gamma

Gamma function

Syntax

Y = gamma(X)

Description

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

Examples

Evaluate Gamma Function

Evaluate the gamma function with a scalar and a vector.

Evaluate $$\Gamma (0.5)$$, which is equal to $$\sqrt{\pi }$$.

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 = 1×8

    0.2701   -0.9453    2.3633   -3.5449    1.7725    0.8862    1.3293    3.3234

Plot Gamma Function

Plot the gamma function and its reciprocal.

Use fplot to plot the gamma function and its reciprocal. 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 reciprocal gamma function has zeros at all negative integer arguments (as well as 0).

fplot(@gamma)
hold on
fplot(@(x) 1./gamma(x))
ylim([-10 10])
legend('\Gamma(x)','1/\Gamma(x)')
hold off
grid on

Input Arguments

X — Input array

scalar | vector | matrix | multidimensional array

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

Data Types: single | double

Limitations

• For double and single data types, the gamma function returns Inf for all values greater than realmax and realmax('single'). The saturation thresholds for positive integers are gamma(172) and gamma(single(36)), where the evaluated gamma functions are greater than the maximum representable values.

More About

Gamma Function

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

$$\Gamma (x)={\displaystyle {\int }_0^{\infty }{e}^{-t}}{t}^{x-1}dt$$

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

$$\Gamma \left(n-1\right)=\frac{\Gamma \left(n\right)}{n-1}.$$

Algorithms

The computation of gamma is based on algorithms outlined in [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.

Extended Capabilities

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

The gamma function fully supports tall arrays. For more information, see Tall Arrays.

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

The gamma function fully supports GPU arrays. To run the function on a GPU, specify the input data as a gpuArray (Parallel Computing Toolbox). For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).

Version History

Introduced before R2006a