MATLAB Function Reference |

**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`

:

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

```
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.

gallery | gca |

© 1994-2005 The MathWorks, Inc.