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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

 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.

 Abramowitz, M. and I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series #55, Dover Publications, 1965, sec. 6.5.