Results of Bessel Functions: Difference between revisions

| [https://dlmf.nist.gov/10.2.E1 10.2.E1] || [[Item:Q3005|<math>z^{2}\deriv[2]{w}{z}+z\deriv{w}{z}+(z^{2}-\nu^{2})w = 0</math>]] || <code>(z)^(2)* diff(w, [z$(2)])+ z*diff(w, z)+((z)^(2)- (nu)^(2))* w = 0</code> || <code>(z)^(2)* D[w, {z, 2}]+ z*D[w, z]+((z)^(2)- (\[Nu])^(2))* w = 0</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>11.31370849-11.31370849*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}</code><br><code>11.31370849-11.31370849*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}</code><br><code>-11.31370849-11.31370849*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}</code><br><code>-11.31370849-11.31370849*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}</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[-11.313708498984761, 11.313708498984761] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-11.313708498984761, 11.313708498984761] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[11.313708498984761, -11.313708498984761] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[11.313708498984761, -11.313708498984761] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.6#Ex11 10.6#Ex11] || [[Item:Q3044|<math>p_{\nu} = \BesselJ{\nu}@{a}\BesselY{\nu}@{b}-\BesselJ{\nu}@{b}\BesselY{\nu}@{a}</math>]] || <code>p[nu]= BesselJ(nu, a)*BesselY(nu, b)- BesselJ(nu, b)*BesselY(nu, a)</code> || <code>Subscript[p, \[Nu]]= BesselJ[\[Nu], a]*BesselY[\[Nu], b]- BesselJ[\[Nu], b]*BesselY[\[Nu], a]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>1.414213562+1.414213562*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), p[nu] = 2^(1/2)+I*2^(1/2)}</code><br><code>1.414213562-1.414213562*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), p[nu] = 2^(1/2)-I*2^(1/2)}</code><br><code>-1.414213562-1.414213562*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), p[nu] = -2^(1/2)-I*2^(1/2)}</code><br><code>-1.414213562+1.414213562*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), p[nu] = -2^(1/2)+I*2^(1/2)}</code><br>... skip entries to safe data<br></div></div> || Skip  

| [https://dlmf.nist.gov/10.6#Ex12 10.6#Ex12] || [[Item:Q3045|<math>q_{\nu} = \BesselJ{\nu}@{a}\BesselY{\nu}'@{b}-\BesselJ{\nu}'@{b}\BesselY{\nu}@{a}</math>]] || <code>q[nu]= BesselJ(nu, a)*subs( temp=b, diff( BesselY(nu, temp), temp$(1) ) )- subs( temp=b, diff( BesselJ(nu, temp), temp$(1) ) )*BesselY(nu, a)</code> || <code>Subscript[q, \[Nu]]= BesselJ[\[Nu], a]*(D[BesselY[\[Nu], temp], {temp, 1}]/.temp-> b)- (D[BesselJ[\[Nu], temp], {temp, 1}]/.temp-> b)*BesselY[\[Nu], a]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>1.189134483+1.639292641*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), q[nu] = 2^(1/2)+I*2^(1/2)}</code><br><code>1.189134483-1.189134483*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), q[nu] = 2^(1/2)-I*2^(1/2)}</code><br><code>-1.639292641-1.189134483*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), q[nu] = -2^(1/2)-I*2^(1/2)}</code><br><code>-1.639292641+1.639292641*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), q[nu] = -2^(1/2)+I*2^(1/2)}</code><br>... skip entries to safe data<br></div></div> || Skip  

| [https://dlmf.nist.gov/10.6#Ex13 10.6#Ex13] || [[Item:Q3046|<math>r_{\nu} = \BesselJ{\nu}'@{a}\BesselY{\nu}@{b}-\BesselJ{\nu}@{b}\BesselY{\nu}'@{a}</math>]] || <code>r[nu]= subs( temp=a, diff( BesselJ(nu, temp), temp$(1) ) )*BesselY(nu, b)- BesselJ(nu, b)*subs( temp=a, diff( BesselY(nu, temp), temp$(1) ) )</code> || <code>Subscript[r, \[Nu]]= (D[BesselJ[\[Nu], temp], {temp, 1}]/.temp-> a)*BesselY[\[Nu], b]- BesselJ[\[Nu], b]*(D[BesselY[\[Nu], temp], {temp, 1}]/.temp-> a)</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>1.639292641+1.189134483*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), r[nu] = 2^(1/2)+I*2^(1/2)}</code><br><code>1.639292641-1.639292641*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), r[nu] = 2^(1/2)-I*2^(1/2)}</code><br><code>-1.189134483-1.639292641*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), r[nu] = -2^(1/2)-I*2^(1/2)}</code><br><code>-1.189134483+1.189134483*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), r[nu] = -2^(1/2)+I*2^(1/2)}</code><br>... skip entries to safe data<br></div></div> || Skip  

| [https://dlmf.nist.gov/10.6#Ex14 10.6#Ex14] || [[Item:Q3047|<math>s_{\nu} = \BesselJ{\nu}'@{a}\BesselY{\nu}'@{b}-\BesselJ{\nu}'@{b}\BesselY{\nu}'@{a}</math>]] || <code>s[nu]= subs( temp=a, diff( BesselJ(nu, temp), temp$(1) ) )*subs( temp=b, diff( BesselY(nu, temp), temp$(1) ) )- subs( temp=b, diff( BesselJ(nu, temp), temp$(1) ) )*subs( temp=a, diff( BesselY(nu, temp), temp$(1) ) )</code> || <code>Subscript[s, \[Nu]]= (D[BesselJ[\[Nu], temp], {temp, 1}]/.temp-> a)*(D[BesselY[\[Nu], temp], {temp, 1}]/.temp-> b)- (D[BesselJ[\[Nu], temp], {temp, 1}]/.temp-> b)*(D[BesselY[\[Nu], temp], {temp, 1}]/.temp-> a)</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>1.414213562+1.414213562*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), s[nu] = 2^(1/2)+I*2^(1/2)}</code><br><code>1.414213562-1.414213562*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), s[nu] = 2^(1/2)-I*2^(1/2)}</code><br><code>-1.414213562-1.414213562*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), s[nu] = -2^(1/2)-I*2^(1/2)}</code><br><code>-1.414213562+1.414213562*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), s[nu] = -2^(1/2)+I*2^(1/2)}</code><br>... skip entries to safe data<br></div></div> || Skip  

| [https://dlmf.nist.gov/10.9.E1 10.9.E1] || [[Item:Q3066|<math>\BesselJ{0}@{z} = \frac{1}{\pi}\int_{0}^{\pi}\cos@{z\sin@@{\theta}}\diff{\theta}</math>]] || <code>BesselJ(0, z)=(1)/(Pi)*int(cos(z*sin(theta)), theta = 0..Pi)</code> || <code>BesselJ[0, z]=Divide[1,Pi]*Integrate[Cos[z*Sin[\[Theta]]], {\[Theta], 0, Pi}]</code> || Successful || Failure || - || Successful

| [https://dlmf.nist.gov/10.9.E1 10.9.E1] || [[Item:Q3066|<math>\frac{1}{\pi}\int_{0}^{\pi}\cos@{z\sin@@{\theta}}\diff{\theta} = \frac{1}{\pi}\int_{0}^{\pi}\cos@{z\cos@@{\theta}}\diff{\theta}</math>]] || <code>(1)/(Pi)*int(cos(z*sin(theta)), theta = 0..Pi)=(1)/(Pi)*int(cos(z*cos(theta)), theta = 0..Pi)</code> || <code>Divide[1,Pi]*Integrate[Cos[z*Sin[\[Theta]]], {\[Theta], 0, Pi}]=Divide[1,Pi]*Integrate[Cos[z*Cos[\[Theta]]], {\[Theta], 0, Pi}]</code> || Successful || Failure || - || Successful

| [https://dlmf.nist.gov/10.9.E2 10.9.E2] || [[Item:Q3067|<math>\BesselJ{n}@{z} = \frac{1}{\pi}\int_{0}^{\pi}\cos@{z\sin@@{\theta}-n\theta}\diff{\theta}</math>]] || <code>BesselJ(n, z)=(1)/(Pi)*int(cos(z*sin(theta)- n*theta), theta = 0..Pi)</code> || <code>BesselJ[n, z]=Divide[1,Pi]*Integrate[Cos[z*Sin[\[Theta]]- n*\[Theta]], {\[Theta], 0, Pi}]</code> || Failure || Failure || Skip || Successful

| [https://dlmf.nist.gov/10.9.E2 10.9.E2] || [[Item:Q3067|<math>\frac{1}{\pi}\int_{0}^{\pi}\cos@{z\sin@@{\theta}-n\theta}\diff{\theta} = \frac{i^{-n}}{\pi}\int_{0}^{\pi}e^{iz\cos@@{\theta}}\cos@{n\theta}\diff{\theta}</math>]] || <code>(1)/(Pi)*int(cos(z*sin(theta)- n*theta), theta = 0..Pi)=((I)^(- n))/(Pi)*int(exp(I*z*cos(theta))*cos(n*theta), theta = 0..Pi)</code> || <code>Divide[1,Pi]*Integrate[Cos[z*Sin[\[Theta]]- n*\[Theta]], {\[Theta], 0, Pi}]=Divide[(I)^(- n),Pi]*Integrate[Exp[I*z*Cos[\[Theta]]]*Cos[n*\[Theta]], {\[Theta], 0, Pi}]</code> || Failure || Failure || Skip || Successful

| [https://dlmf.nist.gov/10.9.E3 10.9.E3] || [[Item:Q3068|<math>\BesselY{0}@{z} = \frac{4}{\pi^{2}}\int_{0}^{\frac{1}{2}\pi}\cos@{z\cos@@{\theta}}\left(\EulerConstant+\ln@{2z\sin^{2}@@{\theta}}\right)\diff{\theta}</math>]] || <code>BesselY(0, z)=(4)/((Pi)^(2))*int(cos(z*cos(theta))*(gamma + ln(2*z*(sin(theta))^(2))), theta = 0..(1)/(2)*Pi)</code> || <code>BesselY[0, z]=Divide[4,(Pi)^(2)]*Integrate[Cos[z*Cos[\[Theta]]]*(EulerGamma + Log[2*z*(Sin[\[Theta]])^(2)]), {\[Theta], 0, Divide[1,2]*Pi}]</code> || Failure || Failure || Skip || Successful

| [https://dlmf.nist.gov/10.9.E4 10.9.E4] || [[Item:Q3069|<math>\BesselJ{\nu}@{z} = \frac{(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\EulerGamma@{\nu+\tfrac{1}{2}}}\int_{0}^{\pi}\cos@{z\cos@@{\theta}}(\sin@@{\theta})^{2\nu}\diff{\theta}</math>]] || <code>BesselJ(nu, z)=(((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*int(cos(z*cos(theta))*(sin(theta))^(2*nu), theta = 0..Pi)</code> || <code>BesselJ[\[Nu], z]=Divide[(Divide[1,2]*z)^(\[Nu]),(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*Integrate[Cos[z*Cos[\[Theta]]]*(Sin[\[Theta]])^(2*\[Nu]), {\[Theta], 0, Pi}]</code> || Failure || Failure || Skip || Successful

| [https://dlmf.nist.gov/10.9.E4 10.9.E4] || [[Item:Q3069|<math>\frac{(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\EulerGamma@{\nu+\tfrac{1}{2}}}\int_{0}^{\pi}\cos@{z\cos@@{\theta}}(\sin@@{\theta})^{2\nu}\diff{\theta} = \frac{2(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\EulerGamma@{\nu+\tfrac{1}{2}}}\int_{0}^{1}(1-t^{2})^{\nu-\frac{1}{2}}\cos@{zt}\diff{t}</math>]] || <code>(((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*int(cos(z*cos(theta))*(sin(theta))^(2*nu), theta = 0..Pi)=(2*((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*int((1 - (t)^(2))^(nu -(1)/(2))* cos(z*t), t = 0..1)</code> || <code>Divide[(Divide[1,2]*z)^(\[Nu]),(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*Integrate[Cos[z*Cos[\[Theta]]]*(Sin[\[Theta]])^(2*\[Nu]), {\[Theta], 0, Pi}]=Divide[2*(Divide[1,2]*z)^(\[Nu]),(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*Integrate[(1 - (t)^(2))^(\[Nu]-Divide[1,2])* Cos[z*t], {t, 0, 1}]</code> || Failure || Failure || Skip || Skip

| [https://dlmf.nist.gov/10.9.E5 10.9.E5] || [[Item:Q3070|<math>\BesselY{\nu}@{z} = \frac{2(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\EulerGamma@{\nu+\tfrac{1}{2}}}\left(\int_{0}^{1}(1-t^{2})^{\nu-\frac{1}{2}}\sin@{zt}\diff{t}-\int_{0}^{\infty}e^{-zt}(1+t^{2})^{\nu-\frac{1}{2}}\diff{t}\right)</math>]] || <code>BesselY(nu, z)=(2*((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*(int((1 - (t)^(2))^(nu -(1)/(2))* sin(z*t), t = 0..1)- int(exp(- z*t)*(1 + (t)^(2))^(nu -(1)/(2)), t = 0..infinity))</code> || <code>BesselY[\[Nu], z]=Divide[2*(Divide[1,2]*z)^(\[Nu]),(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*(Integrate[(1 - (t)^(2))^(\[Nu]-Divide[1,2])* Sin[z*t], {t, 0, 1}]- Integrate[Exp[- z*t]*(1 + (t)^(2))^(\[Nu]-Divide[1,2]), {t, 0, Infinity}])</code> || Successful || Failure || - || Skip

| [https://dlmf.nist.gov/10.9#Ex5 10.9#Ex5] || [[Item:Q3079|<math>\BesselJ{\nu}@{x} = \frac{2(\tfrac{1}{2}x)^{-\nu}}{\pi^{\frac{1}{2}}\EulerGamma@{\tfrac{1}{2}-\nu}}\int_{1}^{\infty}\frac{\sin@{xt}\diff{t}}{(t^{2}-1)^{\nu+\frac{1}{2}}}</math>]] || <code>BesselJ(nu, x)=(2*((1)/(2)*x)^(- nu))/((Pi)^((1)/(2))* GAMMA((1)/(2)- nu))*int((sin(x*t))/(((t)^(2)- 1)^(nu +(1)/(2))), t = 1..infinity)</code> || <code>BesselJ[\[Nu], x]=Divide[2*(Divide[1,2]*x)^(- \[Nu]),(Pi)^(Divide[1,2])* Gamma[Divide[1,2]- \[Nu]]]*Integrate[Divide[Sin[x*t],((t)^(2)- 1)^(\[Nu]+Divide[1,2])], {t, 1, Infinity}]</code> || Successful || Failure || - || Successful

| [https://dlmf.nist.gov/10.9.E17 10.9.E17] || [[Item:Q3085|<math>\BesselJ{\nu}@{z} = \frac{1}{2\pi i}\int_{\infty-\pi i}^{\infty+\pi i}e^{z\sinh@@{t}-\nu t}\diff{t}</math>]] || <code>BesselJ(nu, z)=(1)/(2*Pi*I)*int(exp(z*sinh(t)- nu*t), t = infinity - Pi*I..infinity + Pi*I)</code> || <code>BesselJ[\[Nu], z]=Divide[1,2*Pi*I]*Integrate[Exp[z*Sinh[t]- \[Nu]*t], {t, Infinity - Pi*I, Infinity + Pi*I}]</code> || Failure || Failure || Skip || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>Complex[0.342503927390088, -0.08973210023585859] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.3263028372306598, 4.480608248698951] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-26.41355287980499, 14.935276359740396] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-0.2663767645899945, 0.9702347233898156] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.9.E23 10.9.E23] || [[Item:Q3093|<math>\BesselJ{\nu}@{z} = \frac{1}{2\pi i}\int_{-\infty-ic}^{-\infty+ic}\frac{\EulerGamma@{t}}{\EulerGamma@{\nu-t+1}}(\tfrac{1}{2}z)^{\nu-2t}\diff{t}</math>]] || <code>BesselJ(nu, z)=(1)/(2*Pi*I)*int((GAMMA(t))/(GAMMA(nu - t + 1))*((1)/(2)*z)^(nu - 2*t), t = - infinity - I*c..- infinity + I*c)</code> || <code>BesselJ[\[Nu], z]=Divide[1,2*Pi*I]*Integrate[Divide[Gamma[t],Gamma[\[Nu]- t + 1]]*(Divide[1,2]*z)^(\[Nu]- 2*t), {t, - Infinity - I*c, - Infinity + I*c}]</code> || Failure || Failure || Skip || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>Complex[0.342503927390088, -0.08973210023585859] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.3263028372306598, 4.480608248698951] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-26.41355287980499, 14.935276359740396] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-0.2663767645899945, 0.9702347233898156] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.11.E1 10.11.E1] || [[Item:Q3103|<math>\BesselJ{\nu}@{ze^{m\pi i}} = e^{m\nu\pi i}\BesselJ{\nu}@{z}</math>]] || <code>BesselJ(nu, z*exp(m*Pi*I))= exp(m*nu*Pi*I)*BesselJ(nu, z)</code> || <code>BesselJ[\[Nu], z*Exp[m*Pi*I]]= Exp[m*\[Nu]*Pi*I]*BesselJ[\[Nu], z]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>-.3975453294+30.10329939*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 1}</code><br><code>.3425382206-.8976707513e-1*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 2}</code><br><code>-.3996358010+30.09969700*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3}</code><br><code>-1.326778468-4.481046040*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 2}</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[-0.3975452718986143, 30.10329943602099] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[0.34253822069590145, -0.08976707542141499] <- {Rule[m, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-0.3996357209647074, 30.09969706489566] <- {Rule[m, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-397.25907376207414, -7.293217978872368] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.11.E2 10.11.E2] || [[Item:Q3104|<math>\BesselY{\nu}@{ze^{m\pi i}} = e^{-m\nu\pi i}\BesselY{\nu}@{z}+2i\sin@{m\nu\pi}\cot@{\nu\pi}\BesselJ{\nu}@{z}</math>]] || <code>BesselY(nu, z*exp(m*Pi*I))= exp(- m*nu*Pi*I)*BesselY(nu, z)+ 2*I*sin(m*nu*Pi)*cot(nu*Pi)*BesselJ(nu, z)</code> || <code>BesselY[\[Nu], z*Exp[m*Pi*I]]= Exp[- m*\[Nu]*Pi*I]*BesselY[\[Nu], z]+ 2*I*Sin[m*\[Nu]*Pi]*Cot[\[Nu]*Pi]*BesselJ[\[Nu], z]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>59.96664792+53.22883098*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 1}</code><br><code>-5005.486114+1251.725768*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 2}</code><br><code>10758.9952-438485.5093*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3}</code><br><code>-4.21339-169.756927*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 2}</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[59.966647971782265, 53.22883092179323] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-5005.486119861246, 1251.7257744468322] <- {Rule[m, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[10758.99323773, -438485.5113105696] <- {Rule[m, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[9.175550013151128, -396.6330304507175] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.11.E3 10.11.E3] || [[Item:Q3105|<math>\sin@{\nu\pi}\HankelH{1}{\nu}@{ze^{m\pi i}} = -\sin@{(m-1)\nu\pi}\HankelH{1}{\nu}@{z}-e^{-\nu\pi i}\sin@{m\nu\pi}\HankelH{2}{\nu}@{z}</math>]] || <code>sin(nu*Pi)*HankelH1(nu, z*exp(m*Pi*I))= - sin((m - 1)* nu*Pi)*HankelH1(nu, z)- exp(- nu*Pi*I)*sin(m*nu*Pi)*HankelH2(nu, z)</code> || <code>Sin[\[Nu]*Pi]*HankelH1[\[Nu], z*Exp[m*Pi*I]]= - Sin[(m - 1)* \[Nu]*Pi]*HankelH1[\[Nu], z]- Exp[- \[Nu]*Pi*I]*Sin[m*\[Nu]*Pi]*HankelH2[\[Nu], z]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>3216.976842-3084.273397*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 1}</code><br><code>-5364.683403+219295.3867*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 2}</code><br><code>-17847467.19-5404443.822*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3}</code><br><code>-7000.832672-1549.801603*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 2}</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[3216.976837863537, -3084.273404768022] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-5364.6831188620945, 219295.38712307377] <- {Rule[m, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.7847467293085534*^7, -5404443.760123314] <- {Rule[m, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[4.353713736263555, -84.22475855786759] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.11.E4 10.11.E4] || [[Item:Q3106|<math>\sin@{\nu\pi}\HankelH{2}{\nu}@{ze^{m\pi i}} = e^{\nu\pi i}\sin@{m\nu\pi}\HankelH{1}{\nu}@{z}+\sin@{(m+1)\nu\pi}\HankelH{2}{\nu}@{z}</math>]] || <code>sin(nu*Pi)*HankelH2(nu, z*exp(m*Pi*I))= exp(nu*Pi*I)*sin(m*nu*Pi)*HankelH1(nu, z)+ sin((m + 1)* nu*Pi)*HankelH2(nu, z)</code> || <code>Sin[\[Nu]*Pi]*HankelH2[\[Nu], z*Exp[m*Pi*I]]= Exp[\[Nu]*Pi*I]*Sin[m*\[Nu]*Pi]*HankelH1[\[Nu], z]+ Sin[(m + 1)* \[Nu]*Pi]*HankelH2[\[Nu], z]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>-2503.040664+625.9436263*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 1}</code><br><code>5334.577216-219295.7816*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 2}</code><br><code>17848181.49+5401985.686*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3}</code><br><code>7008.154936+1947.107340*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 2}</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[-2503.040666874715, 625.9436301275297] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[5334.576217054554, -219295.782577897] <- {Rule[m, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.7848181319058534*^7, 5401985.77292121] <- {Rule[m, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[32720.882533131233, -8309.4971554526] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.11#Ex1 10.11#Ex1] || [[Item:Q3107|<math>\HankelH{1}{\nu}@{ze^{\pi i}} = -e^{-\nu\pi i}\HankelH{2}{\nu}@{z}</math>]] || <code>HankelH1(nu, z*exp(Pi*I))= - exp(- nu*Pi*I)*HankelH2(nu, z)</code> || <code>HankelH1[\[Nu], z*Exp[Pi*I]]= - Exp[- \[Nu]*Pi*I]*HankelH2[\[Nu], z]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>-53.62637626+90.06994733*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}</code><br><code>168.4301368-8.694434719*I <- {nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}</code><br><code>-.6260433097+1.882332034*I <- {nu = 2^(1/2)-I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}</code><br><code>1.189100470-.3259126629*I <- {nu = 2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}</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[-53.62637619369183, 90.06994740780324] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-0.6260433113566327, 1.8823320342787686] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.1891004703150516, 0.325912661920741] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[168.43013693288603, 8.694434886635074] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.11#Ex2 10.11#Ex2] || [[Item:Q3108|<math>\HankelH{2}{\nu}@{ze^{-\pi i}} = -e^{\nu\pi i}\HankelH{1}{\nu}@{z}</math>]] || <code>HankelH2(nu, z*exp(- Pi*I))= - exp(nu*Pi*I)*HankelH1(nu, z)</code> || <code>HankelH2[\[Nu], z*Exp[- Pi*I]]= - Exp[\[Nu]*Pi*I]*HankelH1[\[Nu], z]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>-.6260433097-1.882332034*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}</code><br><code>1.189100470+.3259126629*I <- {nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}</code><br><code>-53.62637626-90.06994733*I <- {nu = 2^(1/2)-I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}</code><br><code>168.4301368+8.694434719*I <- {nu = 2^(1/2)-I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}</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[-0.6260433113566327, -1.8823320342787686] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-53.62637619369183, -90.06994740780324] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[168.43013693288603, -8.694434886635074] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.1891004703150516, -0.325912661920741] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.11.E6 10.11.E6] || [[Item:Q3109|<math>\BesselY{n}@{ze^{m\pi i}} = (-1)^{mn}(\BesselY{n}@{z}+2im\BesselJ{n}@{z})</math>]] || <code>BesselY(n, z*exp(m*Pi*I))=(- 1)^(m*n)*(BesselY(n, z)+ 2*I*m*BesselJ(n, z))</code> || <code>BesselY[n, z*Exp[m*Pi*I]]=(- 1)^(m*n)*(BesselY[n, z]+ 2*I*m*BesselJ[n, z])</code> || Failure || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>Complex[-1.199101748008134, 3.9883106077057144] <- {Rule[m, 1], Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.9168980103888886, -0.6611177226809124] <- {Rule[m, 1], Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-0.5768397662292734, -0.34244579398718544] <- {Rule[m, 1], Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.199101748008134, -3.9883106077057144] <- {Rule[m, 2], Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.11.E7 10.11.E7] || [[Item:Q3110|<math>\HankelH{1}{n}@{ze^{m\pi i}} = (-1)^{mn-1}((m-1)\HankelH{1}{n}@{z}+m\HankelH{2}{n}@{z})</math>]] || <code>HankelH1(n, z*exp(m*Pi*I))=(- 1)^(m*n - 1)*((m - 1)*HankelH1(n, z)+ m*HankelH2(n, z))</code> || <code>HankelH1[n, z*Exp[m*Pi*I]]=(- 1)^(m*n - 1)*((m - 1)*HankelH1[n, z]+ m*HankelH2[n, z])</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>-3.988310607-1.199101751*I <- {z = 2^(1/2)+I*2^(1/2), m = 1, n = 1}</code><br><code>.6611177206+1.916898011*I <- {z = 2^(1/2)+I*2^(1/2), m = 1, n = 2}</code><br><code>.3424457937-.5768397666*I <- {z = 2^(1/2)+I*2^(1/2), m = 1, n = 3}</code><br><code>3.988310606+1.199101748*I <- {z = 2^(1/2)+I*2^(1/2), m = 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[-3.988310607705715, -1.1991017480081343] <- {Rule[m, 1], Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[0.6611177226809126, 1.9168980103888886] <- {Rule[m, 1], Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[0.34244579398718544, -0.5768397662292732] <- {Rule[m, 1], Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[3.988310607705715, 1.1991017480081343] <- {Rule[m, 2], Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.11.E8 10.11.E8] || [[Item:Q3111|<math>\HankelH{2}{n}@{ze^{m\pi i}} = (-1)^{mn}(m\HankelH{1}{n}@{z}+(m+1)\HankelH{2}{n}@{z})</math>]] || <code>HankelH2(n, z*exp(m*Pi*I))=(- 1)^(m*n)*(m*HankelH1(n, z)+(m + 1)*HankelH2(n, z))</code> || <code>HankelH2[n, z*Exp[m*Pi*I]]=(- 1)^(m*n)*(m*HankelH1[n, z]+(m + 1)*HankelH2[n, z])</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>3.988310606+1.199101748*I <- {z = 2^(1/2)+I*2^(1/2), m = 1, n = 1}</code><br><code>-.6611177221-1.916898010*I <- {z = 2^(1/2)+I*2^(1/2), m = 1, n = 2}</code><br><code>-.3424457926+.5768397669*I <- {z = 2^(1/2)+I*2^(1/2), m = 1, n = 3}</code><br><code>-3.988310606-1.199101746*I <- {z = 2^(1/2)+I*2^(1/2), m = 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[3.988310607705715, 1.1991017480081343] <- {Rule[m, 1], Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-0.6611177226809126, -1.9168980103888886] <- {Rule[m, 1], Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-0.34244579398718566, 0.5768397662292732] <- {Rule[m, 1], Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-3.988310607705715, -1.1991017480081343] <- {Rule[m, 2], Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.14#Ex1 10.14#Ex1] || [[Item:Q3137|<math>|\BesselJ{\nu}@{x}| <= 1</math>]] || <code>abs(BesselJ(nu, x))< = 1</code> || <code>Abs[BesselJ[\[Nu], x]]< = 1</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>1.148999867 <= 1. <- {nu = 2^(1/2)+I*2^(1/2), x = 3}</code><br><code>1.148999867 <= 1. <- {nu = 2^(1/2)-I*2^(1/2), x = 3}</code><br><code>14.70966635 <= 1. <- {nu = -2^(1/2)-I*2^(1/2), x = 1}</code><br><code>6.163423173 <= 1. <- {nu = -2^(1/2)-I*2^(1/2), x = 2}</code><br>... skip entries to safe data<br></div></div> || Successful  

| [https://dlmf.nist.gov/10.14#Ex2 10.14#Ex2] || [[Item:Q3138|<math>|\BesselJ{\nu}@{x}| <= 2^{-\frac{1}{2}}</math>]] || <code>abs(BesselJ(nu, x))< = (2)^(-(1)/(2))</code> || <code>Abs[BesselJ[\[Nu], x]]< = (2)^(-Divide[1,2])</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>.9422017731 <= .7071067810 <- {nu = 2^(1/2)+I*2^(1/2), x = 2}</code><br><code>1.148999867 <= .7071067810 <- {nu = 2^(1/2)+I*2^(1/2), x = 3}</code><br><code>.9422017731 <= .7071067810 <- {nu = 2^(1/2)-I*2^(1/2), x = 2}</code><br><code>1.148999867 <= .7071067810 <- {nu = 2^(1/2)-I*2^(1/2), x = 3}</code><br>... skip entries to safe data<br></div></div> || Successful  

| [https://dlmf.nist.gov/10.14.E9 10.14.E9] || [[Item:Q3146|<math>|\BesselJ{n}@{nz}| <= 1</math>]] || <code>abs(BesselJ(n, n*z))< = 1</code> || <code>Abs[BesselJ[n, n*z]]< = 1</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>1.041167208 <= 1. <- {z = 2^(1/2)+I*2^(1/2), n = 1}</code><br><code>2.428697298 <= 1. <- {z = 2^(1/2)+I*2^(1/2), n = 2}</code><br><code>6.705297847 <= 1. <- {z = 2^(1/2)+I*2^(1/2), n = 3}</code><br><code>1.041167208 <= 1. <- {z = 2^(1/2)-I*2^(1/2), n = 1}</code><br>... skip entries to safe data<br></div></div> || Successful  

| [https://dlmf.nist.gov/10.15.E1 10.15.E1] || [[Item:Q3147|<math>\pderiv{\BesselJ{+\nu}@{z}}{\nu} = +\BesselJ{+\nu}@{z}\ln@{\tfrac{1}{2}z}-(\tfrac{1}{2}z)^{+\nu}\sum_{k=0}^{\infty}(-1)^{k}\frac{\digamma@{k+1+\nu}}{\EulerGamma@{k+1+\nu}}\frac{(\tfrac{1}{4}z^{2})^{k}}{k!}</math>]] || <code>diff(BesselJ(+ nu, z), nu)= + BesselJ(+ nu, z)*ln((1)/(2)*z)-((1)/(2)*z)^(+ nu)* sum((- 1)^(k)*(Psi(k + 1 + nu))/(GAMMA(k + 1 + nu))*(((1)/(4)*(z)^(2))^(k))/(factorial(k)), k = 0..infinity)</code> || <code>D[BesselJ[+ \[Nu], z], \[Nu]]= + BesselJ[+ \[Nu], z]*Log[Divide[1,2]*z]-(Divide[1,2]*z)^(+ \[Nu])* Sum[(- 1)^(k)*Divide[PolyGamma[k + 1 + \[Nu]],Gamma[k + 1 + \[Nu]]]*Divide[(Divide[1,4]*(z)^(2))^(k),(k)!], {k, 0, Infinity}]</code> || Failure || Failure || Skip || Successful

| [https://dlmf.nist.gov/10.20.E1 10.20.E1] || [[Item:Q3250|<math>\left(\deriv{\zeta}{z}\right)^{2} = \frac{1-z^{2}}{\zeta z^{2}}</math>]] || <code>(diff(zeta, z))^(2)=(1 - (z)^(2))/(zeta*(z)^(2))</code> || <code>(D[\[zeta], z])^(2)=Divide[1 - (z)^(2),\[zeta]*(z)^(2)]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>.4419417384-.2651650430*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2)}</code><br><code>.2651650430+.4419417384*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)-I*2^(1/2)}</code><br><code>-.4419417384+.2651650430*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)-I*2^(1/2)}</code><br><code>-.2651650430-.4419417384*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)+I*2^(1/2)}</code><br>... skip entries to safe data<br></div></div> || Error  

| [https://dlmf.nist.gov/10.22.E10 10.22.E10] || [[Item:Q3384|<math>\int_{0}^{x}t^{\mu}\BesselJ{\nu}@{t}\diff{t} = x^{\mu}\frac{\EulerGamma@{\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}}}{\EulerGamma@{\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}}}\*\sum_{k=0}^{\infty}\frac{(\nu+2k+1)\EulerGamma@{\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}+k}}{\EulerGamma@{\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{2}+k}}\BesselJ{\nu+2k+1}@{x}</math>]] || <code>int((t)^(mu)* BesselJ(nu, t), t = 0..x)= (x)^(mu)*(GAMMA((1)/(2)*nu +(1)/(2)*mu +(1)/(2)))/(GAMMA((1)/(2)*nu -(1)/(2)*mu +(1)/(2)))* sum(((nu + 2*k + 1)* GAMMA((1)/(2)*nu -(1)/(2)*mu +(1)/(2)+ k))/(GAMMA((1)/(2)*nu +(1)/(2)*mu +(3)/(2)+ k))*BesselJ(nu + 2*k + 1, x), k = 0..infinity)</code> || <code>Integrate[(t)^(\[Mu])* BesselJ[\[Nu], t], {t, 0, x}]= (x)^(\[Mu])*Divide[Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[1,2]],Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+Divide[1,2]]]* Sum[Divide[(\[Nu]+ 2*k + 1)* Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+Divide[1,2]+ k],Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[3,2]+ k]]*BesselJ[\[Nu]+ 2*k + 1, x], {k, 0, Infinity}]</code> || Failure || Failure || Skip || Successful

| [https://dlmf.nist.gov/10.22.E11 10.22.E11] || [[Item:Q3385|<math>\int_{0}^{x}\frac{1-\BesselJ{0}@{t}}{t}\diff{t} = \frac{1}{2}\sum_{k=1}^{\infty}\frac{\digamma@{k+1}-\digamma@{1}}{k!}(\tfrac{1}{2}x)^{k}\BesselJ{k}@{x}</math>]] || <code>int((1 - BesselJ(0, t))/(t), t = 0..x)=(1)/(2)*sum((Psi(k + 1)- Psi(1))/(factorial(k))*((1)/(2)*x)^(k)* BesselJ(k, x), k = 1..infinity)</code> || <code>Integrate[Divide[1 - BesselJ[0, t],t], {t, 0, x}]=Divide[1,2]*Sum[Divide[PolyGamma[k + 1]- PolyGamma[1],(k)!]*(Divide[1,2]*x)^(k)* BesselJ[k, x], {k, 1, Infinity}]</code> || Failure || Failure || Skip || Skip

| [https://dlmf.nist.gov/10.22.E14 10.22.E14] || [[Item:Q3388|<math>\int_{0}^{\pi}\BesselJ{2\nu}@{2z\sin@@{\theta}}\cos@{2\mu\theta}\diff{\theta} = \pi\cos@{\mu\pi}\BesselJ{\nu+\mu}@{z}\BesselJ{\nu-\mu}@{z}</math>]] || <code>int(BesselJ(2*nu, 2*z*sin(theta))*cos(2*mu*theta), theta = 0..Pi)= Pi*cos(mu*Pi)*BesselJ(nu + mu, z)*BesselJ(nu - mu, z)</code> || <code>Integrate[BesselJ[2*\[Nu], 2*z*Sin[\[Theta]]]*Cos[2*\[Mu]*\[Theta]], {\[Theta], 0, Pi}]= Pi*Cos[\[Mu]*Pi]*BesselJ[\[Nu]+ \[Mu], z]*BesselJ[\[Nu]- \[Mu], z]</code> || Failure || Failure || Skip || Skip

| [https://dlmf.nist.gov/10.22.E15 10.22.E15] || [[Item:Q3389|<math>\int_{0}^{\pi}\BesselJ{2\nu}@{2z\sin@@{\theta}}\sin@{2\mu\theta}\diff{\theta} = \pi\sin@{\mu\pi}\BesselJ{\nu+\mu}@{z}\BesselJ{\nu-\mu}@{z}</math>]] || <code>int(BesselJ(2*nu, 2*z*sin(theta))*sin(2*mu*theta), theta = 0..Pi)= Pi*sin(mu*Pi)*BesselJ(nu + mu, z)*BesselJ(nu - mu, z)</code> || <code>Integrate[BesselJ[2*\[Nu], 2*z*Sin[\[Theta]]]*Sin[2*\[Mu]*\[Theta]], {\[Theta], 0, Pi}]= Pi*Sin[\[Mu]*Pi]*BesselJ[\[Nu]+ \[Mu], z]*BesselJ[\[Nu]- \[Mu], z]</code> || Failure || Failure || Skip || Skip

| [https://dlmf.nist.gov/10.22.E17 10.22.E17] || [[Item:Q3391|<math>\int_{0}^{\frac{1}{2}\pi}\BesselY{2\nu}@{2z\cos@@{\theta}}\cos@{2\mu\theta}\diff{\theta} = \tfrac{1}{2}\pi\cot@{2\nu\pi}\BesselJ{\nu+\mu}@{z}\BesselJ{\nu-\mu}@{z}-\tfrac{1}{2}\pi\csc@{2\nu\pi}\BesselJ{\mu-\nu}@{z}\BesselJ{-\mu-\nu}@{z}</math>]] || <code>int(BesselY(2*nu, 2*z*cos(theta))*cos(2*mu*theta), theta = 0..(1)/(2)*Pi)=(1)/(2)*Pi*cot(2*nu*Pi)*BesselJ(nu + mu, z)*BesselJ(nu - mu, z)-(1)/(2)*Pi*csc(2*nu*Pi)*BesselJ(mu - nu, z)*BesselJ(- mu - nu, z)</code> || <code>Integrate[BesselY[2*\[Nu], 2*z*Cos[\[Theta]]]*Cos[2*\[Mu]*\[Theta]], {\[Theta], 0, Divide[1,2]*Pi}]=Divide[1,2]*Pi*Cot[2*\[Nu]*Pi]*BesselJ[\[Nu]+ \[Mu], z]*BesselJ[\[Nu]- \[Mu], z]-Divide[1,2]*Pi*Csc[2*\[Nu]*Pi]*BesselJ[\[Mu]- \[Nu], z]*BesselJ[- \[Mu]- \[Nu], z]</code> || Failure || Failure || Skip || Successful

| [https://dlmf.nist.gov/10.22.E18 10.22.E18] || [[Item:Q3392|<math>\int_{0}^{\frac{1}{2}\pi}\BesselY{0}@{2z\sin@@{\theta}}\cos@{2n\theta}\diff{\theta} = \tfrac{1}{2}\pi\BesselJ{n}@{z}\BesselY{n}@{z}</math>]] || <code>int(BesselY(0, 2*z*sin(theta))*cos(2*n*theta), theta = 0..(1)/(2)*Pi)=(1)/(2)*Pi*BesselJ(n, z)*BesselY(n, z)</code> || <code>Integrate[BesselY[0, 2*z*Sin[\[Theta]]]*Cos[2*n*\[Theta]], {\[Theta], 0, Divide[1,2]*Pi}]=Divide[1,2]*Pi*BesselJ[n, z]*BesselY[n, z]</code> || Failure || Failure || Skip || Successful

| [https://dlmf.nist.gov/10.22.E27 10.22.E27] || [[Item:Q3401|<math>\int_{0}^{x}t\BesselJ{\nu-1}^{2}@{t}\diff{t} = 2\sum_{k=0}^{\infty}(\nu+2k)\BesselJ{\nu+2k}^{2}@{x}</math>]] || <code>int(t*(BesselJ(nu - 1, t))^(2), t = 0..x)= 2*sum((nu + 2*k)* (BesselJ(nu + 2*k, x))^(2), k = 0..infinity)</code> || <code>Integrate[t*(BesselJ[\[Nu]- 1, t])^(2), {t, 0, x}]= 2*Sum[(\[Nu]+ 2*k)* (BesselJ[\[Nu]+ 2*k, x])^(2), {k, 0, Infinity}]</code> || Failure || Failure || Skip || Successful

| [https://dlmf.nist.gov/10.22.E28 10.22.E28] || [[Item:Q3402|<math>\int_{0}^{x}t\left(\BesselJ{\nu-1}^{2}@{t}-\BesselJ{\nu+1}^{2}@{t}\right)\diff{t} = 2\nu\BesselJ{\nu}^{2}@{x}</math>]] || <code>int(t*((BesselJ(nu - 1, t))^(2)- (BesselJ(nu + 1, t))^(2)), t = 0..x)= 2*nu*(BesselJ(nu, x))^(2)</code> || <code>Integrate[t*((BesselJ[\[Nu]- 1, t])^(2)- (BesselJ[\[Nu]+ 1, t])^(2)), {t, 0, x}]= 2*\[Nu]*(BesselJ[\[Nu], x])^(2)</code> || Successful || Failure || - || Skip

| [https://dlmf.nist.gov/10.22.E30 10.22.E30] || [[Item:Q3404|<math>\int_{0}^{x}\BesselJ{n}@{t}\BesselJ{n+1}@{t}\diff{t} = \tfrac{1}{2}\left(1-\BesselJ{0}^{2}@{x}\right)-\sum_{k=1}^{n}\BesselJ{k}^{2}@{x}</math>]] || <code>int(BesselJ(n, t)*BesselJ(n + 1, t), t = 0..x)=(1)/(2)*(1 - (BesselJ(0, x))^(2))- sum((BesselJ(k, x))^(2), k = 1..n)</code> || <code>Integrate[BesselJ[n, t]*BesselJ[n + 1, t], {t, 0, x}]=Divide[1,2]*(1 - (BesselJ[0, x])^(2))- Sum[(BesselJ[k, x])^(2), {k, 1, n}]</code> || Failure || Failure || Skip || Successful

| [https://dlmf.nist.gov/10.22.E30 10.22.E30] || [[Item:Q3404|<math>\tfrac{1}{2}\left(1-\BesselJ{0}^{2}@{x}\right)-\sum_{k=1}^{n}\BesselJ{k}^{2}@{x} = \sum_{k=n+1}^{\infty}\BesselJ{k}^{2}@{x}</math>]] || <code>(1)/(2)*(1 - (BesselJ(0, x))^(2))- sum((BesselJ(k, x))^(2), k = 1..n)= sum((BesselJ(k, x))^(2), k = n + 1..infinity)</code> || <code>Divide[1,2]*(1 - (BesselJ[0, x])^(2))- Sum[(BesselJ[k, x])^(2), {k, 1, n}]= Sum[(BesselJ[k, x])^(2), {k, n + 1, Infinity}]</code> || Failure || Failure || Skip || Successful

| [https://dlmf.nist.gov/10.22.E31 10.22.E31] || [[Item:Q3405|<math>\int_{0}^{x}\BesselJ{\mu}@{t}\BesselJ{\nu}@{x-t}\diff{t} = 2\sum_{k=0}^{\infty}(-1)^{k}\BesselJ{\mu+\nu+2k+1}@{x}</math>]] || <code>int(BesselJ(mu, t)*BesselJ(nu, x - t), t = 0..x)= 2*sum((- 1)^(k)* BesselJ(mu + nu + 2*k + 1, x), k = 0..infinity)</code> || <code>Integrate[BesselJ[\[Mu], t]*BesselJ[\[Nu], x - t], {t, 0, x}]= 2*Sum[(- 1)^(k)* BesselJ[\[Mu]+ \[Nu]+ 2*k + 1, x], {k, 0, Infinity}]</code> || Failure || Failure || Skip || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[μ, Rational[-1, 2]], Rule[ν, Rational[-1, 2]], Rule[Integrate[Times[BesselJ[μ, t], BesselJ[ν, Plus[Times[-1, t], x]]], {t, 0, x}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], BesselJ[Plus[1, Times[2, k], μ, ν], x]], {k, 0, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.4142135623730951, 4.242640687119286] <- {Rule[μ, Rational[-1, 2]], Rule[ν, Rational[-1, 2]], Rule[Integrate[Times[BesselJ[μ, t], BesselJ[ν, Plus[Times[-1, t], x]]], {t, 0, x}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], BesselJ[Plus[1, Times[2, k], μ, ν], x]], {k, 0, DirectedInfinity[1]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[4.242640687119286, 4.242640687119286] <- {Rule[μ, Rational[-1, 2]], Rule[ν, Rational[-1, 2]], Rule[Integrate[Times[BesselJ[μ, t], BesselJ[ν, Plus[Times[-1, t], x]]], {t, 0, x}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], BesselJ[Plus[1, Times[2, k], μ, ν], x]], {k, 0, DirectedInfinity[1]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[4.242640687119286, -1.4142135623730951] <- {Rule[μ, Rational[-1, 2]], Rule[ν, Rational[-1, 2]], Rule[Integrate[Times[BesselJ[μ, t], BesselJ[ν, Plus[Times[-1, t], x]]], {t, 0, x}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], BesselJ[Plus[1, Times[2, k], μ, ν], x]], {k, 0, DirectedInfinity[1]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>

| [https://dlmf.nist.gov/10.22.E32 10.22.E32] || [[Item:Q3406|<math>\int_{0}^{x}\BesselJ{\nu}@{t}\BesselJ{1-\nu}@{x-t}\diff{t} = \BesselJ{0}@{x}-\cos@@{x}</math>]] || <code>int(BesselJ(nu, t)*BesselJ(1 - nu, x - t), t = 0..x)= BesselJ(0, x)- cos(x)</code> || <code>Integrate[BesselJ[\[Nu], t]*BesselJ[1 - \[Nu], x - t], {t, 0, x}]= BesselJ[0, x]- Cos[x]</code> || Failure || Failure || Skip || Successful

| [https://dlmf.nist.gov/10.22.E33 10.22.E33] || [[Item:Q3407|<math>\int_{0}^{x}\BesselJ{\nu}@{t}\BesselJ{-\nu}@{x-t}\diff{t} = \sin@@{x}</math>]] || <code>int(BesselJ(nu, t)*BesselJ(- nu, x - t), t = 0..x)= sin(x)</code> || <code>Integrate[BesselJ[\[Nu], t]*BesselJ[- \[Nu], x - t], {t, 0, x}]= Sin[x]</code> || Failure || Failure || Skip || Error

| [https://dlmf.nist.gov/10.22.E36 10.22.E36] || [[Item:Q3410|<math>\frac{1}{\EulerGamma@{\alpha}}\int_{0}^{x}(x-t)^{\alpha-1}\BesselJ{\nu}@{t}\diff{t} = 2^{\alpha}\sum_{k=0}^{\infty}\frac{(\alpha)_{k}}{k!}\BesselJ{\nu+\alpha+2k}@{x}</math>]] || <code>(1)/(GAMMA(alpha))*int((x - t)^(alpha - 1)* BesselJ(nu, t), t = 0..x)= (2)^(alpha)* sum((alpha[k])/(factorial(k))*BesselJ(nu + alpha + 2*k, x), k = 0..infinity)</code> || <code>Divide[1,Gamma[\[Alpha]]]*Integrate[(x - t)^(\[Alpha]- 1)* BesselJ[\[Nu], t], {t, 0, x}]= (2)^(\[Alpha])* Sum[Divide[Subscript[\[Alpha], k],(k)!]*BesselJ[\[Nu]+ \[Alpha]+ 2*k, x], {k, 0, Infinity}]</code> || Failure || Failure || Skip || Successful

| [https://dlmf.nist.gov/10.22.E37 10.22.E37] || [[Item:Q3411|<math>\int_{0}^{1}t\BesselJ{\nu}@{j_{\nu,\ell}t}\BesselJ{\nu}@{j_{\nu,m}t}\diff{t} = \tfrac{1}{2}\left(\BesselJ{\nu}'@{j_{\nu,\ell}}\right)^{2}\Kroneckerdelta{\ell}{m}</math>]] || <code>int(t*BesselJ(nu, j[nu , ell]*t)*BesselJ(nu, j[nu , m]*t), t = 0..1)=(1)/(2)*(subs( temp=j[nu , ell], diff( BesselJ(nu, temp), temp$(1) ) ))^(2)* KroneckerDelta[ell, m]</code> || <code>Integrate[t*BesselJ[\[Nu], Subscript[j, \[Nu], \[ScriptL]]*t]*BesselJ[\[Nu], Subscript[j, \[Nu], m]*t], {t, 0, 1}]=Divide[1,2]*(((D[BesselJ[\[Nu], temp], {temp, 1}]/.temp-> Subscript[j, \[Nu], \[ScriptL]])))^(2)* KroneckerDelta[\[ScriptL], m]</code> || Failure || Failure || Skip || Successful

| [https://dlmf.nist.gov/10.22.E38 10.22.E38] || [[Item:Q3412|<math>\int_{0}^{1}t\BesselJ{\nu}@{\alpha_{\ell}t}\BesselJ{\nu}@{\alpha_{m}t}\diff{t} = \left(\frac{a^{2}}{b^{2}}+\alpha_{\ell}^{2}-\nu^{2}\right)\frac{(\BesselJ{\nu}@{\alpha_{\ell}})^{2}}{2\alpha_{\ell}^{2}}\Kroneckerdelta{\ell}{m}</math>]] || <code>int(t*BesselJ(nu, alpha[ell]*t)*BesselJ(nu, alpha[m]*t), t = 0..1)((BesselJ(nu, alpha[ell]))^(2))/(2*alpha(alpha[ell])^(2))*KroneckerDelta[ell, m]</code> || <code>Integrate[t*BesselJ[\[Nu], Subscript[\[Alpha], \[ScriptL]]*t]*BesselJ[\[Nu], Subscript[\[Alpha], m]*t], {t, 0, 1}]Divide[(BesselJ[\[Nu], Subscript[\[Alpha], \[ScriptL]]])^(2),2*\[Alpha](Subscript[\[Alpha], \[ScriptL]])^(2)]*KroneckerDelta[\[ScriptL], m]</code> || Failure || Failure || Skip || Successful

| [https://dlmf.nist.gov/10.22.E39 10.22.E39] || [[Item:Q3413|<math>\int_{x}^{\infty}\frac{\BesselJ{0}@{t}}{t}\diff{t}+\EulerConstant+\ln@{\tfrac{1}{2}x} = \int_{0}^{x}\frac{1-\BesselJ{0}@{t}}{t}\diff{t}</math>]] || <code>int((BesselJ(0, t))/(t), t = x..infinity)+ gamma + ln((1)/(2)*x)= int((1 - BesselJ(0, t))/(t), t = 0..x)</code> || <code>Integrate[Divide[BesselJ[0, t],t], {t, x, Infinity}]+ EulerGamma + Log[Divide[1,2]*x]= Integrate[Divide[1 - BesselJ[0, t],t], {t, 0, x}]</code> || Successful || Failure || - || Successful

| [https://dlmf.nist.gov/10.22.E39 10.22.E39] || [[Item:Q3413|<math>\int_{0}^{x}\frac{1-\BesselJ{0}@{t}}{t}\diff{t} = \sum_{k=1}^{\infty}(-1)^{k-1}\frac{(\frac{1}{2}x)^{2k}}{2k(k!)^{2}}</math>]] || <code>int((1 - BesselJ(0, t))/(t), t = 0..x)= sum((- 1)^(k - 1)*(((1)/(2)*x)^(2*k))/(2*k*(factorial(k))^(2)), k = 1..infinity)</code> || <code>Integrate[Divide[1 - BesselJ[0, t],t], {t, 0, x}]= Sum[(- 1)^(k - 1)*Divide[(Divide[1,2]*x)^(2*k),2*k*((k)!)^(2)], {k, 1, Infinity}]</code> || Successful || Failure || - || Successful

| [https://dlmf.nist.gov/10.22.E41 10.22.E41] || [[Item:Q3415|<math>\int_{0}^{\infty}\BesselJ{\nu}@{t}\diff{t} = 1</math>]] || <code>int(BesselJ(nu, t), t = 0..infinity)= 1</code> || <code>Integrate[BesselJ[\[Nu], t], {t, 0, Infinity}]= 1</code> || Successful || Failure || - || Successful

| [https://dlmf.nist.gov/10.22.E43 10.22.E43] || [[Item:Q3417|<math>\int_{0}^{\infty}t^{\mu}\BesselJ{\nu}@{t}\diff{t} = 2^{\mu}\frac{\EulerGamma@{\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+\tfrac{1}{2}}}{\EulerGamma@{\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+\tfrac{1}{2}}}</math>]] || <code>int((t)^(mu)* BesselJ(nu, t), t = 0..infinity)= (2)^(mu)*(GAMMA((1)/(2)*nu +(1)/(2)*mu +(1)/(2)))/(GAMMA((1)/(2)*nu -(1)/(2)*mu +(1)/(2)))</code> || <code>Integrate[(t)^(\[Mu])* BesselJ[\[Nu], t], {t, 0, Infinity}]= (2)^(\[Mu])*Divide[Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[1,2]],Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+Divide[1,2]]]</code> || Successful || Failure || - || Successful

| [https://dlmf.nist.gov/10.22.E55 10.22.E55] || [[Item:Q3429|<math>\int_{0}^{\infty}t^{-1}\BesselJ{\nu+2\ell+1}@{t}\BesselJ{\nu+2m+1}@{t}\diff{t} = \frac{\Kroneckerdelta{\ell}{m}}{2(2\ell+\nu+1)}</math>]] || <code>int((t)^(- 1)* BesselJ(nu + 2*ell + 1, t)*BesselJ(nu + 2*m + 1, t), t = 0..infinity)=(KroneckerDelta[ell, m])/(2*(2*ell + nu + 1))</code> || <code>Integrate[(t)^(- 1)* BesselJ[\[Nu]+ 2*\[ScriptL]+ 1, t]*BesselJ[\[Nu]+ 2*m + 1, t], {t, 0, Infinity}]=Divide[KroneckerDelta[\[ScriptL], m],2*(2*\[ScriptL]+ \[Nu]+ 1)]</code> || Failure || Failure || Skip || Successful

| [https://dlmf.nist.gov/10.22.E63 10.22.E63] || [[Item:Q3437|<math>b < a,\\ (2b)^{-1},&b</math>]] || <code>b < a ,(2*b)^(- 1),</code> || <code>b < a ,(2*b)^(- 1),</code> || Error || Failure || - || Error

| [https://dlmf.nist.gov/10.22.E64 10.22.E64] || [[Item:Q3438|<math>b < a,\\ (-1)^{n}/(2a),&b</math>]] || <code>b < a ,(- 1)^(n)/(2*a),</code> || <code>b < a ,(- 1)^(n)/(2*a),</code> || Error || Failure || - || Error

| [https://dlmf.nist.gov/10.22.E65 10.22.E65] || [[Item:Q3439|<math>b < a,0</math>]] || <code>b < a , 0</code> || <code>b < a , 0</code> || Error || Failure || - || -

| [https://dlmf.nist.gov/10.22.E65 10.22.E65] || [[Item:Q3439|<math>a,0 < c</math>]] || <code>a , 0 < c</code> || <code>a , 0 < c</code> || Error || Failure || - || Error

| [https://dlmf.nist.gov/10.22.E65 10.22.E65] || [[Item:Q3439|<math>c <= a,\\ \ln@{c/a},&0</math>]] || <code>c < = a , ln(c/ a),</code> || <code>c < = a , Log[c/ a],</code> || Failure || Failure || Skip || Successful

| [https://dlmf.nist.gov/10.22.E71 10.22.E71] || [[Item:Q3445|<math>\int_{0}^{\infty}\BesselJ{\mu}@{at}\BesselJ{\nu}@{bt}\BesselJ{\nu}@{ct}t^{1-\mu}\diff{t} = \frac{(bc)^{\mu-1}(\sin@@{\phi})^{\mu-\frac{1}{2}}}{(2\pi)^{\frac{1}{2}}a^{\mu}}\FerrersP[\frac{1}{2}-\mu]{\nu-\frac{1}{2}}(\cos@@{\phi})</math>]] || <code>int(BesselJ(mu, a*t)*BesselJ(nu, b*t)*BesselJ(nu, c*t)*(t)^(1 - mu), t = 0..infinity)=((b*c)^(mu - 1)*(sin(phi))^(mu -(1)/(2)))/((2*Pi)^((1)/(2))* (a)^(mu))*LegendreP(nu -(1)/(2), (1)/(2)- mu, cos(phi))</code> || <code>Integrate[BesselJ[\[Mu], a*t]*BesselJ[\[Nu], b*t]*BesselJ[\[Nu], c*t]*(t)^(1 - \[Mu]), {t, 0, Infinity}]=Divide[(b*c)^(\[Mu]- 1)*(Sin[\[Phi]])^(\[Mu]-Divide[1,2]),(2*Pi)^(Divide[1,2])* (a)^(\[Mu])]*LegendreP[\[Nu]-Divide[1,2], Divide[1,2]- \[Mu], Cos[\[Phi]]]</code> || Failure || Failure || Skip || Skip

| [https://dlmf.nist.gov/10.22.E75 10.22.E75] || [[Item:Q3450|<math>a < |b-c|,\\ 0,&|b-c|</math>]] || <code>a <abs(b - c), 0 ,</code> || <code>a <Abs[b - c], 0 ,</code> || Error || Failure || - || -

| [https://dlmf.nist.gov/10.23.E4 10.23.E4] || [[Item:Q3456|<math>\sum_{k=0}^{2n}(-1)^{k}\BesselJ{k}@{z}\BesselJ{2n-k}@{z}\\ +2\sum_{k=1}^{\infty}\BesselJ{k}@{z}\BesselJ{2n+k}@{z} = 0</math>]] || <code>sum((- 1)^(k)* BesselJ(k, z)*BesselJ(2*n - k, z), k = 0..2*n)+ 2*sum(BesselJ(k, z)*BesselJ(2*n + k, z), k = 1..infinity)= 0</code> || <code>Sum[(- 1)^(k)* BesselJ[k, z]*BesselJ[2*n - k, z], {k, 0, 2*n}]+ 2*Sum[BesselJ[k, z]*BesselJ[2*n + k, z], {k, 1, Infinity}]= 0</code> || Failure || Failure || Skip || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>Complex[4.242640687119286, 4.242640687119286] <- {Rule[n, 3], Rule[Sum[Times[BesselJ[k, z], BesselJ[Plus[k, Times[2, n]], z]], {k, 1, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[DifferenceRoot[Function[{, }, {Equal[Plus[Times[-1, Plus[-5, Times[-2, ], Times[2, n]], Power[z, 2], []], Times[Plus[-20, Times[-48, ], Times[-36, Power[, 2]], Times[-8, Power[, 3]], Times[48, n], Times[72, , n], Times[24, Power[, 2], n], Times[-16, Power[n, 2]], Times[-16, , Power[n, 2]], Times[-7, Power[z, 2]], Times[-2, , Power[z, 2]], Times[2, n, Power[z, 2]]], [Plus[1, ]]], Times[-2, Plus[-3, Times[-2, ], Times[2, n]], Plus[22, Times[24, ], Times[6, Power[, 2]], Times[-24, n], Times[-12, , n], Times[8, Power[n, 2]], Times[-1, Power[z, 2]]], [Plus[2, ]]], Times[2, Plus[-5, Times[-2, ], Times[2, n]], Plus[22, Times[24, ], Times[6, Power[, 2]], Times[-24, n], Times[-12, , n], Times[8, Power[n, 2]], Times[-1, Power[z, 2]]], [Plus[3, ]]], Times[Plus[108, Times[144, ], Times[60, Power[, 2]], Times[8, Power[, 3]], Times[-144, n], Times[-120, , n], Times[-24, Power[, 2], n], Times[48, Power[n, 2]], Times[16, , Power[n, 2]], Power[z, 2], Times[2, , Power[z, 2]], Times[-2, n, Power[z, 2]]], [Plus[4, ]]], Times[Plus[-3, Times[-2, ], Times[2, n]], Power[z, 2], [Plus[5, ]]]], 0], Equal[[0], 0], Equal[[1], Times[BesselJ[0, z], BesselJ[Times[2, n], z]]], Equal[[2], Plus[Times[BesselJ[0, z], BesselJ[Times[2, n], z]], Times[-1, BesselJ[1, z], BesselJ[Plus[-1, Times[2, n]], z]]]], Equal[[3], Plus[Times[BesselJ[0, z], BesselJ[Times[2, n], z]], Times[-1, BesselJ[1, z], BesselJ[Plus[-1, Times[2, n]], z]], Times[Power[z, -2], Plus[Times[-1, z, BesselJ[0, z]], Times[2, BesselJ[1, z]]], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]]]], Equal[[4], Plus[Times[BesselJ[0, z], BesselJ[Times[2, n], z]], Times[-1, BesselJ[1, z], BesselJ[Plus[-1, Times[2, n]], z]], Times[Power[z, -2], Plus[Times[-1, z, BesselJ[0, z]], Times[2, BesselJ[1, z]]], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]], Times[Power[z, -2], Plus[Times[z, BesselJ[1, z]], Times[-4, Power[z, -1], Plus[Times[-1, z, BesselJ[0, z]], Times[2, BesselJ[1, z]]]]], Plus[Times[-1, z, BesselJ[Plus[-1, Times[2, n]], z]], Times[-4, Power[z, -1], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]], Times[4, n, Power[z, -1], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]]]]]]}]][Plus[1, Times[2, n]]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[4.242640687119286, 1.4142135623730951] <- {Rule[n, 3], Rule[Sum[Times[BesselJ[k, z], BesselJ[Plus[k, Times[2, n]], z]], {k, 1, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[DifferenceRoot[Function[{, }, {Equal[Plus[Times[-1, Plus[-5, Times[-2, ], Times[2, n]], Power[z, 2], []], Times[Plus[-20, Times[-48, ], Times[-36, Power[, 2]], Times[-8, Power[, 3]], Times[48, n], Times[72, , n], Times[24, Power[, 2], n], Times[-16, Power[n, 2]], Times[-16, , Power[n, 2]], Times[-7, Power[z, 2]], Times[-2, , Power[z, 2]], Times[2, n, Power[z, 2]]], [Plus[1, ]]], Times[-2, Plus[-3, Times[-2, ], Times[2, n]], Plus[22, Times[24, ], Times[6, Power[, 2]], Times[-24, n], Times[-12, , n], Times[8, Power[n, 2]], Times[-1, Power[z, 2]]], [Plus[2, ]]], Times[2, Plus[-5, Times[-2, ], Times[2, n]], Plus[22, Times[24, ], Times[6, Power[, 2]], Times[-24, n], Times[-12, , n], Times[8, Power[n, 2]], Times[-1, Power[z, 2]]], [Plus[3, ]]], Times[Plus[108, Times[144, ], Times[60, Power[, 2]], Times[8, Power[, 3]], Times[-144, n], Times[-120, , n], Times[-24, Power[, 2], n], Times[48, Power[n, 2]], Times[16, , Power[n, 2]], Power[z, 2], Times[2, , Power[z, 2]], Times[-2, n, Power[z, 2]]], [Plus[4, ]]], Times[Plus[-3, Times[-2, ], Times[2, n]], Power[z, 2], [Plus[5, ]]]], 0], Equal[[0], 0], Equal[[1], Times[BesselJ[0, z], BesselJ[Times[2, n], z]]], Equal[[2], Plus[Times[BesselJ[0, z], BesselJ[Times[2, n], z]], Times[-1, BesselJ[1, z], BesselJ[Plus[-1, Times[2, n]], z]]]], Equal[[3], Plus[Times[BesselJ[0, z], BesselJ[Times[2, n], z]], Times[-1, BesselJ[1, z], BesselJ[Plus[-1, Times[2, n]], z]], Times[Power[z, -2], Plus[Times[-1, z, BesselJ[0, z]], Times[2, BesselJ[1, z]]], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]]]], Equal[[4], Plus[Times[BesselJ[0, z], BesselJ[Times[2, n], z]], Times[-1, BesselJ[1, z], BesselJ[Plus[-1, Times[2, n]], z]], Times[Power[z, -2], Plus[Times[-1, z, BesselJ[0, z]], Times[2, BesselJ[1, z]]], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]], Times[Power[z, -2], Plus[Times[z, BesselJ[1, z]], Times[-4, Power[z, -1], Plus[Times[-1, z, BesselJ[0, z]], Times[2, BesselJ[1, z]]]]], Plus[Times[-1, z, BesselJ[Plus[-1, Times[2, n]], z]], Times[-4, Power[z, -1], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]], Times[4, n, Power[z, -1], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]]]]]]}]][Plus[1, Times[2, n]]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[n, 3], Rule[Sum[Times[BesselJ[k, z], BesselJ[Plus[k, Times[2, n]], z]], {k, 1, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[DifferenceRoot[Function[{, }, {Equal[Plus[Times[-1, Plus[-5, Times[-2, ], Times[2, n]], Power[z, 2], []], Times[Plus[-20, Times[-48, ], Times[-36, Power[, 2]], Times[-8, Power[, 3]], Times[48, n], Times[72, , n], Times[24, Power[, 2], n], Times[-16, Power[n, 2]], Times[-16, , Power[n, 2]], Times[-7, Power[z, 2]], Times[-2, , Power[z, 2]], Times[2, n, Power[z, 2]]], [Plus[1, ]]], Times[-2, Plus[-3, Times[-2, ], Times[2, n]], Plus[22, Times[24, ], Times[6, Power[, 2]], Times[-24, n], Times[-12, , n], Times[8, Power[n, 2]], Times[-1, Power[z, 2]]], [Plus[2, ]]], Times[2, Plus[-5, Times[-2, ], Times[2, n]], Plus[22, Times[24, ], Times[6, Power[, 2]], Times[-24, n], Times[-12, , n], Times[8, Power[n, 2]], Times[-1, Power[z, 2]]], [Plus[3, ]]], Times[Plus[108, Times[144, ], Times[60, Power[, 2]], Times[8, Power[, 3]], Times[-144, n], Times[-120, , n], Times[-24, Power[, 2], n], Times[48, Power[n, 2]], Times[16, , Power[n, 2]], Power[z, 2], Times[2, , Power[z, 2]], Times[-2, n, Power[z, 2]]], [Plus[4, ]]], Times[Plus[-3, Times[-2, ], Times[2, n]], Power[z, 2], [Plus[5, ]]]], 0], Equal[[0], 0], Equal[[1], Times[BesselJ[0, z], BesselJ[Times[2, n], z]]], Equal[[2], Plus[Times[BesselJ[0, z], BesselJ[Times[2, n], z]], Times[-1, BesselJ[1, z], BesselJ[Plus[-1, Times[2, n]], z]]]], Equal[[3], Plus[Times[BesselJ[0, z], BesselJ[Times[2, n], z]], Times[-1, BesselJ[1, z], BesselJ[Plus[-1, Times[2, n]], z]], Times[Power[z, -2], Plus[Times[-1, z, BesselJ[0, z]], Times[2, BesselJ[1, z]]], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]]]], Equal[[4], Plus[Times[BesselJ[0, z], BesselJ[Times[2, n], z]], Times[-1, BesselJ[1, z], BesselJ[Plus[-1, Times[2, n]], z]], Times[Power[z, -2], Plus[Times[-1, z, BesselJ[0, z]], Times[2, BesselJ[1, z]]], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]], Times[Power[z, -2], Plus[Times[z, BesselJ[1, z]], Times[-4, Power[z, -1], Plus[Times[-1, z, BesselJ[0, z]], Times[2, BesselJ[1, z]]]]], Plus[Times[-1, z, BesselJ[Plus[-1, Times[2, n]], z]], Times[-4, Power[z, -1], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]], Times[4, n, Power[z, -1], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]]]]]]}]][Plus[1, Times[2, n]]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1.4142135623730951, 4.242640687119286] <- {Rule[n, 3], Rule[Sum[Times[BesselJ[k, z], BesselJ[Plus[k, Times[2, n]], z]], {k, 1, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[DifferenceRoot[Function[{, }, {Equal[Plus[Times[-1, Plus[-5, Times[-2, ], Times[2, n]], Power[z, 2], []], Times[Plus[-20, Times[-48, ], Times[-36, Power[, 2]], Times[-8, Power[, 3]], Times[48, n], Times[72, , n], Times[24, Power[, 2], n], Times[-16, Power[n, 2]], Times[-16, , Power[n, 2]], Times[-7, Power[z, 2]], Times[-2, , Power[z, 2]], Times[2, n, Power[z, 2]]], [Plus[1, ]]], Times[-2, Plus[-3, Times[-2, ], Times[2, n]], Plus[22, Times[24, ], Times[6, Power[, 2]], Times[-24, n], Times[-12, , n], Times[8, Power[n, 2]], Times[-1, Power[z, 2]]], [Plus[2, ]]], Times[2, Plus[-5, Times[-2, ], Times[2, n]], Plus[22, Times[24, ], Times[6, Power[, 2]], Times[-24, n], Times[-12, , n], Times[8, Power[n, 2]], Times[-1, Power[z, 2]]], [Plus[3, ]]], Times[Plus[108, Times[144, ], Times[60, Power[, 2]], Times[8, Power[, 3]], Times[-144, n], Times[-120, , n], Times[-24, Power[, 2], n], Times[48, Power[n, 2]], Times[16, , Power[n, 2]], Power[z, 2], Times[2, , Power[z, 2]], Times[-2, n, Power[z, 2]]], [Plus[4, ]]], Times[Plus[-3, Times[-2, ], Times[2, n]], Power[z, 2], [Plus[5, ]]]], 0], Equal[[0], 0], Equal[[1], Times[BesselJ[0, z], BesselJ[Times[2, n], z]]], Equal[[2], Plus[Times[BesselJ[0, z], BesselJ[Times[2, n], z]], Times[-1, BesselJ[1, z], BesselJ[Plus[-1, Times[2, n]], z]]]], Equal[[3], Plus[Times[BesselJ[0, z], BesselJ[Times[2, n], z]], Times[-1, BesselJ[1, z], BesselJ[Plus[-1, Times[2, n]], z]], Times[Power[z, -2], Plus[Times[-1, z, BesselJ[0, z]], Times[2, BesselJ[1, z]]], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]]]], Equal[[4], Plus[Times[BesselJ[0, z], BesselJ[Times[2, n], z]], Times[-1, BesselJ[1, z], BesselJ[Plus[-1, Times[2, n]], z]], Times[Power[z, -2], Plus[Times[-1, z, BesselJ[0, z]], Times[2, BesselJ[1, z]]], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]], Times[Power[z, -2], Plus[Times[z, BesselJ[1, z]], Times[-4, Power[z, -1], Plus[Times[-1, z, BesselJ[0, z]], Times[2, BesselJ[1, z]]]]], Plus[Times[-1, z, BesselJ[Plus[-1, Times[2, n]], z]], Times[-4, Power[z, -1], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]], Times[4, n, Power[z, -1], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]]]]]]}]][Plus[1, Times[2, n]]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.23#Ex1 10.23#Ex1] || [[Item:Q3458|<math>w = \sqrt{u^{2}+v^{2}-2uv\cos@@{\alpha}}</math>]] || <code>w =sqrt((u)^(2)+ (v)^(2)- 2*u*v*cos(alpha))</code> || <code>w =Sqrt[(u)^(2)+ (v)^(2)- 2*u*v*Cos[\[Alpha]]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>.7483404465-2.552464124*I <- {alpha = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2)}</code><br><code>.7483404465-5.380891248*I <- {alpha = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2)}</code><br><code>-2.080086678-5.380891248*I <- {alpha = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), w = -2^(1/2)-I*2^(1/2)}</code><br><code>-2.080086678-2.552464124*I <- {alpha = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), w = -2^(1/2)+I*2^(1/2)}</code><br>... skip entries to safe data<br></div></div> || Skip  

| [https://dlmf.nist.gov/10.23#Ex3 10.23#Ex3] || [[Item:Q3460|<math>v\sin@@{\alpha} = w\sin@@{\chi}</math>]] || <code>v*sin(alpha)= w*sin(chi)</code> || <code>v*Sin[\[Alpha]]= w*Sin[\[Chi]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>-.853510328+6.085461480*I <- {alpha = 2^(1/2)+I*2^(1/2), chi = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2)}</code><br><code>5.231951152+6.938971808*I <- {alpha = 2^(1/2)+I*2^(1/2), chi = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), w = -2^(1/2)-I*2^(1/2)}</code><br><code>6.085461480+.853510328*I <- {alpha = 2^(1/2)+I*2^(1/2), chi = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), w = -2^(1/2)+I*2^(1/2)}</code><br><code>.853510328-6.085461480*I <- {alpha = 2^(1/2)+I*2^(1/2), chi = 2^(1/2)+I*2^(1/2), v = 2^(1/2)-I*2^(1/2), w = 2^(1/2)+I*2^(1/2)}</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[-0.853510326577255, 0.853510326577255] <- {Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[5.231951158402261, 6.938971811556771] <- {Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[6.085461484979516, 6.085461484979516] <- {Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[0.853510326577255, -0.853510326577255] <- {Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[α, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.24.E1 10.24.E1] || [[Item:Q3476|<math>x^{2}\deriv[2]{w}{x}+x\deriv{w}{x}+(x^{2}+\nu^{2})w = 0</math>]] || <code>(x)^(2)* diff(w, [x$(2)])+ x*diff(w, x)+((x)^(2)+ (nu)^(2))* w = 0</code> || <code>(x)^(2)* D[w, {x, 2}]+ x*D[w, x]+((x)^(2)+ (\[Nu])^(2))* w = 0</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>-4.242640683+7.071067807*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), x = 1}</code><br><code>.2828427124e-8+11.31370849*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), x = 2}</code><br><code>7.071067813+18.38477630*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), x = 3}</code><br><code>7.071067807+4.242640683*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2), x = 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[-4.242640687119286, 7.0710678118654755] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[0.0, 11.313708498984761] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[7.0710678118654755, 18.38477631085024] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[7.0710678118654755, -4.242640687119286] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.24.E3 10.24.E3] || [[Item:Q3479|<math>\EulerGamma@{1+i\nu} = \left(\frac{\pi\nu}{\sinh@{\pi\nu}}\right)^{\frac{1}{2}}e^{i\gamma_{\nu}}</math>]] || <code>GAMMA(1 + I*nu)=((Pi*nu)/(sinh(Pi*nu)))^((1)/(2))* exp(I*gamma[nu])</code> || <code>Gamma[1 + I*\[Nu]]=(Divide[Pi*\[Nu],Sinh[Pi*\[Nu]]])^(Divide[1,2])* Exp[I*Subscript[\[Gamma], \[Nu]]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>-.7864250629e-1-.1325922997*I <- {nu = 2^(1/2)+I*2^(1/2), gamma[nu] = 2^(1/2)+I*2^(1/2)}</code><br><code>1.283131241-.7318661334*I <- {nu = 2^(1/2)+I*2^(1/2), gamma[nu] = 2^(1/2)-I*2^(1/2)}</code><br><code>-1.737329775+.6511700055e-1*I <- {nu = 2^(1/2)+I*2^(1/2), gamma[nu] = -2^(1/2)-I*2^(1/2)}</code><br><code>-.2571691098-.8548601655e-1*I <- {nu = 2^(1/2)+I*2^(1/2), gamma[nu] = -2^(1/2)+I*2^(1/2)}</code><br>... skip entries to safe data<br></div></div> || Successful  

| [https://dlmf.nist.gov/10.24#Ex3 10.24#Ex3] || [[Item:Q3480|<math>\BesselJimag{-\nu}@{x} = \BesselJimag{\nu}@{x}</math>]] || <code>sech((1/2)*Pi*(- nu))*Re(BesselJ(I*(- nu), x))= sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x))</code> || <code>Sech[1/2 Pi - \[Nu]] Re[BesselJ[I - \[Nu], x]]= Sech[1/2 Pi \[Nu]] Re[BesselJ[I \[Nu], x]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>-.4072055387-.5224985000*I <- {nu = 2^(1/2)+I*2^(1/2), x = 1}</code><br><code>-.9080795132-1.165185978*I <- {nu = 2^(1/2)+I*2^(1/2), x = 2}</code><br><code>-.3702824234-.4751212654*I <- {nu = 2^(1/2)+I*2^(1/2), x = 3}</code><br><code>.4072055387-.5224985000*I <- {nu = 2^(1/2)-I*2^(1/2), x = 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[-5.717128116473797, -5.753336678220267] <- {Rule[x, 1], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-3.18027157410826, -3.3753924459653097] <- {Rule[x, 2], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-0.3142029624806735, -0.37398699406023267] <- {Rule[x, 3], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[335.05864396153805, -329.61926001758485] <- {Rule[x, 1], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.24#Ex4 10.24#Ex4] || [[Item:Q3481|<math>\BesselYimag{-\nu}@{x} = \BesselYimag{\nu}@{x}</math>]] || <code>sech((1/2)*Pi*(- nu))*Re(BesselY(I*(- nu), x))= sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x))</code> || <code>Sech[1/2 Pi - \[Nu]] Re[BesselY[I - \[Nu], x]]= Sech[1/2 Pi \[Nu]] Re[BesselY[I \[Nu], x]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>1.996706293+2.562037949*I <- {nu = 2^(1/2)+I*2^(1/2), x = 1}</code><br><code>.9116896387e-1+.1169818245*I <- {nu = 2^(1/2)+I*2^(1/2), x = 2}</code><br><code>-.4855048259-.6229668292*I <- {nu = 2^(1/2)+I*2^(1/2), x = 3}</code><br><code>-1.996706293+2.562037949*I <- {nu = 2^(1/2)-I*2^(1/2), x = 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[3.2898455559215325, 3.8112807679993184] <- {Rule[x, 1], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-1.6593720309671711, -1.6388399049382785] <- {Rule[x, 2], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-2.5986068001960096, -2.7005355423537045] <- {Rule[x, 3], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[104.18379750674467, -102.47171282147995] <- {Rule[x, 1], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.25.E1 10.25.E1] || [[Item:Q3488|<math>z^{2}\deriv[2]{w}{z}+z\deriv{w}{z}-(z^{2}+\nu^{2})w = 0</math>]] || <code>(z)^(2)* diff(w, [z$(2)])+ z*diff(w, z)-((z)^(2)+ (nu)^(2))* w = 0</code> || <code>(z)^(2)* D[w, {z, 2}]+ z*D[w, z]-((z)^(2)+ (\[Nu])^(2))* w = 0</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>11.31370849-11.31370849*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}</code><br><code>11.31370849-11.31370849*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}</code><br><code>-11.31370849-11.31370849*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}</code><br><code>-11.31370849-11.31370849*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}</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[11.313708498984761, -11.313708498984761] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[11.313708498984761, -11.313708498984761] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-11.313708498984761, 11.313708498984761] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-11.313708498984761, 11.313708498984761] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.28.E1 10.28.E1] || [[Item:Q3502|<math>\Wronskian@{\modBesselI{\nu}@{z},\modBesselI{-\nu}@{z}} = \modBesselI{\nu}@{z}\modBesselI{-\nu-1}@{z}-\modBesselI{\nu+1}@{z}\modBesselI{-\nu}@{z}</math>]] || <code>(BesselI(nu, z))*diff(BesselI(- nu, z), x)-diff(BesselI(nu, z), x)*(BesselI(- nu, z))= BesselI(nu, z)*BesselI(- nu - 1, z)- BesselI(nu + 1, z)*BesselI(- nu, z)</code> || <code>Wronskian[{BesselI[\[Nu], z], BesselI[- \[Nu], z]}, x]= BesselI[\[Nu], z]*BesselI[- \[Nu]- 1, z]- BesselI[\[Nu]+ 1, z]*BesselI[- \[Nu], z]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>-11.77116916+6.676770606*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}</code><br><code>-6.676770612-11.77116916*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}</code><br><code>11.77116916-6.676770609*I <- {nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}</code><br><code>6.676770609+11.77116916*I <- {nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}</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[-11.771169167436858, 6.676770630088813] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-6.676770630088807, 11.771169167436854] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[11.771169167436858, -6.676770630088829] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[6.67677063008883, -11.771169167436856] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.28.E2 10.28.E2] || [[Item:Q3503|<math>\Wronskian@{\modBesselK{\nu}@{z},\modBesselI{\nu}@{z}} = \modBesselI{\nu}@{z}\modBesselK{\nu+1}@{z}+\modBesselI{\nu+1}@{z}\modBesselK{\nu}@{z}</math>]] || <code>(BesselK(nu, z))*diff(BesselI(nu, z), x)-diff(BesselK(nu, z), x)*(BesselI(nu, z))= BesselI(nu, z)*BesselK(nu + 1, z)+ BesselI(nu + 1, z)*BesselK(nu, z)</code> || <code>Wronskian[{BesselK[\[Nu], z], BesselI[\[Nu], z]}, x]= BesselI[\[Nu], z]*BesselK[\[Nu]+ 1, z]+ BesselI[\[Nu]+ 1, z]*BesselK[\[Nu], z]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>-.3535533907+.3535533906*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}</code><br><code>-.3535533908-.3535533907*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}</code><br><code>.353553388-.35355339*I <- {nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}</code><br><code>.3535533905+.3535533907*I <- {nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}</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[-0.35355339059327384, 0.3535533905932732] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-0.35355339059327306, 0.3535533905932739] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-0.35355339059327395, 0.35355339059327373] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-0.3535533905932736, 0.3535533905932738] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.32.E1 10.32.E1] || [[Item:Q3521|<math>\frac{1}{\pi}\int_{0}^{\pi}e^{+ z\cos@@{\theta}}\diff{\theta} = \frac{1}{\pi}\int_{0}^{\pi}\cosh@{z\cos@@{\theta}}\diff{\theta}</math>]] || <code>(1)/(Pi)*int(exp(+ z*cos(theta)), theta = 0..Pi)=(1)/(Pi)*int(cosh(z*cos(theta)), theta = 0..Pi)</code> || <code>Divide[1,Pi]*Integrate[Exp[+ z*Cos[\[Theta]]], {\[Theta], 0, Pi}]=Divide[1,Pi]*Integrate[Cosh[z*Cos[\[Theta]]], {\[Theta], 0, Pi}]</code> || Failure || Failure || Skip || Successful

| [https://dlmf.nist.gov/10.32.E1 10.32.E1] || [[Item:Q3521|<math>\frac{1}{\pi}\int_{0}^{\pi}e^{- z\cos@@{\theta}}\diff{\theta} = \frac{1}{\pi}\int_{0}^{\pi}\cosh@{z\cos@@{\theta}}\diff{\theta}</math>]] || <code>(1)/(Pi)*int(exp(- z*cos(theta)), theta = 0..Pi)=(1)/(Pi)*int(cosh(z*cos(theta)), theta = 0..Pi)</code> || <code>Divide[1,Pi]*Integrate[Exp[- z*Cos[\[Theta]]], {\[Theta], 0, Pi}]=Divide[1,Pi]*Integrate[Cosh[z*Cos[\[Theta]]], {\[Theta], 0, Pi}]</code> || Failure || Failure || Skip || Successful

| [https://dlmf.nist.gov/10.32.E2 10.32.E2] || [[Item:Q3522|<math>\frac{(\frac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\EulerGamma@{\nu+\frac{1}{2}}}\int_{0}^{\pi}e^{+ z\cos@@{\theta}}(\sin@@{\theta})^{2\nu}\diff{\theta} = \frac{(\frac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\EulerGamma@{\nu+\frac{1}{2}}}\int_{-1}^{1}(1-t^{2})^{\nu-\frac{1}{2}}e^{+ zt}\diff{t}</math>]] || <code>(((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*int(exp(+ z*cos(theta))*(sin(theta))^(2*nu), theta = 0..Pi)=(((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*int((1 - (t)^(2))^(nu -(1)/(2))* exp(+ z*t), t = - 1..1)</code> || <code>Divide[(Divide[1,2]*z)^(\[Nu]),(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*Integrate[Exp[+ z*Cos[\[Theta]]]*(Sin[\[Theta]])^(2*\[Nu]), {\[Theta], 0, Pi}]=Divide[(Divide[1,2]*z)^(\[Nu]),(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*Integrate[(1 - (t)^(2))^(\[Nu]-Divide[1,2])* Exp[+ z*t], {t, - 1, 1}]</code> || Failure || Failure || Skip || Error

| [https://dlmf.nist.gov/10.32.E2 10.32.E2] || [[Item:Q3522|<math>\frac{(\frac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\EulerGamma@{\nu+\frac{1}{2}}}\int_{0}^{\pi}e^{- z\cos@@{\theta}}(\sin@@{\theta})^{2\nu}\diff{\theta} = \frac{(\frac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\EulerGamma@{\nu+\frac{1}{2}}}\int_{-1}^{1}(1-t^{2})^{\nu-\frac{1}{2}}e^{- zt}\diff{t}</math>]] || <code>(((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*int(exp(- z*cos(theta))*(sin(theta))^(2*nu), theta = 0..Pi)=(((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*int((1 - (t)^(2))^(nu -(1)/(2))* exp(- z*t), t = - 1..1)</code> || <code>Divide[(Divide[1,2]*z)^(\[Nu]),(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*Integrate[Exp[- z*Cos[\[Theta]]]*(Sin[\[Theta]])^(2*\[Nu]), {\[Theta], 0, Pi}]=Divide[(Divide[1,2]*z)^(\[Nu]),(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*Integrate[(1 - (t)^(2))^(\[Nu]-Divide[1,2])* Exp[- z*t], {t, - 1, 1}]</code> || Failure || Failure || Skip || Error

| [https://dlmf.nist.gov/10.32.E10 10.32.E10] || [[Item:Q3530|<math>\modBesselK{\nu}@{z} = \tfrac{1}{2}(\tfrac{1}{2}z)^{\nu}\int_{0}^{\infty}\exp@{-t-\frac{z^{2}}{4t}}\frac{\diff{t}}{t^{\nu+1}}</math>]] || <code>BesselK(nu, z)=(1)/(2)*((1)/(2)*z)^(nu)* int(exp(- t -((z)^(2))/(4*t))*(1)/((t)^(nu + 1)), t = 0..infinity)</code> || <code>BesselK[\[Nu], z]=Divide[1,2]*(Divide[1,2]*z)^(\[Nu])* Integrate[Exp[- t -Divide[(z)^(2),4*t]]*Divide[1,(t)^(\[Nu]+ 1)], {t, 0, Infinity}]</code> || Successful || Failure || - || Skip

| [https://dlmf.nist.gov/10.32.E12 10.32.E12] || [[Item:Q3532|<math>\modBesselI{\nu}@{z} = \frac{1}{2\pi i}\int_{\infty-i\pi}^{\infty+i\pi}e^{z\cosh@@{t}-\nu t}\diff{t}</math>]] || <code>BesselI(nu, z)=(1)/(2*Pi*I)*int(exp(z*cosh(t)- nu*t), t = infinity - I*Pi..infinity + I*Pi)</code> || <code>BesselI[\[Nu], z]=Divide[1,2*Pi*I]*Integrate[Exp[z*Cosh[t]- \[Nu]*t], {t, Infinity - I*Pi, Infinity + I*Pi}]</code> || Failure || Failure || Skip || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>Complex[0.47377882604348887, 0.17987673448701852] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-2.5712028891376235, 2.011728577446344] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-5.630703522037926, 7.3730343474306625] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[0.4462596814449855, 3.2604536086998377] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.34.E1 10.34.E1] || [[Item:Q3542|<math>\modBesselI{\nu}@{ze^{m\pi i}} = e^{m\nu\pi i}\modBesselI{\nu}@{z}</math>]] || <code>BesselI(nu, z*exp(m*Pi*I))= exp(m*nu*Pi*I)*BesselI(nu, z)</code> || <code>BesselI[\[Nu], z*Exp[m*Pi*I]]= Exp[m*\[Nu]*Pi*I]*BesselI[\[Nu], z]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>-25.46648651+34.76058054*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 1}</code><br><code>.4738478497+.1798644481*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 2}</code><br><code>-25.46593127+34.75464498*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3}</code><br><code>-2.571651015-2.011784848*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 2}</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[-25.466486497459893, 34.76058068352855] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[0.47384785056443346, 0.17986444785635414] <- {Rule[m, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-25.465931246406512, 34.754645199849094] <- {Rule[m, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-223.08150600961898, -165.2079311070147] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.34.E2 10.34.E2] || [[Item:Q3543|<math>\modBesselK{\nu}@{ze^{m\pi i}} = e^{-m\nu\pi i}\modBesselK{\nu}@{z}-\pi i\sin@{m\nu\pi}\csc@{\nu\pi}\modBesselI{\nu}@{z}</math>]] || <code>BesselK(nu, z*exp(m*Pi*I))= exp(- m*nu*Pi*I)*BesselK(nu, z)- Pi*I*sin(m*nu*Pi)*csc(nu*Pi)*BesselI(nu, z)</code> || <code>BesselK[\[Nu], z*Exp[m*Pi*I]]= Exp[- m*\[Nu]*Pi*I]*BesselK[\[Nu], z]- Pi*I*Sin[m*\[Nu]*Pi]*Csc[\[Nu]*Pi]*BesselI[\[Nu], z]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>-23.72816996-16.20095675*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 1}</code><br><code>1974.016674-1497.794581*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 2}</code><br><code>78036.43688+195659.8555*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3}</code><br><code>-346.8741250+807.6398259*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 2}</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[-23.728169968169517, -16.200956740051907] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1974.0166738862135, -1497.7945856695665] <- {Rule[m, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[78036.4381344157, 195659.85598804062] <- {Rule[m, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-16.563048824383813, -6.675705970582722] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.34.E3 10.34.E3] || [[Item:Q3544|<math>\modBesselI{\nu}@{ze^{m\pi i}} = (i/\pi)\left(+ e^{m\nu\pi i}\modBesselK{\nu}@{ze^{+\pi i}}- e^{(m- 1)\nu\pi i}\modBesselK{\nu}@{z}\right)</math>]] || <code>BesselI(nu, z*exp(m*Pi*I))=(I/ Pi)*(+ exp(m*nu*Pi*I)*BesselK(nu, z*exp(+ Pi*I))- exp((m - 1)* nu*Pi*I)*BesselK(nu, z))</code> || <code>BesselI[\[Nu], z*Exp[m*Pi*I]]=(I/ Pi)*(+ Exp[m*\[Nu]*Pi*I]*BesselK[\[Nu], z*Exp[+ Pi*I]]- Exp[(m - 1)* \[Nu]*Pi*I]*BesselK[\[Nu], z])</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>-25.36470622+34.79539330*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 1}</code><br><code>.4739237916+.1786015012*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 2}</code><br><code>-25.46594582+34.75464807*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3}</code><br><code>-2.571651015-2.011784848*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 2}</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[-25.36470621174673, 34.79539343891294] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[0.47392379247751504, 0.17860150099441258] <- {Rule[m, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-25.465945802770374, 34.75464829402328] <- {Rule[m, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[257.07974135711925, -110.41346285737623] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.34.E3 10.34.E3] || [[Item:Q3544|<math>\modBesselI{\nu}@{ze^{m\pi i}} = (i/\pi)\left(- e^{m\nu\pi i}\modBesselK{\nu}@{ze^{-\pi i}}+ e^{(m+ 1)\nu\pi i}\modBesselK{\nu}@{z}\right)</math>]] || <code>BesselI(nu, z*exp(m*Pi*I))=(I/ Pi)*(- exp(m*nu*Pi*I)*BesselK(nu, z*exp(- Pi*I))+ exp((m + 1)* nu*Pi*I)*BesselK(nu, z))</code> || <code>BesselI[\[Nu], z*Exp[m*Pi*I]]=(I/ Pi)*(- Exp[m*\[Nu]*Pi*I]*BesselK[\[Nu], z*Exp[- Pi*I]]+ Exp[(m + 1)* \[Nu]*Pi*I]*BesselK[\[Nu], z])</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>-25.46648651+34.76058054*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 1}</code><br><code>.4738478497+.1798644481*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 2}</code><br><code>-25.46593127+34.75464498*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3}</code><br><code>-.5311818950e-1+.4060217813e-1*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 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[-25.466486497459893, 34.76058068352855] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[0.47384785056443346, 0.17986444785635414] <- {Rule[m, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-25.465931246406512, 34.754645199849094] <- {Rule[m, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-223.0815060096189, -165.20793110701467] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.34.E4 10.34.E4] || [[Item:Q3545|<math>\modBesselK{\nu}@{ze^{m\pi i}} = \csc@{\nu\pi}\left(+\sin@{m\nu\pi}\modBesselK{\nu}@{ze^{+\pi i}}-\sin@{(m- 1)\nu\pi}\modBesselK{\nu}@{z}\right)</math>]] || <code>BesselK(nu, z*exp(m*Pi*I))= csc(nu*Pi)*(+ sin(m*nu*Pi)*BesselK(nu, z*exp(+ Pi*I))- sin((m - 1)* nu*Pi)*BesselK(nu, z))</code> || <code>BesselK[\[Nu], z*Exp[m*Pi*I]]= Csc[\[Nu]*Pi]*(+ Sin[m*\[Nu]*Pi]*BesselK[\[Nu], z*Exp[+ Pi*I]]- Sin[(m - 1)* \[Nu]*Pi]*BesselK[\[Nu], z])</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>109.3129522+79.68557470*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 2}</code><br><code>-9027.967286+7136.744811*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3}</code><br><code>-346.8741247+807.6398251*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 2}</code><br><code>-58340.79702-46700.99889*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 3}</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[109.31295240645538, 79.68557469528692] <- {Rule[m, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-9027.967291383398, 7136.744876012727] <- {Rule[m, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-346.8741237701426, -807.6398268342901] <- {Rule[m, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-58340.79750295953, 46700.99881352048] <- {Rule[m, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.34.E4 10.34.E4] || [[Item:Q3545|<math>\modBesselK{\nu}@{ze^{m\pi i}} = \csc@{\nu\pi}\left(-\sin@{m\nu\pi}\modBesselK{\nu}@{ze^{-\pi i}}+\sin@{(m+ 1)\nu\pi}\modBesselK{\nu}@{z}\right)</math>]] || <code>BesselK(nu, z*exp(m*Pi*I))= csc(nu*Pi)*(- sin(m*nu*Pi)*BesselK(nu, z*exp(- Pi*I))+ sin((m + 1)* nu*Pi)*BesselK(nu, z))</code> || <code>BesselK[\[Nu], z*Exp[m*Pi*I]]= Csc[\[Nu]*Pi]*(- Sin[m*\[Nu]*Pi]*BesselK[\[Nu], z*Exp[- Pi*I]]+ Sin[(m + 1)* \[Nu]*Pi]*BesselK[\[Nu], z])</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>-23.72816993-16.20095676*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 1}</code><br><code>1974.016672-1497.794570*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 2}</code><br><code>78036.44012+195659.8571*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3}</code><br><code>-16.56304884+6.675705955*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 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[-23.728169968169517, -16.2009567400519] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[1974.0166738862144, -1497.7945856695726] <- {Rule[m, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[78036.43813441524, 195659.85598804036] <- {Rule[m, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-16.563048824383813, -6.675705970582721] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.34.E5 10.34.E5] || [[Item:Q3546|<math>\modBesselK{n}@{ze^{m\pi i}} = (-1)^{mn}\modBesselK{n}@{z}+(-1)^{n(m-1)-1}m\pi i\modBesselI{n}@{z}</math>]] || <code>BesselK(n, z*exp(m*Pi*I))=(- 1)^(m*n)* BesselK(n, z)+(- 1)^(n*(m - 1)- 1)* m*Pi*I*BesselI(n, z)</code> || <code>BesselK[n, z*Exp[m*Pi*I]]=(- 1)^(m*n)* BesselK[n, z]+(- 1)^(n*(m - 1)- 1)* m*Pi*I*BesselI[n, z]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>-6.264823649+1.883544620*I <- {z = 2^(1/2)+I*2^(1/2), m = 1, n = 1}</code><br><code>-3.011056351-1.038481291*I <- {z = 2^(1/2)+I*2^(1/2), m = 1, n = 2}</code><br><code>-.5379125927-.9060977874*I <- {z = 2^(1/2)+I*2^(1/2), m = 1, n = 3}</code><br><code>6.264823651-1.883544623*I <- {z = 2^(1/2)+I*2^(1/2), m = 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[-6.264823652701258, 1.8835446212245166] <- {Rule[m, 1], Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-3.0110563535593116, -1.0384812903661844] <- {Rule[m, 1], Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-0.537912595321433, -0.9060977859421694] <- {Rule[m, 1], Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[6.264823652701258, -1.8835446212245166] <- {Rule[m, 2], Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.34.E6 10.34.E6] || [[Item:Q3547|<math>\modBesselK{n}@{ze^{m\pi i}} = +(-1)^{n(m-1)}m\modBesselK{n}@{ze^{+\pi i}}-(-1)^{nm}(m- 1)\modBesselK{n}@{z}</math>]] || <code>BesselK(n, z*exp(m*Pi*I))= +(- 1)^(n*(m - 1))* m*BesselK(n, z*exp(+ Pi*I))-(- 1)^(n*m)*(m - 1)* BesselK(n, z)</code> || <code>BesselK[n, z*Exp[m*Pi*I]]= +(- 1)^(n*(m - 1))* m*BesselK[n, z*Exp[+ Pi*I]]-(- 1)^(n*m)*(m - 1)* BesselK[n, z]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>-6.264823650+1.883544622*I <- {z = 2^(1/2)+I*2^(1/2), m = 2, n = 1}</code><br><code>3.011056351+1.038481289*I <- {z = 2^(1/2)+I*2^(1/2), m = 2, n = 2}</code><br><code>-.5379125964-.9060977833*I <- {z = 2^(1/2)+I*2^(1/2), m = 2, n = 3}</code><br><code>6.264823652-1.883544629*I <- {z = 2^(1/2)+I*2^(1/2), m = 3, 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[-6.264823652701258, 1.8835446212245168] <- {Rule[m, 2], Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[3.0110563535593116, 1.0384812903661842] <- {Rule[m, 2], Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-0.537912595321433, -0.9060977859421694] <- {Rule[m, 2], Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[6.264823652701258, -1.8835446212245168] <- {Rule[m, 3], Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.34.E6 10.34.E6] || [[Item:Q3547|<math>\modBesselK{n}@{ze^{m\pi i}} = -(-1)^{n(m-1)}m\modBesselK{n}@{ze^{-\pi i}}+(-1)^{nm}(m+ 1)\modBesselK{n}@{z}</math>]] || <code>BesselK(n, z*exp(m*Pi*I))= -(- 1)^(n*(m - 1))* m*BesselK(n, z*exp(- Pi*I))+(- 1)^(n*m)*(m + 1)* BesselK(n, z)</code> || <code>BesselK[n, z*Exp[m*Pi*I]]= -(- 1)^(n*(m - 1))* m*BesselK[n, z*Exp[- Pi*I]]+(- 1)^(n*m)*(m + 1)* BesselK[n, z]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>-6.264823648+1.883544620*I <- {z = 2^(1/2)+I*2^(1/2), m = 1, n = 1}</code><br><code>-3.011056351-1.038481291*I <- {z = 2^(1/2)+I*2^(1/2), m = 1, n = 2}</code><br><code>-.537912592-.9060977875*I <- {z = 2^(1/2)+I*2^(1/2), m = 1, n = 3}</code><br><code>6.264823649-1.883544623*I <- {z = 2^(1/2)+I*2^(1/2), m = 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[-6.264823652701258, 1.8835446212245168] <- {Rule[m, 1], Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-3.0110563535593116, -1.0384812903661842] <- {Rule[m, 1], Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-0.537912595321433, -0.9060977859421694] <- {Rule[m, 1], Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[6.264823652701258, -1.8835446212245168] <- {Rule[m, 2], Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.34#Ex1 10.34#Ex1] || [[Item:Q3548|<math>\modBesselI{\nu}@{\conj{z}} = \conj{\modBesselI{\nu}@{z}}</math>]] || <code>BesselI(nu, conjugate(z))= conjugate(BesselI(nu, z))</code> || <code>BesselI[\[Nu], Conjugate[z]]= Conjugate[BesselI[\[Nu], z]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>-3.044981713-1.831851844*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}</code><br><code>3.044981713-1.831851844*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}</code><br><code>25.45117532+34.79009675*I <- {nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}</code><br><code>-25.45117532+34.79009675*I <- {nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}</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[-3.0449817151811125, -1.8318518429593253] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[3.0449817151811125, 1.8318518429593253] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[6.0769632034829115, 4.112580738730825] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-6.0769632034829115, -4.112580738730825] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.34#Ex2 10.34#Ex2] || [[Item:Q3549|<math>\modBesselK{\nu}@{\conj{z}} = \conj{\modBesselK{\nu}@{z}}</math>]] || <code>BesselK(nu, conjugate(z))= conjugate(BesselK(nu, z))</code> || <code>BesselK[\[Nu], Conjugate[z]]= Conjugate[BesselK[\[Nu], z]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>-.2418444739-.2420650681*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}</code><br><code>.2418444739-.2420650681*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}</code><br><code>-9.672734607+.86628375e-1*I <- {nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}</code><br><code>9.672734607+.86628375e-1*I <- {nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}</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[-0.2418444736872933, -0.24206506816430606] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[0.2418444736872933, 0.24206506816430606] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-0.2418444736872933, -0.24206506816430606] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[0.2418444736872933, 0.24206506816430606] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}</code><br>... skip entries to safe data<br></div></div>  

| [https://dlmf.nist.gov/10.35.E3 10.35.E3] || [[Item:Q3552|<math>e^{z\sin@@{\theta}} = \modBesselI{0}@{z}+2\sum_{k=0}^{\infty}(-1)^{k}\modBesselI{2k+1}@{z}\sin@{(2k+1)\theta}+2\sum_{k=1}^{\infty}(-1)^{k}\modBesselI{2k}@{z}\cos@{2k\theta}</math>]] || <code>exp(z*sin(theta))= BesselI(0, z)+ 2*sum((- 1)^(k)* BesselI(2*k + 1, z)*sin((2*k + 1)* theta), k = 0..infinity)+ 2*sum((- 1)^(k)* BesselI(2*k, z)*cos(2*k*theta), k = 1..infinity)</code> || <code>Exp[z*Sin[\[Theta]]]= BesselI[0, z]+ 2*Sum[(- 1)^(k)* BesselI[2*k + 1, z]*Sin[(2*k + 1)* \[Theta]], {k, 0, Infinity}]+ 2*Sum[(- 1)^(k)* BesselI[2*k, z]*Cos[2*k*\[Theta]], {k, 1, Infinity}]</code> || Failure || Failure || Skip || Skip

| [https://dlmf.nist.gov/10.37.E1 10.37.E1] || [[Item:Q3559|<math>|\modBesselK{\nu}@{z}| < |\modBesselK{\mu}@{z}|</math>]] || <code>abs(BesselK(nu, z))<abs(BesselK(mu, z))</code> || <code>Abs[BesselK[\[Nu], z]]<Abs[BesselK[\[Mu], z]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Fail<div class="mw-collapsible-content"><code>.3485691514 < .3485691514 <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}</code><br><code>.1206554296 < .1206554296 <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}</code><br><code>1.592895962 < 1.592895962 <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}</code><br><code>10.33727274 < 10.33727274 <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}</code><br>... skip entries to safe data<br></div></div> || Successful