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gammainc

Incomplete gamma function

Syntax

Y = gammainc(X,A)
Y = gammainc(X,A,tail)
Y = gammainc(X,A,'scaledlower')
Y = gammainc(X,A,'scaledupper')

Definitions

The incomplete gamma function is

$P\left(a,x\right)=\frac{1}{\Gamma \left(a\right)}{\int }_{0}^{x}{e}^{-t}{t}^{a-1}dt\text{\hspace{0.17em}},$

where $\Gamma \left(a\right)$ is the gamma function, gamma(a).

 Note:   The syntax gammainc(X,A) is equivalent to the function P(A,X) defined above, where X is the limit of integration in each case.

For any A ≥ 0, gammainc(X,A) approaches 1 as X approaches infinity. For small X and A, gammainc(X,A) is approximately equal to X^A, so gammainc(0,0) = 1.

Description

Y = gammainc(X,A) returns the incomplete gamma function of corresponding elements of X and A. The elements of A must be nonnegative. Furthermore, X and A must be real and the same size (or either can be scalar).

Y = gammainc(X,A,tail) specifies the tail of the incomplete gamma function. The choices for tail are 'lower' (the default) and 'upper'. The upper incomplete gamma function is defined as:

$Q\left(a,x\right)=\frac{1}{\Gamma \left(a\right)}\underset{x}{\overset{\infty }{\int }}{e}^{-t}{t}^{a-1}dt=1-P\left(a,x\right).$

When the upper tail value is close to 0, the 'upper' option provides a way to compute that value more accurately than by subtracting the lower tail value from 1.

Y = gammainc(X,A,'scaledlower') and Y = gammainc(X,A,'scaledupper') return the incomplete gamma function, scaled by

$\Gamma \left(a+1\right)\left(\frac{{e}^{x}}{{x}^{a}}\right).$

These functions are unbounded above, but are useful for values of X and A where gammainc(X,A,'lower') or gammainc(X,A,'upper') underflow to zero.

 Note:   When X is negative, Y can be inaccurate for abs(X)>A+1. This applies to all syntaxes.

References

[1] Cody, J., An Overview of Software Development for Special Functions, Lecture Notes in Mathematics, 506, Numerical Analysis Dundee, G. A. Watson (ed.), Springer Verlag, Berlin, 1976.

[2] Abramowitz, M. and I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series #55, Dover Publications, 1965, sec. 6.5.