Beta function

  • Thread starter Ikastun
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  • #1
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Homework Statement



∫(0,1) √x/√[3]1-x

Homework Equations



[itex]\Gamma[/itex]p[itex]\Gamma[/itex]q/[itex]\Gamma[/itex]p+q

The Attempt at a Solution



p-1=1/2 →p=3/2
q-1=-1/3 →q=2/3

β(3/2,2/3)=[itex]\Gamma[/itex](3/2) [itex]\Gamma[/itex](2/3)/[itex]\Gamma[/itex](13/6)

[itex]\Gamma[/itex]3/2=1/2[itex]\Gamma[/itex](1/2)=√π/2
[itex]\Gamma[/itex]2/3=-1/3
[itex]\Gamma[/itex]13/6=7/6 1/6=7/36

β(3/2,2/3)=-6√π/7
 

Answers and Replies

  • #2
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10,583
[itex]\Gamma[/itex]2/3=-1/3
[itex]\Gamma[/itex]13/6=7/6 1/6=7/36
That looks wrong.

Can you explain what you want to calculate, how you attempt to do this and where your problem is?
 
  • #3
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Hello and thank you for answering.

My problem begins with the part you quote. I don't know how to properly use the recursive formula in those cases.
Regarding my attempt to calculate the integral, what I wrote above is everything.
 
  • #4
34,465
10,583
##\Gamma(\frac{13}{6})=\frac{7}{6}\Gamma(\frac{7}{6}) =\frac{7}{36}\Gamma(\frac{1}{6})##
For some values, an analytic expression is known, in general this doesn't work and you have to live with the expressions (or find a numerical approximation).

Regarding my attempt to calculate the integral, what I wrote above is everything.
There is some connections between the formulas that you could explain.
 

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