gamma function | Physics Library

The gamma function is

∫ ∞ −t x−1 Γ (x) = e t dt 0

where x ∈ ℂ ∖{0,−1,−2,…}.

The Gamma function satisfies

Γ (x + 1 ) = x Γ (x)

Therefore, for integer values of x = n,

Γ (n ) = (n − 1)!

Some values of the gamma function for small arguments are:

Γ (1∕5 ) = 4.5909 Γ (1∕4) = 3.6256 Γ (1∕3 ) = 2.6789 Γ (2∕5) = 2.2182 Γ (3∕5 ) = 1.4892 Γ (2∕3) = 1.3541 Γ (3∕4 ) = 1.2254 Γ (4∕5) = 1.1642

and the ever-useful Γ(1∕2) = √ π- . These values allow a quick calculation of

Γ (n + f)

Where n is a natural number and f is any fractional value for which the Gamma function’s value is known. Since Γ(x + 1) = xΓ(x), we have

Γ (n + f) = (n + f − 1 )Γ (n + f − 1) = (n + f − 1 )(n + f − 2)Γ (n + f − 2) .. . = (n + f − 1 )(n + f − 2)⋅⋅⋅(f)Γ (f )

Which is easy to calculate if we know Γ(f).

The gamma function has a meromorphic continuation to the entire complex plane with poles at the non-positive integers. It satisfies the product formula

∞ ( ) e−γz-∏ z- −1 z∕n Γ (z) = z 1 + n e n=1

where γ is Euler’s constant, and the functional equation

--π--- Γ (z)Γ (1 − z) = sinπz .

This entry is a derivative of the gamma function article from PlanetMath. Author of the orginial article: akrowne. History page of the original is here

"gamma function" is owned by bloftin.
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