Lamé Functions - 29.8 Integral Equations
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| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|---|
| 29.8.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x = k^{2}\Jacobiellsnk@{z}{k}\Jacobiellsnk@{z_{1}}{k}\Jacobiellsnk@{z_{2}}{k}\Jacobiellsnk@{z_{3}}{k}-\frac{k^{2}}{{k^{\prime}}^{2}}\Jacobiellcnk@{z}{k}\Jacobiellcnk@{z_{1}}{k}\Jacobiellcnk@{z_{2}}{k}\Jacobiellcnk@{z_{3}}{k}+\frac{1}{{k^{\prime}}^{2}}\Jacobielldnk@{z}{k}\Jacobielldnk@{z_{1}}{k}\Jacobielldnk@{z_{2}}{k}\Jacobielldnk@{z_{3}}{k}}
x = k^{2}\Jacobiellsnk@{z}{k}\Jacobiellsnk@{z_{1}}{k}\Jacobiellsnk@{z_{2}}{k}\Jacobiellsnk@{z_{3}}{k}-\frac{k^{2}}{{k^{\prime}}^{2}}\Jacobiellcnk@{z}{k}\Jacobiellcnk@{z_{1}}{k}\Jacobiellcnk@{z_{2}}{k}\Jacobiellcnk@{z_{3}}{k}+\frac{1}{{k^{\prime}}^{2}}\Jacobielldnk@{z}{k}\Jacobielldnk@{z_{1}}{k}\Jacobielldnk@{z_{2}}{k}\Jacobielldnk@{z_{3}}{k} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | x = (k)^(2)* JacobiSN(x + y*I, k)*JacobiSN(x + y*I[1], k)*JacobiSN(x + y*I[2], k)*JacobiSN(x + y*I[3], k)-((k)^(2))/(1 - (k)^(2))*JacobiCN(x + y*I, k)*JacobiCN(x + y*I[1], k)*JacobiCN(x + y*I[2], k)*JacobiCN(x + y*I[3], k)+(1)/(1 - (k)^(2))*JacobiDN(x + y*I, k)*JacobiDN(x + y*I[1], k)*JacobiDN(x + y*I[2], k)*JacobiDN(x + y*I[3], k)
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x == (k)^(2)* JacobiSN[x + y*I, (k)^2]*JacobiSN[Subscript[x + y*I, 1], (k)^2]*JacobiSN[Subscript[x + y*I, 2], (k)^2]*JacobiSN[Subscript[x + y*I, 3], (k)^2]-Divide[(k)^(2),1 - (k)^(2)]*JacobiCN[x + y*I, (k)^2]*JacobiCN[Subscript[x + y*I, 1], (k)^2]*JacobiCN[Subscript[x + y*I, 2], (k)^2]*JacobiCN[Subscript[x + y*I, 3], (k)^2]+Divide[1,1 - (k)^(2)]*JacobiDN[x + y*I, (k)^2]*JacobiDN[Subscript[x + y*I, 1], (k)^2]*JacobiDN[Subscript[x + y*I, 2], (k)^2]*JacobiDN[Subscript[x + y*I, 3], (k)^2]
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Failure | Aborted | Error | Failed [54 / 54]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[x, 1.5], Rule[y, -1.5]}
Result: Plus[1.5, Times[Complex[1.6461554600232724, 0.45267954815584505], JacobiCN[Subscript[Complex[1.5, -1.5], 1], 4.0], JacobiCN[Subscript[Complex[1.5, -1.5], 2], 4.0], JacobiCN[Subscript[Complex[1.5, -1.5], 3], 4.0]], Times[Complex[0.619203045121549, 0.30086290583863873], JacobiDN[Subscript[Complex[1.5, -1.5], 1], 4.0], JacobiDN[Subscript[Complex[1.5, -1.5], 2], 4.0], JacobiDN[Subscript[Complex[1.5, -1.5], 3], 4.0]], Times[Complex[-2.0469921952210957, 3.2763330530501245], JacobiSN[Subscript[Complex[1.5, -1.5], 1], 4.0], JacobiSN[Subscript[Complex[1.5, -1.5], 2], 4.0], JacobiSN[Subscript[Complex[1.5, -1.5], 3], 4.0]]]
Test Values: {Rule[k, 2], Rule[x, 1.5], Rule[y, -1.5]}
... skip entries to safe data |
| 29.8.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \mu w(z_{1})w(z_{2})w(z_{3}) = \int_{-2\compellintKk@@{k}}^{2\compellintKk@@{k}}\FerrersP[]{\nu}@{x}w(z)\diff{z}}
\mu w(z_{1})w(z_{2})w(z_{3}) = \int_{-2\compellintKk@@{k}}^{2\compellintKk@@{k}}\FerrersP[]{\nu}@{x}w(z)\diff{z} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | mu*w*(x + y*I[1])*w*(x + y*I[2])*w*(x + y*I[3]) = int(LegendreP(nu, x)*w*((x + y*I)), (x + y*I) = - 2*EllipticK(k)..2*EllipticK(k))
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\[Mu]*w*(Subscript[x + y*I, 1])*w*(Subscript[x + y*I, 2])*w*(Subscript[x + y*I, 3]) == Integrate[LegendreP[\[Nu], x]*w*((x + y*I)), {(x + y*I), - 2*EllipticK[(k)^2], 2*EllipticK[(k)^2]}, GenerateConditions->None]
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Error | Failure | - | Failed [300 / 300]
Result: Plus[Times[-1.0, NIntegrate[Complex[2.8644916021274596, 0.010098545944192239]
Test Values: {Complex[1.5, -1.5], DirectedInfinity[], DirectedInfinity[]}]], Times[Complex[-0.49999999999999994, 0.8660254037844387], Subscript[Complex[1.5, -1.5], 1], Subscript[Complex[1.5, -1.5], 2], Subscript[Complex[1.5, -1.5], 3]]], {Rule[k, 1], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[y, -1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Plus[Times[-1.0, NIntegrate[Complex[2.8644916021274596, 0.010098545944192239]
Test Values: {Complex[1.5, -1.5], Times[-2, EllipticK[4]], Times[2, EllipticK[4]]}]], Times[Complex[-0.49999999999999994, 0.8660254037844387], Subscript[Complex[1.5, -1.5], 1], Subscript[Complex[1.5, -1.5], 2], Subscript[Complex[1.5, -1.5], 3]]], {Rule[k, 2], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[y, -1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
... skip entries to safe data |
| 29.8.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \mu = \frac{2\sigma\tau}{\Wronskian@{w,w_{2}}}}
\mu = \frac{2\sigma\tau}{\Wronskian@{w,w_{2}}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | mu = (2*sigma*tau)/((w)*diff(w[2], w)-diff(w, w)*(w[2]))
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\[Mu] == Divide[2*\[Sigma]*\[Tau],Wronskian[{w, Subscript[w, 2]}, w]]
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Failure | Failure | Failed [300 / 300] Result: 2.598076212+1.500000000*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, sigma = 1/2*3^(1/2)+1/2*I, tau = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, w[2] = 1/2*3^(1/2)+1/2*I}
Result: 1.866025404-1.232050808*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, sigma = 1/2*3^(1/2)+1/2*I, tau = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, w[2] = -1/2+1/2*I*3^(1/2)}
... skip entries to safe data |
Failed [300 / 300]
Result: Plus[Complex[0.8660254037844387, 0.49999999999999994], Times[Complex[-1.0000000000000002, -1.7320508075688772], Power[Plus[Complex[-0.8660254037844387, -0.49999999999999994], Times[Complex[0.8660254037844387, 0.49999999999999994], Derivative[1, 0][Subscript][Complex[0.8660254037844387, 0.49999999999999994], 2.0]]], -1]]]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[σ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[w, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Plus[Complex[0.8660254037844387, 0.49999999999999994], Times[Complex[-1.0000000000000002, -1.7320508075688772], Power[Plus[Complex[0.4999999999999998, -0.8660254037844387], Times[Complex[0.8660254037844387, 0.49999999999999994], Derivative[1, 0][Subscript][Complex[0.8660254037844387, 0.49999999999999994], 2.0]]], -1]]]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[σ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[w, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
| 29.8#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle w(z+2\compellintKk@@{k}) = \sigma w(z)}
w(z+2\compellintKk@@{k}) = \sigma w(z) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | w(z + 2*EllipticK(k)) = sigma*w(z)
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w[z + 2*EllipticK[(k)^2]] == \[Sigma]*w[z]
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Failure | Failure | Error | Failed [300 / 300]
Result: DirectedInfinity[]
Test Values: {Rule[k, 1], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[σ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Complex[3.038160455456161, -1.1586967532026022]
Test Values: {Rule[k, 2], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[σ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
... skip entries to safe data |
| 29.8#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle w_{2}(z+2\compellintKk@@{k}) = \tau w(z)+\sigma w_{2}(z)}
w_{2}(z+2\compellintKk@@{k}) = \tau w(z)+\sigma w_{2}(z) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | w[2](z + 2*EllipticK(k)) = tau*w(z)+ sigma*w[2](z)
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Subscript[w, 2][z + 2*EllipticK[(k)^2]] == \[Tau]*w[z]+ \[Sigma]*Subscript[w, 2][z]
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Failure | Failure | Error | Failed [300 / 300]
Result: DirectedInfinity[]
Test Values: {Rule[k, 1], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[σ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[w, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Complex[3.038160455456161, -2.1586967532026025]
Test Values: {Rule[k, 2], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[σ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[w, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
... skip entries to safe data |
| 29.8.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle y = \frac{1}{k^{\prime}}\Jacobielldnk@{z}{k}\Jacobielldnk@{z_{1}}{k}}
y = \frac{1}{k^{\prime}}\Jacobielldnk@{z}{k}\Jacobielldnk@{z_{1}}{k} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | y = (1)/(sqrt(1 - (k)^(2)))*JacobiDN(x + y*I, k)*JacobiDN(x + y*I[1], k)
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y == Divide[1,Sqrt[1 - (k)^(2)]]*JacobiDN[x + y*I, (k)^2]*JacobiDN[Subscript[x + y*I, 1], (k)^2]
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Failure | Aborted | Error | Failed [54 / 54]
Result: DirectedInfinity[]
Test Values: {Rule[k, 1], Rule[x, 1.5], Rule[y, -1.5]}
Result: Plus[-1.5, Times[Complex[-0.5211098390253335, 1.072491134351887], JacobiDN[Subscript[Complex[1.5, -1.5], 1], 4.0]]]
Test Values: {Rule[k, 2], Rule[x, 1.5], Rule[y, -1.5]}
... skip entries to safe data |