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2023.0.0 - 日本語


gamma function, complete or incomplete normalized


y = gamma(u)
y = gamma(x, a)
y = gamma(x, a, b)
y = gamma(x, .., "upper")


array of positive or negative real numbers

gamma(u) and gamma(x,…) can be overloaded for complex numbers with %s_gamma_user(), and for other a types with the usual overload naming rule.

x, a, b
arrays of positive real numbers. If at least one input is not scalar, scalar ones are expanded to its size. If several inputs are not scalar, they must have the same size.

array of real numbers, with the size of u or of (the non-scalar) x, a, or b.


gamma(…) computes and yields the complete or incomplete gamma function for each element of its input(s), in an element-wise way. The complete gamma function extends the factorial one to non-integer real positive or negative numbers, as gamma(u+1)=u*gamma(u).

gamma(u) computes

Γ(u)= ∫_0→∞ t^{u-1}.exp(-t).dt

Incomplete normalized integrals

gamma(x, a) computes the integral

P(x,a)= ∫_0→x t^{a-1}.exp(-t).dt / Γ(a)

gamma(x, a, b) computes the generalized integral

P(x,a,b)= ∫_0→x t^{a-1}.exp(-bt).dt . b^a / Γ(a)

gamma(x, a, "upper") computes accurately the complementary integral

Q(x,a)= ∫_x→∞ t^{a-1}.exp(-t).dt / Γ(a) = 1-P(x,a)
even for big x and P(x,a)→1. Finally,

gamma(x, a, b, "upper") computes the generalized complementary integral

Q(x,a,b)= ∫_x→∞ t^{a-1}.exp(-bt).dt . b^a / Γ(a)

The inverse incomplete normalized gamma function can be computed with x = cdfgam("X", a, b, y, 1-y), that is the x bound such that y=∫_0→x t^{a-1}.exp(-bt).dt . b^a / Γ(a)

Calling x = cdfgam("X", a, b, z-1, z) with z=1-y will be preferred when 0.5 < y < 1, to get a full accuracy on x.


Gamma as the extension of the factorial function to non-integer numbers:

[gamma(2:7) ; factorial(1:6)]
gamma(1.5:7) ./ gamma(0.5:6)
--> [gamma(2:7) ; factorial(1:6)]
 ans  =
   1.   2.   6.   24.   120.   720.
   1.   2.   6.   24.   120.   720.

--> gamma(1.5:7)
 ans  =
   0.8862269   1.3293404   3.323351   11.631728   52.342778   287.88528

--> gamma(1.5:7) ./ gamma(0.5:6)
 ans  =
   0.5   1.5   2.5   3.5   4.5   5.5

Graph of the Gamma function around 0:

[a, b] = (-3, 5);
x = linspace(a,b,40000);
y = gamma(x);
plot2d(x, y, style=0, axesflag=5, rect=[a,-10,b,10])
title("$\Gamma(u)$", "fontsize",3.5)

Incomplete normalized P(x,a) gamma function:

x = 0.1:0.2:8;
a = 0.1:0.2:7;
[X, A] = ndgrid(x, a);
P = gamma(X,A);
gcf().color_map = coolcolormap(100);
title("$P(x,a)=\frac{1}{\Gamma(a)}\int_0^x\! t^{a-1}e^{-t}\,dt$","fontsize",3.5)
xlabel(["" "a"], "fontsize",2)
ylabel("x", "fontsize",2)
zlabel("P(x,a)", "fontsize",2)

Incomplete generalized normalized P(x,a,b) function:

a = 0.1:0.2:8;
b = 0.1:0.2:7;
[A, B] = ndgrid(a, b);
P = gamma(1,A,B);
gcf().color_map = parulacolormap(100);
title("$P(x,a,b)=\frac{b^a}{\Gamma(a)}\int_0^x\! t^{a-1}e^{-b\,t}\,dt\quad for\quad x=1$","fontsize",3.7)
xlabel("b", "fontsize",2)
ylabel("a", "fontsize",2)
gca().rotation_angles = [58 75];

See also

  • gammaln — ガンマ関数の対数.
  • dlgamma — ガンマ関数, psi関数の微分
  • cdfgam — 累積分布関数: ガンマ分布
  • factorial — factorial function : product of the n first positive integers


5.4.0 Overloading allowed for list, mlist, tlist and hypermatrix types.
  • The input can now be an hypermatrix.
  • gamma can now be overloaded for complex numbers.
6.1.1 gamma(x,..) incomplete versions added.
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Last updated:
Tue Mar 07 09:28:47 CET 2023