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{{DISPLAYTITLE:Incomplete Gamma and Related Functions - 8.11 Asymptotic Approximations and Expansions}}

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! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! scope="col" style="position: sticky; top: 0;" | Constraints 
! scope="col" style="position: sticky; top: 0;" | Maple
! scope="col" style="position: sticky; top: 0;" | Mathematica
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/8.11.E2 8.11.E2] || <math qid="Q2573">\incGamma@{a}{z} = z^{a-1}e^{-z}\left(\sum_{k=0}^{n-1}\frac{u_{k}}{z^{k}}+R_{n}(a,z)\right)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{a}{z} = z^{a-1}e^{-z}\left(\sum_{k=0}^{n-1}\frac{u_{k}}{z^{k}}+R_{n}(a,z)\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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])</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br>... skip entries to safe data</div></div> || Error
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| [https://dlmf.nist.gov/8.11.E4 8.11.E4] || <math qid="Q2575">\incgamma@{a}{z} = z^{a}e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\Pochhammersym{a}{k+1}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incgamma@{a}{z} = z^{a}e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\Pochhammersym{a}{k+1}}</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a)-GAMMA(a, z) = (z)^(a)* exp(- z)*sum(((z)^(k))/(pochhammer(a, k + 1)), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, 0, z] == (z)^(a)* Exp[- z]*Sum[Divide[(z)^(k),Pochhammer[a, k + 1]], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
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| [https://dlmf.nist.gov/8.11#Ex1 8.11#Ex1] || <math qid="Q2579">b_{0}(\lambda) = 1</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>b_{0}(\lambda) = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">b[0](lambda) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[b, 0][\[Lambda]] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/8.11#Ex2 8.11#Ex2] || <math qid="Q2580">b_{1}(\lambda) = \lambda</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>b_{1}(\lambda) = \lambda</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">b[1](lambda) = lambda</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[b, 1][\[Lambda]] == \[Lambda]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/8.11#Ex3 8.11#Ex3] || <math qid="Q2581">b_{2}(\lambda) = \lambda(2\lambda+1)</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>b_{2}(\lambda) = \lambda(2\lambda+1)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">b[2](lambda) = lambda*(2*lambda + 1)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[b, 2][\[Lambda]] == \[Lambda]*(2*\[Lambda]+ 1)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/8.11.E15 8.11.E15] || <math qid="Q2588">S_{n}(x) = \frac{\incgamma@{n+1}{nx}}{(nx)^{n}e^{-nx}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>S_{n}(x) = \frac{\incgamma@{n+1}{nx}}{(nx)^{n}e^{-nx}}</syntaxhighlight> || <math>\realpart@@{(n+1)} > 0</math> || <syntaxhighlight lang=mathematica>S[n](x) = (GAMMA(n + 1)-GAMMA(n + 1, n*x))/((n*x)^(n)* exp(- n*x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[S, n][x] == Divide[Gamma[n + 1, 0, n*x],(n*x)^(n)* Exp[- n*x]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [90 / 90]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.22087941e-1+.7500000000*I
Test Values: {x = 1.5, S[n] = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.275525655+.7500000000*I
Test Values: {x = 1.5, S[n] = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [90 / 90]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/8.11.E19 8.11.E19] || <math qid="Q2592">d_{k}(x) = \frac{(-1)^{k}b_{k}(x)}{(1-x)^{2k+1}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>d_{k}(x) = \frac{(-1)^{k}b_{k}(x)}{(1-x)^{2k+1}}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">d[k](x) = ((- 1)^(k)* b[k](x))/((1 - x)^(2*k + 1))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[d, k][x] == Divide[(- 1)^(k)* Subscript[b, k][x],(1 - x)^(2*k + 1)]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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