# Gamma Function Calculator

\( \)\( \)\( \)\( \)
An easy to use calculator to compute Gamma function \( \Gamma \; (z) \) defined by the integral
\[ \displaystyle \Gamma (z) = \int_0^{\infty} \; t^{z-1}e^{-t} \; dt \]
is presented.

The graph of the error function \( \Gamma (x) \), for real \( x \), is shown below as well as its reciprocal \( \dfrac{1}{\Gamma (x)} \) .

## Properties of Gamma Function

It can be shown that for positive integers
\[ \Gamma(n) = (n - 1)! \]
hence

\( \quad \Gamma(1) = (1 - 1)! = 0! = 1 \)

\( \quad \Gamma(2) = (2 - 1)! = 1! = 1 \)

\( \quad \Gamma(3) = (3 - 1)! = 2! = 2 \)

\( \quad \Gamma(4) = (4 - 1)! = 3! = 6 \)

valeus that can be checked in the graph above.

The Gamma function \( \Gamma(z) \) is used to extend the factorial function to complex numbers.

Using imptoper integrals and integration by parts, it can be shown that
\[ \Gamma(z+1) = z \Gamma(z) \]
for \( z \) in the complex plane sucth that \( \Re (z) \gt 0 \)

## Use Calculator

Enter the the real and imagianry parts \( Re \; z\) and \(Im \; z \) respectively of the argument \( z \) of the Gamma function, and the number of decimal places desired, then click "Calculate".

Answer

## More References and Links