# Documentation

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# gammainc

Incomplete gamma function

## Syntax

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

## Description

`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:

`$\text{gammainc}\left(\text{x,a,'upper'}\right)=\frac{1}{\Gamma \left(a\right)}\underset{x}{\overset{\infty }{\int }}{t}^{a-1}{e}^{-t}dt$`

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

`$\Gamma \left(a+1\right)\left(\frac{{e}^{x}}{{x}^{a}}\right).$`

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.

 Note:   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

`$\text{gammainc}\left(\text{x,a}\right)=\frac{1}{\Gamma \left(a\right)}{\int }_{0}^{x}{t}^{a-1}{e}^{-t}dt$`

where $\Gamma \left(a\right)$ 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```.

## 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.