Elliptic Integrals - 19.12 Asymptotic Approximations
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| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
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| 19.12.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \compellintKk@{k} = \sum_{m=0}^{\infty}\frac{\Pochhammersym{\tfrac{1}{2}}{m}\Pochhammersym{\tfrac{1}{2}}{m}}{m!\;m!}{k^{\prime}}^{2m}\left(\ln@@{\left(\frac{1}{k^{\prime}}\right)}+d(m)\right)}
\compellintKk@{k} = \sum_{m=0}^{\infty}\frac{\Pochhammersym{\tfrac{1}{2}}{m}\Pochhammersym{\tfrac{1}{2}}{m}}{m!\;m!}{k^{\prime}}^{2m}\left(\ln@@{\left(\frac{1}{k^{\prime}}\right)}+d(m)\right) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 < |k^{\prime}|, |k^{\prime}| < 1} | EllipticK(k) = sum((pochhammer((1)/(2), m)*pochhammer((1)/(2), m))/(factorial(m)*factorial(m))*(sqrt(1 - (k)^(2)))^(2*m)*(ln((1)/(sqrt(1 - (k)^(2))))+ d(m)), m = 0..infinity)
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EllipticK[(k)^2] == Sum[Divide[Pochhammer[Divide[1,2], m]*Pochhammer[Divide[1,2], m],(m)!*(m)!]*(Sqrt[1 - (k)^(2)])^(2*m)*(Log[Divide[1,Sqrt[1 - (k)^(2)]]]+ d[m]), {m, 0, Infinity}, GenerateConditions->None]
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Failure | Failure | Error | Skip - No test values generated |
| 19.12.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \compellintEk@{k} = 1+\frac{1}{2}\sum_{m=0}^{\infty}\frac{\Pochhammersym{\tfrac{1}{2}}{m}\Pochhammersym{\tfrac{3}{2}}{m}}{\Pochhammersym{2}{m}m!}{k^{\prime}}^{2m+2}\*\left(\ln@@{\left(\frac{1}{k^{\prime}}\right)}+d(m)-\frac{1}{(2m+1)(2m+2)}\right)}
\compellintEk@{k} = 1+\frac{1}{2}\sum_{m=0}^{\infty}\frac{\Pochhammersym{\tfrac{1}{2}}{m}\Pochhammersym{\tfrac{3}{2}}{m}}{\Pochhammersym{2}{m}m!}{k^{\prime}}^{2m+2}\*\left(\ln@@{\left(\frac{1}{k^{\prime}}\right)}+d(m)-\frac{1}{(2m+1)(2m+2)}\right) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |k^{\prime}| < 1} | EllipticE(k) = 1 +(1)/(2)*sum((pochhammer((1)/(2), m)*pochhammer((3)/(2), m))/(pochhammer(2, m)*factorial(m))*(sqrt(1 - (k)^(2)))^(2*m + 2)*(ln((1)/(sqrt(1 - (k)^(2))))+ d(m)-(1)/((2*m + 1)*(2*m + 2))), m = 0..infinity)
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EllipticE[(k)^2] == 1 +Divide[1,2]*Sum[Divide[Pochhammer[Divide[1,2], m]*Pochhammer[Divide[3,2], m],Pochhammer[2, m]*(m)!]*(Sqrt[1 - (k)^(2)])^(2*m + 2)*(Log[Divide[1,Sqrt[1 - (k)^(2)]]]+ d[m]-Divide[1,(2*m + 1)*(2*m + 2)]), {m, 0, Infinity}, GenerateConditions->None]
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Error | Failure | - | Failed [10 / 10]
Result: Indeterminate
Test Values: {Rule[d, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[k, 1]}
Result: Indeterminate
Test Values: {Rule[d, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]], Rule[k, 1]}
... skip entries to safe data |
| 19.12#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle d(m) = \digamma@{1+m}-\digamma@{\tfrac{1}{2}+m}}
d(m) = \digamma@{1+m}-\digamma@{\tfrac{1}{2}+m} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | d(m) = Psi(1 + m)- Psi((1)/(2)+ m)
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d[m] == PolyGamma[1 + m]- PolyGamma[Divide[1,2]+ m]
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Failure | Failure | Failed [30 / 30] Result: .4797310429+.5000000000*I
Test Values: {d = 1/2*3^(1/2)+1/2*I, m = 1}
Result: 1.512423114+1.*I
Test Values: {d = 1/2*3^(1/2)+1/2*I, m = 2}
... skip entries to safe data |
Failed [30 / 30]
Result: Complex[0.04671834077232884, 0.24999999999999997]
Test Values: {Rule[d, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[m, 1]}
Result: Complex[0.6463977093312149, 0.49999999999999994]
Test Values: {Rule[d, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[m, 2]}
... skip entries to safe data |
| 19.12#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle d(m+1) = d(m)-\frac{2}{(2m+1)(2m+2)}}
d(m+1) = d(m)-\frac{2}{(2m+1)(2m+2)} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | d(m + 1) = d(m)-(2)/((2*m + 1)*(2*m + 2)) |
d[m + 1] == d[m]-Divide[2,(2*m + 1)*(2*m + 2)] |
Skipped - no semantic math | Skipped - no semantic math | - | - |