# Gamma function

Calculation of the Gamma function using Lanczos approximation

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#### Timur

Created: 2015-03-12 12:32:06, Last updated: 2021-03-18 20:35:12

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Gamma function is defined as
$\Gamma(x) = \displaystyle\int_0^\infty t^{x-1}e^{-t} dt$

In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1 to real and complex numbers. In particular, for positive integer gamma function is
$\Gamma(n)=(n-1)!$

The main property of gamma function is its recurrence relation:
$\Gamma(z+1)=z\Gamma(z)$

The Lanczos approximation is used to calculate the Gamma function numerically. The Lanczos approximation is a method for computing the Gamma function numerically, published by Cornelius Lanczos in 1964. You can check this link Lanczos approximation for details

Why is it needed? Wikipedia says:
Opening a random page in an advanced table of formulas may be as likely to spot the gamma function as a trigonometric function. One author describes the gamma function as "Arguably, the most common special function, or the least 'special' of them. The other transcendental functions listed below are called 'special' because you could conceivably avoid some of them by staying away from many specialized mathematical topics. On the other hand, the gamma function y = Γ(x) is most difficult to avoid."

Here it is needed because the gamma function is part of the distribution function formula for Student's distribution, which I need for the next calculator.

#### Gamma function

Digits after the decimal point: 2
Gamma function

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