Incomplete Gamma and Related Functions - 8.20 Asymptotic Expansions of
| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|---|
| 8.20.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \genexpintE{p}@{z} = \frac{e^{-z}}{z}\left(\sum_{k=0}^{n-1}(-1)^{k}\frac{\Pochhammersym{p}{k}}{z^{k}}+(-1)^{n}\frac{\Pochhammersym{p}{n}e^{z}}{z^{n-1}}\genexpintE{n+p}@{z}\right)}
\genexpintE{p}@{z} = \frac{e^{-z}}{z}\left(\sum_{k=0}^{n-1}(-1)^{k}\frac{\Pochhammersym{p}{k}}{z^{k}}+(-1)^{n}\frac{\Pochhammersym{p}{n}e^{z}}{z^{n-1}}\genexpintE{n+p}@{z}\right) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Ei(p, z) = (exp(- z))/(z)*(sum((- 1)^(k)*(pochhammer(p, k))/((z)^(k)), k = 0..n - 1)+(- 1)^(n)*(pochhammer(p, n)*exp(z))/((z)^(n - 1))*Ei(n + p, z))
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ExpIntegralE[p, z] == Divide[Exp[- z],z]*(Sum[(- 1)^(k)*Divide[Pochhammer[p, k],(z)^(k)], {k, 0, n - 1}, GenerateConditions->None]+(- 1)^(n)*Divide[Pochhammer[p, n]*Exp[z],(z)^(n - 1)]*ExpIntegralE[n + p, z])
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Failure | Successful | Manual Skip! | Failed [7 / 70]
Result: Indeterminate
Test Values: {Rule[n, 3], Rule[p, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Indeterminate
Test Values: {Rule[n, 3], Rule[p, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
| 8.20.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle A_{k+1}(\lambda) = (1-2k\lambda)A_{k}(\lambda)+\lambda(\lambda+1)\deriv{A_{k}(\lambda)}{\lambda}}
A_{k+1}(\lambda) = (1-2k\lambda)A_{k}(\lambda)+\lambda(\lambda+1)\deriv{A_{k}(\lambda)}{\lambda} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | A[k + 1](lambda) = (1 - 2*k*lambda)*A[k](lambda)+ lambda*(lambda + 1)*diff(A[k](lambda), lambda)
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Subscript[A, k + 1][\[Lambda]] == (1 - 2*k*\[Lambda])*Subscript[A, k][\[Lambda]]+ \[Lambda]*(\[Lambda]+ 1)*D[Subscript[A, k][\[Lambda]], \[Lambda]]
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Failure | Failure | Failed [300 / 300] Result: -.5000000000+.133974596*I
Test Values: {lambda = 1/2*3^(1/2)+1/2*I, A[k] = 1/2*3^(1/2)+1/2*I, A[k+1] = 1/2*3^(1/2)+1/2*I, k = 1, k = 3}
Result: -.4999999993+2.133974597*I
Test Values: {lambda = 1/2*3^(1/2)+1/2*I, A[k] = 1/2*3^(1/2)+1/2*I, A[k+1] = 1/2*3^(1/2)+1/2*I, k = 2, k = 3}
... skip entries to safe data |
Failed [300 / 300]
Result: Complex[-0.49999999999999944, 4.133974596215561]
Test Values: {Rule[k, 3], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[A, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[A, Plus[1, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Complex[-1.8660254037844384, 3.7679491924311224]
Test Values: {Rule[k, 3], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[A, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[A, Plus[1, k]], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
| 8.20#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle A_{1}(\lambda) = 1}
A_{1}(\lambda) = 1 |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | A[1](lambda) = 1 |
Subscript[A, 1][\[Lambda]] == 1 |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 8.20#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle A_{2}(\lambda) = 1-2\lambda}
A_{2}(\lambda) = 1-2\lambda |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | A[2](lambda) = 1 - 2*lambda |
Subscript[A, 2][\[Lambda]] == 1 - 2*\[Lambda] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 8.20#Ex3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle A_{3}(\lambda) = 1-8\lambda+6\lambda^{2}}
A_{3}(\lambda) = 1-8\lambda+6\lambda^{2} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | A[3](lambda) = 1 - 8*lambda + 6*(lambda)^(2) |
Subscript[A, 3][\[Lambda]] == 1 - 8*\[Lambda]+ 6*\[Lambda]^(2) |
Skipped - no semantic math | Skipped - no semantic math | - | - |