The Gamma Function

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The gamma function is a generalized version of the factorial function.

gamma function

The gamma function comes in many forms. This integral representation is nice because the connection with factorials can be made quickly with integration by parts. (Γ(z)=(z-1)!)

It is sometimes useful to evaluate the Γ function in the form of an infinite product. To obtain this form, first insert a defnition of e.

gamma function

Then substitute s=t/n.

gamma function

Then integrate by parts.

gamma function

Factor n! from the denominator.

gamma function

Rewrite nz as the exponential of a logarithm and factor an exponential in terms of the Euler-Mascheroni constant.

gamma function

The following definition of the Euler-Mascheroni constant has been used.

euler-mascheroni constant definition

euler-mascheroni constant approximate value

Why bother? Now it's quite easy to evaluate the Γ function in some circumstances. For example, consider an imaginary number iy.

reflection formula

Used in conjunction with the following product for the sine function,

infinite product for sine,

the product becomes

reflection formula

This result is closely related to Euler's reflection formula.

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