Gamma function

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\frac{d}{dx} \sin x=?\, This article/section deals with mathematical concepts appropriate for a student in late high school or early university.

The Gamma funciton is defined as

\Gamma(z) = \int_{0}^{\infty}t^{z-1}e^{-t}dt

Relations and values

Factorial

Using integration by parts,

\Gamma(z) = \int_{0}^{\infty}t^{z-1}d(e^{-t}dt)

= \left[t^{z-1}(-e^{-t}) \right]_0^{\infty}- \int_{0}^{\infty}(z-1)t^{(z-2)}(-e^{-t})dt.

At t = 0, tz − 1( − et) goes to 0. Using L'Hopital's rule, it's easy to show that \lim_{t\to \infty} \frac{t^{z-1}}{e^{t}} = 0.

So,

=(z-1) \int_{0}^{\infty}t^{(z-1)-1}e^{-t}dt
= (z − 1)Γ(z − 1).

Thus,

Γ(z) = (z − 1)Γ(z − 1).

Using this, and the the fact that Γ(1) = 1, then we can get the factorial function. For n a positive integer,

Γ(n) = (n − 1)!.

z=1/2

For z=1/2,

 \Gamma(1/2) = \int_{0}^{\infty}t^{-1/2}e^{-t}dt

Substituting x = t1 / 2,

=\int_{0}^{\infty} (\frac{1}{x}) e^{-x^2} (2xdx )
=2 \int_{0}^{\infty} e^{-x^2}dx
= \int_{-\infty}^{\infty} e^{-x^2}dx

where in the last line we used the fact that e^{-x^2} is an even function. The integral is called the Gaussian integral and has a well known value of \sqrt\pi.

Thus,

 \Gamma(1/2) = \sqrt\pi.
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