Changes

The '''Gamma funciton''' is defined as  

The '''Gamma funciton''' is defined as  

===Factorial===

===Factorial===

Using [[Integration_by_parts|integration by parts]],

Using [[integration by parts]],

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

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

:<math> \Gamma (z) = (z-1) \Gamma(z-1).</math>

:<math> \Gamma (z) = (z-1) \Gamma(z-1).</math>

Using this, and the the fact that <math>\Gamma(0)=1</math>, then we can get the [[Factorial|factorial]]. For <math>n</math> a positive integer,

Using this, and the fact that <math>\Gamma(1)=1</math>, then we can get the [[factorial]] function. For <math>n</math> a positive integer,

:<math> \Gamma(n) = (n-1)!. </math>

:<math> \Gamma(n) = (n-1)!. </math>

:<math>= \int_{-\infty}^{\infty} e^{-x^2}dx </math>

:<math>= \int_{-\infty}^{\infty} e^{-x^2}dx </math>

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

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

Thus,

Thus,

[[category:mathematics]]

[[Category:Mathematics]]