# GAMMAINC

## Purpose:

Computes the regularized incomplete gamma function.

## Syntax:

GAMMAINC(x, a, "tail")

x

-

A scalar or series, the integration limit.

a

-

A scalar or series, the exponential factor.

"tail"

-

Optional. A string, the upper or lower function flag.

 "lower": lower incomplete gamma function (default) "upper": upper incomplete gamma function

## Returns:

A scalar, series, or table.

## Example:

gammainc(10, 2)

returns 0.999501.

## Example:

gammainc(3+2i, 2)

returns 0.992332 + 0.222522i.

## Example:

gammainc(1..4, 0..3)

returns the series {1, 0.864665, 0.800852, 0.761897}

which is equivalent to:

{gammainc(1,0),gammainc(2,1),gammainc(3,2),gammainc(4,3)}

## Example:

gammainc(0..0.01..1, 3);

xlabel("x");ylabel("P(3, x)");label("Incomplete Gamma Function")

returns 101 samples of P(3, x).

## Remarks:

The lower incomplete gamma function is defined as:

where a is the exponential factor and x is the integration limit.

GAMMAINC(x, a) computes the regularized lower incomplete gamma function defined as:

where Q(a, x) is the regularized upper incomplete gamma function and Γ(a) is the gamma function implemented by GAMMA. Note that for GAMMAINC, the first argument is x, the integration limit.

The upper incomplete gamma function is defined as:

GAMMAINC(x, a, "upper") computes the regularized incomplete gamma function:

Again, the first argument, x, is the integration limit.

Both x and a may be complex.

If the inputs x and a are series, they must be of the same size.

GAMMAP implements P(a, x), the regularized lower incomplete gamma function.

GAMMAQ implements Q(a, x), the regularized upper incomplete gamma function.

The arguments for GAMMAP and GAMMAQ are reversed from GAMMAINC.

The regularized upper incomplete gamma function is also known as the regularized complementary incomplete gamma function.

See GAMMAINCINV to compute the inverse incomplete gamma function.

BETAINC

EXPINT

GAMMA

GAMMAINCINV

GAMMAP

GAMMAQ

GRANDGAMMA