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Y = gamma(A)
Y = gammainc(X,A)
Y = gammainc(X,A,tail)
Y = gammaln(A)
The gamma function is defined by the integral:
The gamma function interpolates the factorial function. For integer n:
gamma(n+1) = n! = prod(1:n)
The incomplete gamma function is:
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.
Y = gamma(A) returns the gamma function at the elements of A. A must be real.
Y = gammainc(X,A) returns the incomplete gamma function of corresponding elements of X and A. Arguments 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 when X is non-negative. The choices are for tail are 'lower' (the default) and 'upper'. The upper incomplete gamma function is defined as
1 - gammainc(x,a)
Note When X is negative, Y can be inaccurate for abs(X)>A+1. |
Y = gammaln(A) returns the logarithm of the gamma function, gammaln(A) = log(gamma(A)). The gammaln command avoids the underflow and overflow that may occur if it is computed directly using log(gamma(A)).
The computations of gamma and gammaln are based on algorithms outlined in [1]. Several different minimax rational approximations are used depending upon the value of A. Computation of the incomplete gamma function is based on the algorithm in [2].
[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.
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