Difference between revisions of "Beta"
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[[Beta is symmetric]]<br /> | [[Beta is symmetric]]<br /> | ||
[[B(x,y)=integral (t^(x-1)+t^(y-1))(1+t)^(-x-y) dt]]<br /> | [[B(x,y)=integral (t^(x-1)+t^(y-1))(1+t)^(-x-y) dt]]<br /> | ||
| + | [[B(x,y+1)=(y/x)B(x+1,y)]]<br /> | ||
=Videos= | =Videos= | ||
Revision as of 22:53, 24 June 2017
The beta function $B$ (note: $B$ is capital $\beta$ in Greek) is defined by the following formula for $\mathrm{Re}(x)>0$ and $\mathrm{Re}(y)>0$: $$B(x,y)=\displaystyle\int_0^1 t^{x-1}(1-t)^{y-1} \mathrm{d}t.$$
Properties
Partial derivative of beta function
Beta in terms of gamma
Beta in terms of sine and cosine
Beta as improper integral
Beta is symmetric
B(x,y)=integral (t^(x-1)+t^(y-1))(1+t)^(-x-y) dt
B(x,y+1)=(y/x)B(x+1,y)
Videos
Beta integral function - basic identity (5 December 2011)
Beta Function - Gamma Function Relation Part 1 (5 December 2011)
Beta Function - Gamma Function Relation Part 2 (5 December 2011)
Beta function - Part 1 (14 February 2012)
Mod-04 Lec-09 Analytic continuation and the gamma function (Part I) (3 June 2014)
Gamma function - Part 10 - Beta function (31 July 2012)
Beta function (19 September 2012)
Gamma Function, Transformation of Gamma Function, Beta Function, Transformation of Beta Function (30 October 2012)
Beta Integral: Even Powers Of Sine Function (26 December 2012)
References
- 1953: Harry Bateman: Higher Transcendental Functions Volume I ... (previous) ... (next): $\S 1.5 (1)$
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $6.2.1$
