= Gamma Function =

The '''gamma function''' is the generalization of a factorial.

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== Description ==

For a positive integer ''x'', the function can be defined as ''Γ(x) = (x-1)!''.

For a complex number ''z'' with a positive real part, the function can be defined as:

{{attachment:formula.svg}}

It should be clear that this formula evaluates to ''1'' when ''z = 1'', matching the other definition (''0! = 1'').

Furthermore, evaluating when ''z = α + 1'' leads to a recursive solution.

{{attachment:alpha1.svg}}

Using [[Calculus/IntegrationByParts|integration by parts]] with ''u = x^a^'' and ''dv = e^-x^ dx'':

{{attachment:alpha2.svg}}

{{attachment:alpha3.svg}}

Which can be succinctly rewritten as:

{{attachment:alpha4.svg}}

Again, this matches the other definition (''n! = n*(n-1)!'')



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CategoryRicottone