Gamma function - Revision history

DavidB4-bot: /* Factorial */Spelling/Grammar Check, typos fixed: the the → the

2016-07-18T17:46:25Z

Factorial: Spelling/Grammar Check, typos fixed: the the → the

DavidB4-bot: clean up & uniformity

2016-07-13T13:17:02Z

clean up & uniformity

JLauttamus: add template

2010-06-06T15:48:23Z

add template

ReligiousRight at 14:41, June 6, 2010

2010-06-06T14:41:01Z

ReligiousRight at 14:37, June 6, 2010

2010-06-06T14:37:01Z

ReligiousRight at 14:36, June 6, 2010

2010-06-06T14:36:11Z

ReligiousRight at 21:05, June 5, 2010

2010-06-05T21:05:05Z

ReligiousRight at 20:31, June 5, 2010

2010-06-05T20:31:39Z

ReligiousRight at 20:30, June 5, 2010

2010-06-05T20:30:34Z

ReligiousRight: Created page with 'The '''Gamma funciton''' is defined as $\Gamma(z) = \int_{0}^{\infty}t^{z-1}e^{-t}dt$ Using integration by parts, $\Gamma(z) = \int_{…'$

2010-06-05T20:25:36Z

Created page with 'The '''Gamma funciton''' is defined as <math>\Gamma(z) = \int_{0}^{\infty}t^{z-1}e^{-t}dt</math> Using integration by parts, <math>\Gamma(z) = \int_{…'

New page

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

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

Using [[Integration_by_parts|integration by parts]],

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

@@ Line 22: / Line 22: @@
 :<math> \Gamma (z) = (z-1) \Gamma(z-1).</math>
-Using this, and the the fact that <math>\Gamma(1)=1</math>, then we can get the [[factorial]] function. 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>

@@ Line 1: / Line 1: @@
-{{Template:Math-h}}
+{{Math-h}}
 The '''Gamma funciton''' is defined as
@@ Line 8: / Line 8: @@
 ===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>
@@ Line 22: / Line 22: @@
 :<math> \Gamma (z) = (z-1) \Gamma(z-1).</math>
-Using this, and the the fact that <math>\Gamma(1)=1</math>, then we can get the [[Factorial|factorial]] function. For <math>n</math> a positive integer,
+Using this, and the 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>
@@ Line 45: / Line 45: @@
-[[category:mathematics]]
+[[Category:Mathematics]]

← Older revision		Revision as of 15:48, June 6, 2010
Line 1:		Line 1:
		+	{{Template:Math-h}}
		+
	The '''Gamma funciton''' is defined as		The '''Gamma funciton''' is defined as

@@ Line 35: / Line 35: @@
 :<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 an even function. 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,

@@ Line 35: / Line 35: @@
 :<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 the function <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,

@@ Line 3: / Line 3: @@
 <math>\Gamma(z) = \int_{0}^{\infty}t^{z-1}e^{-t}dt</math>
 Using [[Integration_by_parts|integration by parts]],
 <math>\Gamma(z) = \int_{0}^{\infty}t^{z-1}d(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>= \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) \Gamma(z-1).</math>
 Thus,
-:<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,
 :<math> \Gamma(n) = (n-1)!. </math>
 [[category:mathematics]]