1}\mathrm{erfc}}\left(z\right% )=\frac{2}{\sqrt{\pi}}e^{-z^{2}}}}$ erfc(- 1, z)=(2)/(sqrt(Pi))*exp(- (z)^(2)) I^(- 1)*Erfc[z]=Divide[2,Sqrt[Pi]]*Exp[- (z)^(2)] Successful Failure -
Fail
Complex[1.011483603950918, -0.8436484572858769] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.46363208502383696, 0.8642718813368542] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.46363208502383696, -2.864271881336854] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.011483603950918, -1.1563515427141229] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
7.18#Ex2 ${\displaystyle{\displaystyle\mathop{\mathrm{i}^{0}\mathrm{erfc}}\left(z\right)% =\operatorname{erfc}z}}$ erfc(0, z)= erfc(z) I^(0)*Erfc[z]= Erfc[z] Successful Successful - -
7.18.E2 ${\displaystyle{\displaystyle\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)% =\int_{z}^{\infty}\mathop{\mathrm{i}^{n-1}\mathrm{erfc}}\left(t\right)\mathrm{% d}t}}$ erfc(n, z)= int(erfc(n - 1, t), t = z..infinity) I^(n)*Erfc[z]= Integrate[I^(n - 1)*Erfc[t], {t, z, Infinity}] Failure Failure Skip
Fail
Complex[-0.30711932433427286, -0.06448523556221403] <- {Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.06448523556221401, -0.30711932433427286] <- {Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.30711932433427286, 0.06448523556221403] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.2407321945928082, 0.04386181151123664] <- {Rule[n, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
7.18.E2 ${\displaystyle{\displaystyle\int_{z}^{\infty}\mathop{\mathrm{i}^{n-1}\mathrm{% erfc}}\left(t\right)\mathrm{d}t=\frac{2}{\sqrt{\pi}}\int_{z}^{\infty}\frac{(t-% z)^{n}}{n!}e^{-t^{2}}\mathrm{d}t}}$ int(erfc(n - 1, t), t = z..infinity)=(2)/(sqrt(Pi))*int(((t - z)^(n))/(factorial(n))*exp(- (t)^(2)), t = z..infinity) Integrate[I^(n - 1)*Erfc[t], {t, z, Infinity}]=Divide[2,Sqrt[Pi]]*Integrate[Divide[(t - z)^(n),(n)!]*Exp[- (t)^(2)], {t, z, Infinity}] Failure Failure Skip
Fail
Complex[-0.06643066657209085, 0.02648998567028575] <- {Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.036107850584238765, -0.054264273946754926] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.04191638050136022, 0.039897144071178225] <- {Rule[n, 2], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.03610785058423853, 0.054264273946755606] <- {Rule[n, 3], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
7.18.E3 ${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{i}^{% n}\mathrm{erfc}}\left(z\right)=-\mathop{\mathrm{i}^{n-1}\mathrm{erfc}}\left(z% \right)}}$ diff(erfc(n, z), z)= - erfc(n - 1, z) D[I^(n)*Erfc[z], z]= - I^(n - 1)*Erfc[z] Successful Failure -
Fail
Complex[-0.8436484572858769, -1.011483603950918] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.8642718813368542, -0.46363208502383696] <- {Rule[n, 3], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.864271881336854, -0.46363208502383696] <- {Rule[n, 3], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.1563515427141229, -1.011483603950918] <- {Rule[n, 3], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
7.18.E4