# gamma

Gamma function

## Description

example

gamma(X) returns the gamma function of a symbolic variable or symbolic expression X.

## Examples

### Gamma Function for Numeric and Symbolic Arguments

Depending on its arguments, gamma returns floating-point or exact symbolic results.

Compute the gamma function for these numbers. Because these numbers are not symbolic objects, you get floating-point results.

A = gamma([-11/3, -7/5, -1/2, 1/3, 1, 4])
A =
0.2466    2.6593   -3.5449    2.6789    1.0000    6.0000

Compute the gamma function for the numbers converted to symbolic objects. For many symbolic (exact) numbers, gamma returns unresolved symbolic calls.

symA = gamma(sym([-11/3, -7/5, -1/2, 1/3, 1, 4]))
symA =
[ (27*pi*3^(1/2))/(440*gamma(2/3)), gamma(-7/5),...
-2*pi^(1/2), (2*pi*3^(1/2))/(3*gamma(2/3)), 1, 6]

Use vpa to approximate symbolic results with floating-point numbers:

vpa(symA)
ans =
[ 0.24658411512650858900694446388517,...
2.6592718728800305399898810505738,...
-3.5449077018110320545963349666823,...
2.6789385347077476336556929409747,...
1.0, 6.0]

### Plot Gamma Function

Plot the gamma function and add grid lines.

syms x
fplot(gamma(x))
grid on

### Handle Expressions Containing Gamma Function

Many functions, such as diff, limit, and simplify, can handle expressions containing gamma.

Differentiate the gamma function, and then substitute the variable t with the value 1:

syms t
u = diff(gamma(t^3 + 1))
u1 = subs(u, t, 1)
u =
3*t^2*gamma(t^3 + 1)*psi(t^3 + 1)

u1 =
3 - 3*eulergamma

Approximate the result using vpa:

vpa(u1)
ans =
1.2683530052954014181804637297528

Compute the limit of the following expression that involves the gamma function:

syms x
limit(x/gamma(x), x, inf)
ans =
0

Simplify the following expression:

syms x
simplify(gamma(x)*gamma(1 - x))
ans =
pi/sin(pi*x)

## Input Arguments

collapse all

Input, specified as symbolic number, variable, expression, function, or as a vector or matrix of symbolic numbers, variables, expressions, or functions.

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### Gamma Function

The following integral defines the gamma function:

$\Gamma \left(z\right)=\underset{0}{\overset{\infty }{\int }}{t}^{z-1}{e}^{-t}dt.$

## Version History

Introduced before R2006a