View source for 28.28

{{DISPLAYTITLE:Mathieu Functions and Hill’s Equation - 28.28 Integrals, Integral Representations, and Integral Equations}}

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! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! scope="col" style="position: sticky; top: 0;" | Constraints 
! scope="col" style="position: sticky; top: 0;" | Maple
! scope="col" style="position: sticky; top: 0;" | Mathematica
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/28.28.E1 28.28.E1] || <math qid="Q8422">w = \cosh@@{z}\cos@@{t}\cos@@{\alpha}+\sinh@@{z}\sin@@{t}\sin@@{\alpha}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>w = \cosh@@{z}\cos@@{t}\cos@@{\alpha}+\sinh@@{z}\sin@@{t}\sin@@{\alpha}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>w = cosh(z)*cos(t)*cos(alpha)+ sinh(z)*sin(t)*sin(alpha)</syntaxhighlight> || <syntaxhighlight lang=mathematica>w == Cosh[z]*Cos[t]*Cos[\[Alpha]]+ Sinh[z]*Sin[t]*Sin[\[Alpha]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [299 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.714222282+1.165028049*I
Test Values: {alpha = 3/2, t = -3/2, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .5264627339+1.356668447*I
Test Values: {alpha = 3/2, t = -3/2, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [298 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.7142222818783819, 1.165028048919159]
Test Values: {Rule[t, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.2004296775262544, 0.7916410797173274]
Test Values: {Rule[t, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/28.28.E10 28.28.E10] || <math qid="Q8434">0 < \phase@{h(\cosh@@{z}+ 1)}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>0 < \phase@{h(\cosh@@{z}+ 1)}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>0 < argument(h*(cosh(z)+ 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>0 < Arg[h*(Cosh[z]+ 1)]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [35 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 0. < -.8396703302
Test Values: {h = 1/2-1/2*I*3^(1/2), z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 0. < -1.272675688
Test Values: {h = 1/2-1/2*I*3^(1/2), z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [35 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: False
Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: False
Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/28.28.E10 28.28.E10] || <math qid="Q8434">0 < \phase@{h(\cosh@@{z}- 1)}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>0 < \phase@{h(\cosh@@{z}- 1)}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>0 < argument(h*(cosh(z)- 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>0 < Arg[h*(Cosh[z]- 1)]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [35 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 0. < -1.643566335
Test Values: {h = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 0. < -1.643566335
Test Values: {h = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [35 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: False
Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: False
Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/28.28.E10 28.28.E10] || <math qid="Q8434">\phase@{h(\cosh@@{z}+ 1)} < \pi</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\phase@{h(\cosh@@{z}+ 1)} < \pi</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>argument(h*(cosh(z)+ 1)) < Pi</syntaxhighlight> || <syntaxhighlight lang=mathematica>Arg[h*(Cosh[z]+ 1)] < Pi</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 3.141592654 < 3.141592654
Test Values: {h = -3/2, z = 3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 3.141592654 < 3.141592654
Test Values: {h = -3/2, z = 1/2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: False
Test Values: {Rule[h, -1.5], Rule[z, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: False
Test Values: {Rule[h, -1.5], Rule[z, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/28.28.E10 28.28.E10] || <math qid="Q8434">\phase@{h(\cosh@@{z}- 1)} < \pi</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\phase@{h(\cosh@@{z}- 1)} < \pi</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>argument(h*(cosh(z)- 1)) < Pi</syntaxhighlight> || <syntaxhighlight lang=mathematica>Arg[h*(Cosh[z]- 1)] < Pi</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 3.141592654 < 3.141592654
Test Values: {h = -3/2, z = 3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 3.141592654 < 3.141592654
Test Values: {h = -3/2, z = 1/2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: False
Test Values: {Rule[h, -1.5], Rule[z, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: False
Test Values: {Rule[h, -1.5], Rule[z, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/28.28#Ex4 28.28#Ex4] || <math qid="Q8442">R(z,t) = \left(\tfrac{1}{2}(\cosh@{2z}+\cos@{2t})\right)^{\ifrac{1}{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>R(z,t) = \left(\tfrac{1}{2}(\cosh@{2z}+\cos@{2t})\right)^{\ifrac{1}{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>R(z , t) = ((1)/(2)*(cosh(2*z)+ cos(2*t)))^((1)/(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>R[z , t] == (Divide[1,2]*(Cosh[2*z]+ Cos[2*t]))^(Divide[1,2])</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: (.8660254040+.5000000000*I)*(.8660254040+.5000000000*I, -1.500000000)-.8604472605-.6693200135*I
Test Values: {R = 1/2*3^(1/2)+1/2*I, t = -3/2, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: (.8660254040+.5000000000*I)*(-.5000000000+.8660254040*I, -1.500000000)-.3385916178+.8564557052*I
Test Values: {R = 1/2*3^(1/2)+1/2*I, t = -3/2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Error
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| [https://dlmf.nist.gov/28.28#Ex5 28.28#Ex5] || <math qid="Q8443">R(z,0) = \cosh@@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>R(z,0) = \cosh@@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>R(z , 0) = cosh(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>R[z , 0] == Cosh[z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: (.8660254040+.5000000000*I)*(.8660254040+.5000000000*I, 0.)-1.227765517-.4690753764*I
Test Values: {R = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: (.8660254040+.5000000000*I)*(-.5000000000+.8660254040*I, 0.)-.7305430189+.3969495503*I
Test Values: {R = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Error
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| [https://dlmf.nist.gov/28.28#Ex6 28.28#Ex6] || <math qid="Q8444">e^{2\iunit\phi} = \dfrac{\cosh@{z+\iunit t}}{\cosh@{z-\iunit t}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{2\iunit\phi} = \dfrac{\cosh@{z+\iunit t}}{\cosh@{z-\iunit t}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>exp(2*I*phi) = (cosh(z + I*t))/(cosh(z - I*t))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[2*I*\[Phi]] == Divide[Cosh[z + I*t],Cosh[z - I*t]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .9781641542+.5339822543*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, t = -3/2, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.021212458+.2569827752*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, t = -3/2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.978164154574313, 0.5339822543847044]
Test Values: {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.1328205399920523, 0.022001382090719362]
Test Values: {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/28.28#Ex7 28.28#Ex7] || <math qid="Q8445">\phi(z,0) = 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\phi(z,0) = 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">phi(z , 0) = 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">\[Phi][z , 0] == 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/28.28.E28 28.28.E28] || <math qid="Q8455">\alpha^{(1)}_{\nu,m} = \dfrac{1}{2\pi}\int_{0}^{2\pi}\sin@@{t}\Mathieume{\nu}@{t}{h^{2}}\Mathieume{-\nu-2m-1}@{t}{h^{2}}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\alpha^{(1)}_{\nu,m} = \dfrac{1}{2\pi}\int_{0}^{2\pi}\sin@@{t}\Mathieume{\nu}@{t}{h^{2}}\Mathieume{-\nu-2m-1}@{t}{h^{2}}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>(Subscript[\[Alpha], \[Nu], m])^(1) == Divide[1,2*Pi]*Integrate[Sin[t]*Sqrt[2]*MathieuC[\[Nu], (h)^(2), t]*Sqrt[2]*MathieuC[- \[Nu]- 2*m - 1, (h)^(2), t], {t, 0, 2*Pi}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Skipped - Because timed out
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| [https://dlmf.nist.gov/28.28.E41 28.28.E41] || <math qid="Q8472">\dfrac{\cosh@@{z}}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\sin@@{t}\Mathieuse{n}@{t}{h^{2}}\Mathieuce{m}@{t}{h^{2}}}{\sinh^{2}@@{z}+\sin^{2}@@{t}}\diff{t} = (-1)^{p+1}\iunit h\widehat{\beta}_{n,m}\radMathieuDsc{0}@{n}{m}{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\dfrac{\cosh@@{z}}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\sin@@{t}\Mathieuse{n}@{t}{h^{2}}\Mathieuce{m}@{t}{h^{2}}}{\sinh^{2}@@{z}+\sin^{2}@@{t}}\diff{t} = (-1)^{p+1}\iunit h\widehat{\beta}_{n,m}\radMathieuDsc{0}@{n}{m}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(cosh(z))/((Pi)^(2))*int((sin(t)*MathieuSE(n, (h)^(2), t)*MathieuCE(m, (h)^(2), t))/((sinh(z))^(2)+ (sin(t))^(2)), t = 0..2*Pi) = (- 1)^(p + 1)* I*h*((1)/(2*Pi)*int(sin(t)*MathieuSE(n, (h)^(2), t)*MathieuCE(m, (h)^(2), t), t = 0..2*Pi))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Cosh[z],(Pi)^(2)]*Integrate[Divide[Sin[t]*MathieuS[n, (h)^(2), t]*MathieuC[m, (h)^(2), t],(Sinh[z])^(2)+ (Sin[t])^(2)], {t, 0, 2*Pi}, GenerateConditions->None] == (- 1)^(p + 1)* I*h*(Divide[1,2*Pi]*Integrate[Sin[t]*MathieuS[n, (h)^(2), t]*MathieuC[m, (h)^(2), t], {t, 0, 2*Pi}, GenerateConditions->None])</syntaxhighlight> || Missing Macro Error || Missing Macro Error || - || -
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| [https://dlmf.nist.gov/28.28.E42 28.28.E42] || <math qid="Q8473">\dfrac{\sinh@@{z}}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\cos@@{t}\Mathieuse{n}'@{t}{h^{2}}\Mathieuce{m}@{t}{h^{2}}}{\sinh^{2}@@{z}+\sin^{2}@@{t}}\diff{t} = (-1)^{p}\iunit h\widehat{\beta}_{n,m}\radMathieuDsc{1}@{n}{m}{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\dfrac{\sinh@@{z}}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\cos@@{t}\Mathieuse{n}'@{t}{h^{2}}\Mathieuce{m}@{t}{h^{2}}}{\sinh^{2}@@{z}+\sin^{2}@@{t}}\diff{t} = (-1)^{p}\iunit h\widehat{\beta}_{n,m}\radMathieuDsc{1}@{n}{m}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(sinh(z))/((Pi)^(2))*int((cos(t)*subs( temp=t, diff( MathieuSE(n, (h)^(2), temp), temp$(1) ) )*MathieuCE(m, (h)^(2), t))/((sinh(z))^(2)+ (sin(t))^(2)), t = 0..2*Pi) = (- 1)^(p)* I*h*((1)/(2*Pi)*int(sin(t)*MathieuSE(n, (h)^(2), t)*MathieuCE(m, (h)^(2), t), t = 0..2*Pi))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Sinh[z],(Pi)^(2)]*Integrate[Divide[Cos[t]*(D[MathieuS[n, (h)^(2), temp], {temp, 1}]/.temp-> t)*MathieuC[m, (h)^(2), t],(Sinh[z])^(2)+ (Sin[t])^(2)], {t, 0, 2*Pi}, GenerateConditions->None] == (- 1)^(p)* I*h*(Divide[1,2*Pi]*Integrate[Sin[t]*MathieuS[n, (h)^(2), t]*MathieuC[m, (h)^(2), t], {t, 0, 2*Pi}, GenerateConditions->None])</syntaxhighlight> || Missing Macro Error || Missing Macro Error || - || -
|- 
| [https://dlmf.nist.gov/28.28.E44 28.28.E44] || <math qid="Q8475">\dfrac{1}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\sin@{2t}\Mathieuse{n}@{t}{h^{2}}\Mathieuce{m}@{t}{h^{2}}}{\sinh^{2}@@{z}+\sin^{2}@@{t}}\diff{t} = (-1)^{p}\iunit\widehat{\gamma}_{n,m}\radMathieuDsc{0}@{n}{m}{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\dfrac{1}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\sin@{2t}\Mathieuse{n}@{t}{h^{2}}\Mathieuce{m}@{t}{h^{2}}}{\sinh^{2}@@{z}+\sin^{2}@@{t}}\diff{t} = (-1)^{p}\iunit\widehat{\gamma}_{n,m}\radMathieuDsc{0}@{n}{m}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(1)/((Pi)^(2))*int((sin(2*t)*MathieuSE(n, (h)^(2), t)*MathieuCE(m, (h)^(2), t))/((sinh(z))^(2)+ (sin(t))^(2)), t = 0..2*Pi) = (- 1)^(p)* I*((1)/(2*Pi)*int(subs( temp=t, diff( MathieuSE(n, (h)^(2), temp), temp$(1) ) )*MathieuCE(m, (h)^(2), t), t = 0..2*Pi))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,(Pi)^(2)]*Integrate[Divide[Sin[2*t]*MathieuS[n, (h)^(2), t]*MathieuC[m, (h)^(2), t],(Sinh[z])^(2)+ (Sin[t])^(2)], {t, 0, 2*Pi}, GenerateConditions->None] == (- 1)^(p)* I*(Divide[1,2*Pi]*Integrate[(D[MathieuS[n, (h)^(2), temp], {temp, 1}]/.temp-> t)*MathieuC[m, (h)^(2), t], {t, 0, 2*Pi}, GenerateConditions->None])</syntaxhighlight> || Missing Macro Error || Missing Macro Error || - || -
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| [https://dlmf.nist.gov/28.28.E45 28.28.E45] || <math qid="Q8476">\dfrac{\sinh@{2z}}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\Mathieuse{n}'@{t}{h^{2}}\Mathieuce{m}@{t}{h^{2}}}{\sinh^{2}@@{z}+\sin^{2}@@{t}}\diff{t} = (-1)^{p+1}\iunit\widehat{\gamma}_{n,m}\radMathieuDsc{1}@{n}{m}{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\dfrac{\sinh@{2z}}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\Mathieuse{n}'@{t}{h^{2}}\Mathieuce{m}@{t}{h^{2}}}{\sinh^{2}@@{z}+\sin^{2}@@{t}}\diff{t} = (-1)^{p+1}\iunit\widehat{\gamma}_{n,m}\radMathieuDsc{1}@{n}{m}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(sinh(2*z))/((Pi)^(2))*int((subs( temp=t, diff( MathieuSE(n, (h)^(2), temp), temp$(1) ) )*MathieuCE(m, (h)^(2), t))/((sinh(z))^(2)+ (sin(t))^(2)), t = 0..2*Pi) = (- 1)^(p + 1)* I*((1)/(2*Pi)*int(subs( temp=t, diff( MathieuSE(n, (h)^(2), temp), temp$(1) ) )*MathieuCE(m, (h)^(2), t), t = 0..2*Pi))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Sinh[2*z],(Pi)^(2)]*Integrate[Divide[(D[MathieuS[n, (h)^(2), temp], {temp, 1}]/.temp-> t)*MathieuC[m, (h)^(2), t],(Sinh[z])^(2)+ (Sin[t])^(2)], {t, 0, 2*Pi}, GenerateConditions->None] == (- 1)^(p + 1)* I*(Divide[1,2*Pi]*Integrate[(D[MathieuS[n, (h)^(2), temp], {temp, 1}]/.temp-> t)*MathieuC[m, (h)^(2), t], {t, 0, 2*Pi}, GenerateConditions->None])</syntaxhighlight> || Missing Macro Error || Missing Macro Error || - || -
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