I am confused about incomplete gamma function calculation in Mathematica and MATLAB:
For example, in Mathematica:
Gamma[5,3] = 19.56
But in MATLAB:
gammainc(5,3) = 0.8753
I don't know which one is correct, MATLAB result or Mathematica result?
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$\begingroup$ Also, this relationship is valid: gammainc(a,b,'upper')*gamma(b) in Matlab gives the same result as Gamma[b,a] in Mathematica. $\endgroup$– Masoud MoeiniCommented Jan 11, 2022 at 23:37
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1 Answer
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In Mathematica, Gamma[a, z] refers to the upper incomplete Gamma function, given by
$$\Gamma(a,z)=\int_z^\infty t^{a-1}e^{-t}\,dt$$
whereas in MATLAB, gammainc(z, a) refers to the regularized lower incomplete Gamma function
$$P(a,z)=\frac{1}{\Gamma(a)}\int_0^z t^{a-1}e^{-t}\, dt$$
Obviously, they give different results. To get MATLAB's result in Mathematica, use the following:
gammainc[a_, z_] := Gamma[a, 0, z]/Gamma[a]
N@gammainc[3, 5]
(* 0.875348 *)
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4$\begingroup$ In fact, you could simplify things slightly:
gammainc[a_, z_] := GammaRegularized[a, 0, z]; it has been anticipated that people would frequently want the scaled version. $\endgroup$ Commented May 10, 2015 at 10:12