Incomplete Gamma and Related Functions - 8.7 Series Expansions
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| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|---|
| 8.7.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \scincgamma@{a}{z} = e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\EulerGamma@{a+k+1}}}
\scincgamma@{a}{z} = e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\EulerGamma@{a+k+1}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(a+k+1)} > 0, \realpart@@{a} > 0} | (z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a) = exp(- z)*sum(((z)^(k))/(GAMMA(a + k + 1)), k = 0..infinity)
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Error
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Successful | Missing Macro Error | - | - |
| 8.7.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\EulerGamma@{a+k+1}} = \frac{1}{\EulerGamma@{a}}\sum_{k=0}^{\infty}\frac{(-z)^{k}}{k!(a+k)}}
e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\EulerGamma@{a+k+1}} = \frac{1}{\EulerGamma@{a}}\sum_{k=0}^{\infty}\frac{(-z)^{k}}{k!(a+k)} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(a+k+1)} > 0, \realpart@@{a} > 0} | exp(- z)*sum(((z)^(k))/(GAMMA(a + k + 1)), k = 0..infinity) = (1)/(GAMMA(a))*sum(((- z)^(k))/(factorial(k)*(a + k)), k = 0..infinity)
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Exp[- z]*Sum[Divide[(z)^(k),Gamma[a + k + 1]], {k, 0, Infinity}, GenerateConditions->None] == Divide[1,Gamma[a]]*Sum[Divide[(- z)^(k),(k)!*(a + k)], {k, 0, Infinity}, GenerateConditions->None]
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Successful | Successful | - | Successful [Tested: 21] |
| 8.7.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \incgamma@{a}{x+y}-\incgamma@{a}{x} = \incGamma@{a}{x}-\incGamma@{a}{x+y}}
\incgamma@{a}{x+y}-\incgamma@{a}{x} = \incGamma@{a}{x}-\incGamma@{a}{x+y} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |y| < |x|, \realpart@@{a} > 0} | GAMMA(a)-GAMMA(a, x + y)- GAMMA(a)-GAMMA(a, x) = GAMMA(a, x)- GAMMA(a, x + y)
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Gamma[a, 0, x + y]- Gamma[a, 0, x] == Gamma[a, x]- Gamma[a, x + y]
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Failure | Successful | Failed [18 / 18] Result: -.6941375518
Test Values: {a = 1.5, x = 1.5, y = -.5}
Result: -.6941375518
Test Values: {a = 1.5, x = 1.5, y = .5}
... skip entries to safe data |
Successful [Tested: 18] |
| 8.7.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \incGamma@{a}{x}-\incGamma@{a}{x+y} = e^{-x}x^{a-1}\sum_{n=0}^{\infty}\frac{\Pochhammersym{1-a}{n}}{(-x)^{n}}(1-e^{-y}e_{n}(y))}
\incGamma@{a}{x}-\incGamma@{a}{x+y} = e^{-x}x^{a-1}\sum_{n=0}^{\infty}\frac{\Pochhammersym{1-a}{n}}{(-x)^{n}}(1-e^{-y}e_{n}(y)) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |y| < |x|, \realpart@@{a} > 0} | GAMMA(a, x)- GAMMA(a, x + y) = exp(- x)*(x)^(a - 1)* sum((pochhammer(1 - a, n))/((- x)^(n))*(1 - exp(- y)*exp(1)[n]*(y)), n = 0..infinity)
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Gamma[a, x]- Gamma[a, x + y] == Exp[- x]*(x)^(a - 1)* Sum[Divide[Pochhammer[1 - a, n],(- x)^(n)]*(1 - Exp[- y]*Subscript[E, n]*(y)), {n, 0, Infinity}, GenerateConditions->None]
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Error | Failure | Skip - symbolical successful subtest | Skipped - Because timed out |
| 8.7.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \incGamma@{a}{z} = \EulerGamma@{a}-\sum_{k=0}^{\infty}\frac{(-1)^{k}z^{a+k}}{k!(a+k)}}
\incGamma@{a}{z} = \EulerGamma@{a}-\sum_{k=0}^{\infty}\frac{(-1)^{k}z^{a+k}}{k!(a+k)} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{a} > 0} | GAMMA(a, z) = GAMMA(a)- sum(((- 1)^(k)* (z)^(a + k))/(factorial(k)*(a + k)), k = 0..infinity)
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Gamma[a, z] == Gamma[a]- Sum[Divide[(- 1)^(k)* (z)^(a + k),(k)!*(a + k)], {k, 0, Infinity}, GenerateConditions->None]
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Successful | Successful | - | Successful [Tested: 7] |
| 8.7.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerGamma@{a}-\sum_{k=0}^{\infty}\frac{(-1)^{k}z^{a+k}}{k!(a+k)} = \EulerGamma@{a}\left(1-z^{a}e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\EulerGamma@{a+k+1}}\right)}
\EulerGamma@{a}-\sum_{k=0}^{\infty}\frac{(-1)^{k}z^{a+k}}{k!(a+k)} = \EulerGamma@{a}\left(1-z^{a}e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\EulerGamma@{a+k+1}}\right) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{a} > 0, \realpart@@{(a+k+1)} > 0} | GAMMA(a)- sum(((- 1)^(k)* (z)^(a + k))/(factorial(k)*(a + k)), k = 0..infinity) = GAMMA(a)*(1 - (z)^(a)* exp(- z)*sum(((z)^(k))/(GAMMA(a + k + 1)), k = 0..infinity))
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Gamma[a]- Sum[Divide[(- 1)^(k)* (z)^(a + k),(k)!*(a + k)], {k, 0, Infinity}, GenerateConditions->None] == Gamma[a]*(1 - (z)^(a)* Exp[- z]*Sum[Divide[(z)^(k),Gamma[a + k + 1]], {k, 0, Infinity}, GenerateConditions->None])
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Successful | Successful | - | Successful [Tested: 7] |
| 8.7.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \incGamma@{a}{x} = x^{a}e^{-x}\sum_{n=0}^{\infty}\frac{\LaguerrepolyL[a]{n}@{x}}{n+1}}
\incGamma@{a}{x} = x^{a}e^{-x}\sum_{n=0}^{\infty}\frac{\LaguerrepolyL[a]{n}@{x}}{n+1} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x > 0} | GAMMA(a, x) = (x)^(a)* exp(- x)*sum((LaguerreL(n, a, x))/(n + 1), n = 0..infinity)
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Gamma[a, x] == (x)^(a)* Exp[- x]*Sum[Divide[LaguerreL[n, a, x],n + 1], {n, 0, Infinity}, GenerateConditions->None]
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Missing Macro Error | Failure | - | Failed [18 / 18]
Result: Plus[0.03483445061608508, Times[-0.1214566752420326, NSum[Times[Power[Plus[1, n], -1], LaguerreL[n, -1.5, 1.5]]
Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[a, -1.5], Rule[x, 1.5]}
Result: Plus[0.7498909754592095, Times[-1.7155277699214138, NSum[Times[Power[Plus[1, n], -1], LaguerreL[n, -1.5, 0.5]]
Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[a, -1.5], Rule[x, 0.5]}
... skip entries to safe data |