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gamma

Gamma function

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Syntax

Y = gamma(X)

Description

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

example

Examples

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Open Live Script

Evaluate the gamma function with a scalar and a vector.

Evaluate Γ(0.5), which is equal to π.

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
Open Live Script

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

Figure contains an axes object. The axes object contains 2 objects of type functionline. These objects represent \Gamma(x), 1/\Gamma(x).

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

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

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Gamma Function

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

Γ(x)=∫0∞e−ttx−1dt

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

Γ(n−1)=Γ(n)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

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

Version History

Introduced before R2006a

See Also

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