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gamma

Gamma function

Syntax

``Y = gamma(X)``

Description

example

````Y = gamma(X)` returns the `gamma` function evaluated at the elements of `X`. ```

Examples

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Evaluate the gamma function with a scalar and a vector.

Evaluate , which is equal to .

`y = gamma(0.5)`
```y = 1.7725 ```

Evaluate several values of the gamma function between `[-3.5 3.5]`.

```x = -3.5:3.5; y = gamma(x)```
```y = 0.2701 -0.9453 2.3633 -3.5449 1.7725 0.8862 1.3293 3.3234 ```

Plot the gamma function and its inverse.

Use `fplot` to plot the gamma function and its inverse. The gamma function increases quickly for positive arguments and has simple poles at all negative integer arguments (as well as 0). The function does not have any zeros. Conversely, the inverse gamma function has zeros at all negative integer arguments (as well as 0).

```fplot(@gamma) hold on fplot(@(x) 1./gamma(x)) legend('\Gamma(x)','1/\Gamma(x)') hold off grid on```

Input Arguments

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Input array, specified as a scalar, vector, matrix, or multidimensional array. The elements of `X` must be real.

Data Types: `single` | `double`

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Gamma Function

The `gamma` function is defined for real `x > 0` by the integral:

`$\Gamma \left(x\right)={\int }_{0}^{\infty }{e}^{-t}{t}^{x-1}dt$`

The `gamma` function interpolates the `factorial` function. For integer `n`:

`gamma(n+1) = factorial(n) = prod(1:n)`

The domain of the `gamma` function extends to negative real numbers by analytic continuation, with simple poles at the negative integers. This extension arises from repeated application of the recursion relation

`$\Gamma \left(n-1\right)=\frac{\Gamma \left(n\right)}{n-1}\text{\hspace{0.17em}}.$`

Algorithms

The computation of `gamma` is based on algorithms outlined in [1]. Several different minimax rational approximations are used depending upon the value of `A`.

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.