Difference between revisions of "Gamma function"
| Line 8: | Line 8: | ||
:<math>= \left[t^{z-1}(-e^{-t}) \right]_0^{\infty}- \int_{0}^{\infty}(z-1)t^{(z-2)}(-e^{-t})dt</math> | :<math>= \left[t^{z-1}(-e^{-t}) \right]_0^{\infty}- \int_{0}^{\infty}(z-1)t^{(z-2)}(-e^{-t})dt</math> | ||
:<math>=(z-1) \int_{0}^{\infty}t^{(z-1)-1}e^{-t}dt</math> | :<math>=(z-1) \int_{0}^{\infty}t^{(z-1)-1}e^{-t}dt</math> | ||
| − | :<math>=(z-1) \Gamma(z-1)</math> | + | :<math>=(z-1) \Gamma(z-1).</math> |
Thus, | Thus, | ||
:<math> \Gamma (z) = (z-1) \Gamma(z-1)</math> | :<math> \Gamma (z) = (z-1) \Gamma(z-1)</math> | ||
| Line 15: | Line 15: | ||
:<math> \Gamma(n) = (n-1)!. </math> | :<math> \Gamma(n) = (n-1)!. </math> | ||
| + | |||
| + | [[category:mathematics]] | ||
Revision as of 20:31, June 5, 2010
The Gamma funciton is defined as
Using integration by parts,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(z) = \int_{0}^{\infty}t^{z-1}d(e^{-t}dt)}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \left[t^{z-1}(-e^{-t}) \right]_0^{\infty}- \int_{0}^{\infty}(z-1)t^{(z-2)}(-e^{-t})dt}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =(z-1) \int_{0}^{\infty}t^{(z-1)-1}e^{-t}dt}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =(z-1) \Gamma(z-1).}
Thus,
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma (z) = (z-1) \Gamma(z-1)}
Using this, and the the fact that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(0)=1} , then we can get the factorial. For Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} a positive integer,
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(n) = (n-1)!. }