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Incomplete gamma function


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


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:


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


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.


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

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Incomplete Gamma Function

The incomplete gamma function is


where Γ(a) is the gamma function, gamma(a).

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.


[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.

Extended Capabilities

Introduced before R2006a