Incomplete Gamma and Related Functions - 8.11 Asymptotic Approximations and Expansions
| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|---|
| 8.11.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \incGamma@{a}{z} = z^{a-1}e^{-z}\left(\sum_{k=0}^{n-1}\frac{u_{k}}{z^{k}}+R_{n}(a,z)\right)}
\incGamma@{a}{z} = z^{a-1}e^{-z}\left(\sum_{k=0}^{n-1}\frac{u_{k}}{z^{k}}+R_{n}(a,z)\right) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | GAMMA(a, z) = (z)^(a - 1)* exp(- z)*(sum(((- 1)^(k)* pochhammer(1 - a, k))/((z)^(k)), k = 0..n - 1)+ R[n](a , z))
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Gamma[a, z] == (z)^(a - 1)* Exp[- z]*(Sum[Divide[(- 1)^(k)* Pochhammer[1 - a, k],(z)^(k)], {k, 0, n - 1}, GenerateConditions->None]+ Subscript[R, n][a , z])
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Failure | Failure | Failed [300 / 300] Result: .1072320848e-1-.1480251451*I+(.9924715785e-1+.4087434498*I)*(1.000000000+0.*I+(.8660254040+.5000000000*I)*(-1.5, .8660254040+.5000000000*I))
Test Values: {a = -1.5, z = 1/2*3^(1/2)+1/2*I, R[n] = 1/2*3^(1/2)+1/2*I, n = 1, n = 3}
Result: .1072320848e-1-.1480251451*I+(.9924715785e-1+.4087434498*I)*(-1.165063509+1.250000000*I+(.8660254040+.5000000000*I)*(-1.5, .8660254040+.5000000000*I))
Test Values: {a = -1.5, z = 1/2*3^(1/2)+1/2*I, R[n] = 1/2*3^(1/2)+1/2*I, n = 2, n = 3}
... skip entries to safe data |
Error |
| 8.11.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \incgamma@{a}{z} = z^{a}e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\Pochhammersym{a}{k+1}}}
\incgamma@{a}{z} = z^{a}e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\Pochhammersym{a}{k+1}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{a} > 0} | GAMMA(a)-GAMMA(a, z) = (z)^(a)* exp(- z)*sum(((z)^(k))/(pochhammer(a, k + 1)), k = 0..infinity)
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Gamma[a, 0, z] == (z)^(a)* Exp[- z]*Sum[Divide[(z)^(k),Pochhammer[a, k + 1]], {k, 0, Infinity}, GenerateConditions->None]
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Successful | Successful | - | Successful [Tested: 7] |
| 8.11#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle b_{0}(\lambda) = 1}
b_{0}(\lambda) = 1 |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | b[0](lambda) = 1 |
Subscript[b, 0][\[Lambda]] == 1 |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 8.11#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle b_{1}(\lambda) = \lambda}
b_{1}(\lambda) = \lambda |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | b[1](lambda) = lambda |
Subscript[b, 1][\[Lambda]] == \[Lambda] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 8.11#Ex3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle b_{2}(\lambda) = \lambda(2\lambda+1)}
b_{2}(\lambda) = \lambda(2\lambda+1) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | b[2](lambda) = lambda*(2*lambda + 1) |
Subscript[b, 2][\[Lambda]] == \[Lambda]*(2*\[Lambda]+ 1) |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 8.11.E15 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle S_{n}(x) = \frac{\incgamma@{n+1}{nx}}{(nx)^{n}e^{-nx}}}
S_{n}(x) = \frac{\incgamma@{n+1}{nx}}{(nx)^{n}e^{-nx}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(n+1)} > 0} | S[n](x) = (GAMMA(n + 1)-GAMMA(n + 1, n*x))/((n*x)^(n)* exp(- n*x))
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Subscript[S, n][x] == Divide[Gamma[n + 1, 0, n*x],(n*x)^(n)* Exp[- n*x]]
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Failure | Failure | Failed [90 / 90] Result: -.22087941e-1+.7500000000*I
Test Values: {x = 1.5, S[n] = 1/2*3^(1/2)+1/2*I, n = 1}
Result: -1.275525655+.7500000000*I
Test Values: {x = 1.5, S[n] = 1/2*3^(1/2)+1/2*I, n = 2}
... skip entries to safe data |
Failed [90 / 90]
Result: Complex[-0.02208794121538471, 0.7499999999999999]
Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[Subscript[S, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Complex[-1.2755256550317124, 0.7499999999999999]
Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[Subscript[S, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
... skip entries to safe data |
| 8.11.E19 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle d_{k}(x) = \frac{(-1)^{k}b_{k}(x)}{(1-x)^{2k+1}}}
d_{k}(x) = \frac{(-1)^{k}b_{k}(x)}{(1-x)^{2k+1}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | d[k](x) = ((- 1)^(k)* b[k](x))/((1 - x)^(2*k + 1)) |
Subscript[d, k][x] == Divide[(- 1)^(k)* Subscript[b, k][x],(1 - x)^(2*k + 1)] |
Skipped - no semantic math | Skipped - no semantic math | - | - |