MATLAB Function Reference
gamma, gammainc, gammaln

Gamma functions

Syntax

• ```Y = gamma(A)                Gamma function
Y = gammainc(X,A)           Incomplete gamma function
Y = gammainc(X,A,tail)      Tail of the incomplete gamma function
Y = gammaln(A)              Logarithm of gamma function
```

Definition

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`.

Description

```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))`.

Algorithm

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].

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.

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