Difference between revisions of "Beta"

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(Videos)
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=Videos=
 
=Videos=
[https://www.youtube.com/watch?v=SYfLj-koGJ0 Beta function - Part 1]
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[https://www.youtube.com/watch?v=SYfLj-koGJ0 Beta function - Part 1]<br />
[https://www.youtube.com/watch?v=3vBBh0SDpqM Beta function]
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[https://www.youtube.com/watch?v=3vBBh0SDpqM Beta function]<br />
[https://www.youtube.com/watch?v=HP0kCw2FWJk Beta integral function - basic identity]
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[https://www.youtube.com/watch?v=HP0kCw2FWJk Beta integral function - basic identity]<br />
[https://www.youtube.com/watch?v=Korx_G7eySQ Gamma function - Part 10 - Beta function]
+
[https://www.youtube.com/watch?v=Korx_G7eySQ Gamma function - Part 10 - Beta function]<br />
[https://www.youtube.com/watch?v=C3PmT6oNEew Mod-04 Lec-09 Analytic continuation and the gamma function (Part I)]
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[https://www.youtube.com/watch?v=C3PmT6oNEew Mod-04 Lec-09 Analytic continuation and the gamma function (Part I)]<br />
[https://www.youtube.com/watch?v=w0g9Ff9V7rs Gamma Function, Transformation of Gamma Function, Beta Function, Transformation of Beta Function]
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[https://www.youtube.com/watch?v=w0g9Ff9V7rs Gamma Function, Transformation of Gamma Function, Beta Function, Transformation of Beta Function]<br />
[https://www.youtube.com/watch?v=VlROhSMiI2A Beta Function - Gamma Function Relation Part 1]
+
[https://www.youtube.com/watch?v=VlROhSMiI2A Beta Function - Gamma Function Relation Part 1]<br />
[https://www.youtube.com/watch?v=fFxcrCaYR3k Beta Function - Gamma Function Relation Part 2]
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[https://www.youtube.com/watch?v=fFxcrCaYR3k Beta Function - Gamma Function Relation Part 2]<br />
[https://www.youtube.com/watch?v=uK0KUXVDqlQ Beta Integral: Even Powers Of Sine Function]
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[https://www.youtube.com/watch?v=uK0KUXVDqlQ Beta Integral: Even Powers Of Sine Function]<br />
  
 
=References=
 
=References=
 
Bell. Special Functions <br />
 
Bell. Special Functions <br />
 
[http://web.mst.edu/~lmhall/SPFNS/spfns.pdf Special functions by Leon Hall]
 
[http://web.mst.edu/~lmhall/SPFNS/spfns.pdf Special functions by Leon Hall]

Revision as of 21:23, 27 April 2015

The beta function $B$ (note: $B$ is capital $\beta$ in Greek) is defined by the formula $$B(x,y)=\displaystyle\int_0^1 t^{x-1}(1-t)^{y-1}dt.$$

Properties

Theorem: $B(x,y)=B(y,x)$

Proof:

Theorem: (i) $B(x+1,y)=\dfrac{x}{x+y} B(x,y)$
(ii) $B(x,y+1)=\dfrac{y}{x+y}B(x,y)$

Proof:

Theorem

The following formula holds: $$\dfrac{\partial}{\partial x} B(x,y)=B(x,y) \left( \dfrac{\Gamma'(x)}{\Gamma(x)} - \dfrac{\Gamma'(x+y)}{\Gamma(x+y)} \right) = B(x,y)(\psi(x) - \psi(x+y)),$$ where $B$ denotes the Beta function, $\Gamma$ denotes the gamma function, and $\psi$ denotes the digamma function.

Proof

References

Theorem

The following formula holds: $$B(x,y)=\dfrac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)},$$ where $B$ denotes the beta function and $\Gamma$ denotes the gamma function.

Proof

References

Theorem

The following formula holds: $$B(x,y)=2 \displaystyle\int_0^{\frac{\pi}{2}} (\sin t)^{2x-1}(\cos t)^{2y-1} \mathrm{d}t,$$ where $B$ denotes the beta function, $\sin$ denotes the sine function, and $\cos$ denotes the cosine function.

Proof

From the definition, $$B(x,y)=\displaystyle\int_0^1 \xi^{x-1} (1-\xi)^{y-1} \mathrm{d}\xi.$$ Let $\xi=\sin^2(t)$. Then $d\xi = 2\sin(t)\cos(t)$. Also if $\xi=0$ then $0=\sin^2(t)$ implies that $t=\arcsin(0)=0$ and if $\xi=1$, then $1=\sin^2(t)$ implies $t=\arcsin(1)=\dfrac{\pi}{2}$. Therefore using substitution and the Pythagorean identity for sin and cos, $$\begin{array}{ll} B(x,y) &= \displaystyle\int_0^1 \xi^{x-1}(1-\xi)^{y-1} \mathrm{d}\xi \\ &= \displaystyle\int_0^{\frac{\pi}{2}} (\sin(t))^{2x-2} (1-\sin^2(t))^{y-1} 2 \sin(t)\cos(t) \mathrm{d}t \\ &= 2 \displaystyle\int_0^{\frac{\pi}{2}} (\sin(t))^{2x-1} (\cos(t))^{2y-1} \mathrm{d}t, \end{array}$$ as was to be shown. █

References

Videos

Beta function - Part 1
Beta function
Beta integral function - basic identity
Gamma function - Part 10 - Beta function
Mod-04 Lec-09 Analytic continuation and the gamma function (Part I)
Gamma Function, Transformation of Gamma Function, Beta Function, Transformation of Beta Function
Beta Function - Gamma Function Relation Part 1
Beta Function - Gamma Function Relation Part 2
Beta Integral: Even Powers Of Sine Function

References

Bell. Special Functions
Special functions by Leon Hall