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gamma

Gamma function

Syntax

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

Figure contains an axes. The axes contains an object of type functionline.

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

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

More About

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Gamma Function

The following integral defines the gamma function:

Γ(z)=0tz1etdt.

Introduced before R2006a