Theta Functions - 20.10 Integrals

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Mathematica
20.10.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}x^{s-1}\Jacobithetatau{2}@{0}{ix^{2}}\diff{x} = 2^{s}(1-2^{-s})\pi^{-s/2}\EulerGamma@{\tfrac{1}{2}s}\Riemannzeta@{s}}
\int_{0}^{\infty}x^{s-1}\Jacobithetatau{2}@{0}{ix^{2}}\diff{x} = 2^{s}(1-2^{-s})\pi^{-s/2}\EulerGamma@{\tfrac{1}{2}s}\Riemannzeta@{s}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(\tfrac{1}{2}s)} > 0} 
int((x)^(s - 1)* JacobiTheta2(0,exp(I*Pi*I*(x)^(2))), x = 0..infinity) = (2)^(s)*(1 - (2)^(- s))*(Pi)^(- s/2)* GAMMA((1)/(2)*s)*Zeta(s)

Integrate[(x)^(s - 1)* EllipticTheta[2, 0, Exp[I*Pi*(I*(x)^(2))]], {x, 0, Infinity}, GenerateConditions->None] == (2)^(s)*(1 - (2)^(- s))*(Pi)^(- s/2)* Gamma[Divide[1,2]*s]*Zeta[s]
Failure Failure Error
Failed [1 / 3]
Result: Plus[1.6473133477045354, NIntegrate[Times[Power[x, -0.5], EllipticTheta[2, 0, Power[E, Times[-1, Pi, Power[x, 2]]]]]
Test Values: {x, 0, DirectedInfinity[1]}]], {Rule[s, 0.5]}

20.10.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}x^{s-1}(\Jacobithetatau{3}@{0}{ix^{2}}-1)\diff{x} = \pi^{-s/2}\EulerGamma@{\tfrac{1}{2}s}\Riemannzeta@{s}}
\int_{0}^{\infty}x^{s-1}(\Jacobithetatau{3}@{0}{ix^{2}}-1)\diff{x} = \pi^{-s/2}\EulerGamma@{\tfrac{1}{2}s}\Riemannzeta@{s}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(\tfrac{1}{2}s)} > 0} 
int((x)^(s - 1)*(JacobiTheta3(0,exp(I*Pi*I*(x)^(2)))- 1), x = 0..infinity) = (Pi)^(- s/2)* GAMMA((1)/(2)*s)*Zeta(s)

Integrate[(x)^(s - 1)*(EllipticTheta[3, 0, Exp[I*Pi*(I*(x)^(2))]]- 1), {x, 0, Infinity}, GenerateConditions->None] == (Pi)^(- s/2)* Gamma[Divide[1,2]*s]*Zeta[s]
Failure Failure Skipped - Because timed out Skipped - Because timed out
20.10.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}x^{s-1}(1-\Jacobithetatau{4}@{0}{ix^{2}})\diff{x} = (1-2^{1-s})\pi^{-s/2}\EulerGamma@{\tfrac{1}{2}s}\Riemannzeta@{s}}
\int_{0}^{\infty}x^{s-1}(1-\Jacobithetatau{4}@{0}{ix^{2}})\diff{x} = (1-2^{1-s})\pi^{-s/2}\EulerGamma@{\tfrac{1}{2}s}\Riemannzeta@{s}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(\tfrac{1}{2}s)} > 0} 
int((x)^(s - 1)*(1 - JacobiTheta4(0,exp(I*Pi*I*(x)^(2)))), x = 0..infinity) = (1 - (2)^(1 - s))*(Pi)^(- s/2)* GAMMA((1)/(2)*s)*Zeta(s)

Integrate[(x)^(s - 1)*(1 - EllipticTheta[4, 0, Exp[I*Pi*(I*(x)^(2))]]), {x, 0, Infinity}, GenerateConditions->None] == (1 - (2)^(1 - s))*(Pi)^(- s/2)* Gamma[Divide[1,2]*s]*Zeta[s]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
20.10.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}e^{-st}\Jacobithetatau{1}@{\frac{\beta\pi}{2\ell}}{\frac{i\pi t}{\ell^{2}}}\diff{t} = \int_{0}^{\infty}e^{-st}\Jacobithetatau{2}@{\frac{(1+\beta)\pi}{2\ell}}{\frac{i\pi t}{\ell^{2}}}\diff{t}}
\int_{0}^{\infty}e^{-st}\Jacobithetatau{1}@{\frac{\beta\pi}{2\ell}}{\frac{i\pi t}{\ell^{2}}}\diff{t} = \int_{0}^{\infty}e^{-st}\Jacobithetatau{2}@{\frac{(1+\beta)\pi}{2\ell}}{\frac{i\pi t}{\ell^{2}}}\diff{t}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(exp(- s*t)*JacobiTheta1((beta*Pi)/(2*ell),exp(I*Pi*(I*Pi*t)/((ell)^(2)))), t = 0..infinity) = int(exp(- s*t)*JacobiTheta2(((1 + beta)*Pi)/(2*ell),exp(I*Pi*(I*Pi*t)/((ell)^(2)))), t = 0..infinity)

Integrate[Exp[- s*t]*EllipticTheta[1, Divide[\[Beta]*Pi,2*\[ScriptL]], Exp[I*Pi*(Divide[I*Pi*t,\[ScriptL]^(2)])]], {t, 0, Infinity}, GenerateConditions->None] == Integrate[Exp[- s*t]*EllipticTheta[2, Divide[(1 + \[Beta])*Pi,2*\[ScriptL]], Exp[I*Pi*(Divide[I*Pi*t,\[ScriptL]^(2)])]], {t, 0, Infinity}, GenerateConditions->None]
Failure Failure Error
Failed [9 / 9]
Result: Plus[Times[-1.0, stIntegrate[Times[2.718281828459045, EllipticTheta[1, 2.356194490192345, Power[2.718281828459045, Times[-9.869604401089358, t]]]]
Test Values: {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], stIntegrate[Times[2.718281828459045, EllipticTheta[2, 3.9269908169872414, Power[2.718281828459045, Times[-9.869604401089358, t]]]], {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[ℓ, 1], Rule[β, Rational[3, 2]]}


Result: Plus[Times[-1.0, stIntegrate[Times[2.718281828459045, EllipticTheta[1, 1.1780972450961724, Power[2.718281828459045, Times[-2.4674011002723395, t]]]]
Test Values: {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], stIntegrate[Times[2.718281828459045, EllipticTheta[2, 1.9634954084936207, Power[2.718281828459045, Times[-2.4674011002723395, t]]]], {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[ℓ, 2], Rule[β, Rational[3, 2]]}

... skip entries to safe data
20.10.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}e^{-st}\Jacobithetatau{2}@{\frac{(1+\beta)\pi}{2\ell}}{\frac{i\pi t}{\ell^{2}}}\diff{t} = -\frac{\ell}{\sqrt{s}}\sinh@{\beta\sqrt{s}}\sech@{\ell\sqrt{s}}}
\int_{0}^{\infty}e^{-st}\Jacobithetatau{2}@{\frac{(1+\beta)\pi}{2\ell}}{\frac{i\pi t}{\ell^{2}}}\diff{t} = -\frac{\ell}{\sqrt{s}}\sinh@{\beta\sqrt{s}}\sech@{\ell\sqrt{s}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(exp(- s*t)*JacobiTheta2(((1 + beta)*Pi)/(2*ell),exp(I*Pi*(I*Pi*t)/((ell)^(2)))), t = 0..infinity) = -(ell)/(sqrt(s))*sinh(beta*sqrt(s))*sech(ell*sqrt(s))

Integrate[Exp[- s*t]*EllipticTheta[2, Divide[(1 + \[Beta])*Pi,2*\[ScriptL]], Exp[I*Pi*(Divide[I*Pi*t,\[ScriptL]^(2)])]], {t, 0, Infinity}, GenerateConditions->None] == -Divide[\[ScriptL],Sqrt[s]]*Sinh[\[Beta]*Sqrt[s]]*Sech[\[ScriptL]*Sqrt[s]]
Failure Failure Error
Failed [54 / 54]
Result: Plus[2.32235875408619, Times[-1.0, stIntegrate[Times[2.718281828459045, EllipticTheta[2, 3.9269908169872414, Power[2.718281828459045, Times[-9.869604401089358, t]]]]
Test Values: {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, Rational[-3, 2]], Rule[ℓ, 1], Rule[β, Rational[3, 2]]}


Result: Plus[-2.046254548704581, Times[-1.0, stIntegrate[Times[2.718281828459045, EllipticTheta[2, 1.9634954084936207, Power[2.718281828459045, Times[-2.4674011002723395, t]]]]
Test Values: {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, Rational[-3, 2]], Rule[ℓ, 2], Rule[β, Rational[3, 2]]}

... skip entries to safe data
20.10.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}e^{-st}\Jacobithetatau{3}@{\frac{(1+\beta)\pi}{2\ell}}{\frac{i\pi t}{\ell^{2}}}\diff{t} = \int_{0}^{\infty}e^{-st}\Jacobithetatau{4}@{\frac{\beta\pi}{2\ell}}{\frac{i\pi t}{\ell^{2}}}\diff{t}}
\int_{0}^{\infty}e^{-st}\Jacobithetatau{3}@{\frac{(1+\beta)\pi}{2\ell}}{\frac{i\pi t}{\ell^{2}}}\diff{t} = \int_{0}^{\infty}e^{-st}\Jacobithetatau{4}@{\frac{\beta\pi}{2\ell}}{\frac{i\pi t}{\ell^{2}}}\diff{t}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(exp(- s*t)*JacobiTheta3(((1 + beta)*Pi)/(2*ell),exp(I*Pi*(I*Pi*t)/((ell)^(2)))), t = 0..infinity) = int(exp(- s*t)*JacobiTheta4((beta*Pi)/(2*ell),exp(I*Pi*(I*Pi*t)/((ell)^(2)))), t = 0..infinity)

Integrate[Exp[- s*t]*EllipticTheta[3, Divide[(1 + \[Beta])*Pi,2*\[ScriptL]], Exp[I*Pi*(Divide[I*Pi*t,\[ScriptL]^(2)])]], {t, 0, Infinity}, GenerateConditions->None] == Integrate[Exp[- s*t]*EllipticTheta[4, Divide[\[Beta]*Pi,2*\[ScriptL]], Exp[I*Pi*(Divide[I*Pi*t,\[ScriptL]^(2)])]], {t, 0, Infinity}, GenerateConditions->None]
Failure Failure Error
Failed [3 / 3]
Result: Plus[Times[-1.0, stIntegrate[Times[2.718281828459045, EllipticTheta[3, Times[3.9269908169872414, Power[ℓ, -1]], Power[2.718281828459045, Times[-9.869604401089358, t, Power[ℓ, -2]]]]]
Test Values: {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], stIntegrate[Times[2.718281828459045, EllipticTheta[4, Times[2.356194490192345, Power[ℓ, -1]], Power[2.718281828459045, Times[-9.869604401089358, t, Power[ℓ, -2]]]]], {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[β, 1.5]}


Result: Plus[Times[-1.0, stIntegrate[Times[2.718281828459045, EllipticTheta[3, Times[2.356194490192345, Power[ℓ, -1]], Power[2.718281828459045, Times[-9.869604401089358, t, Power[ℓ, -2]]]]]
Test Values: {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], stIntegrate[Times[2.718281828459045, EllipticTheta[4, Times[0.7853981633974483, Power[ℓ, -1]], Power[2.718281828459045, Times[-9.869604401089358, t, Power[ℓ, -2]]]]], {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[β, 0.5]}

... skip entries to safe data
20.10.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}e^{-st}\Jacobithetatau{4}@{\frac{\beta\pi}{2\ell}}{\frac{i\pi t}{\ell^{2}}}\diff{t} = \frac{\ell}{\sqrt{s}}\cosh@{\beta\sqrt{s}}\csch@{\ell\sqrt{s}}}
\int_{0}^{\infty}e^{-st}\Jacobithetatau{4}@{\frac{\beta\pi}{2\ell}}{\frac{i\pi t}{\ell^{2}}}\diff{t} = \frac{\ell}{\sqrt{s}}\cosh@{\beta\sqrt{s}}\csch@{\ell\sqrt{s}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(exp(- s*t)*JacobiTheta4((beta*Pi)/(2*ell),exp(I*Pi*(I*Pi*t)/((ell)^(2)))), t = 0..infinity) = (ell)/(sqrt(s))*cosh(beta*sqrt(s))*csch(ell*sqrt(s))

Integrate[Exp[- s*t]*EllipticTheta[4, Divide[\[Beta]*Pi,2*\[ScriptL]], Exp[I*Pi*(Divide[I*Pi*t,\[ScriptL]^(2)])]], {t, 0, Infinity}, GenerateConditions->None] == Divide[\[ScriptL],Sqrt[s]]*Cosh[\[Beta]*Sqrt[s]]*Csch[\[ScriptL]*Sqrt[s]]
Failure Failure Error
Failed [18 / 18]
Result: Plus[Times[Complex[-0.21488876057872602, 0.0], ℓ, Csc[Times[Complex[1.224744871391589, 0.0], ℓ]]], Times[-1.0, stIntegrate[Times[2.718281828459045, EllipticTheta[4, Times[2.356194490192345, Power[ℓ, -1]], Power[2.718281828459045, Times[-9.869604401089358, t, Power[ℓ, -2]]]]]
Test Values: {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, -1.5], Rule[β, 1.5]}


Result: Plus[Times[Complex[0.668128228457918, 0.0], ℓ, Csc[Times[Complex[1.224744871391589, 0.0], ℓ]]], Times[-1.0, stIntegrate[Times[2.718281828459045, EllipticTheta[4, Times[0.7853981633974483, Power[ℓ, -1]], Power[2.718281828459045, Times[-9.869604401089358, t, Power[ℓ, -2]]]]]
Test Values: {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, -1.5], Rule[β, 0.5]}

... skip entries to safe data
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