Proof!

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December 16th 2009, 06:34 PM
Legendre

Proof!

Prove that:

$\int_{0}^{\infty}(\sinh{x})^{\alpha-1}P_{v}^{-\mu}(\cosh{x})\,dx = \frac{2^{-1-\mu}\Gamma\left(\frac{1}{2}\alpha+\frac{1}{2}\mu\r ight)\Gamma\left(\frac{1}{2}v-\frac{1}{2}\alpha+1\right)\Gamma\left(\frac{1}{2}-\frac{1}{2}\alpha-\frac{1}{2}v\right)}{\Gamma\left(\frac{1}{2}\mu+\f rac{1}{2}v+1\right)\Gamma\left(\frac{1}{2}+\frac{1 }{2}\mu-\frac{1}{2}v\right)\Gamma\left(1+\frac{1}{2}\mu-\frac{1}{2}\alpha\right)}$

Code:

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December 16th 2009, 08:46 PM
Drexel28

Quote:

Originally Posted by Legendre

Prove that:

$\int_{0}^{\infty}(\sinh{x})^{\alpha-1}P_{v}^{-\mu}(\cosh{x})\,dx = \frac{2^{-1-\mu}\Gamma\left(\frac{1}{2}\alpha+\frac{1}{2}\mu\r ight)\Gamma\left(\frac{1}{2}v-\frac{1}{2}\alpha+1\right)\Gamma\left(\frac{1}{2}-\frac{1}{2}\alpha-\frac{1}{2}v\right)}{\Gamma\left(\frac{1}{2}\mu+\f rac{1}{2}v+1\right)\Gamma\left(\frac{1}{2}+\frac{1 }{2}\mu-\frac{1}{2}v\right)\Gamma\left(1+\frac{1}{2}\mu-\frac{1}{2}\alpha\right)}$

Code:

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Challenge problems should be something that requires you to be clever, nto something that takes two weeks to do because it is so dang laborious. Does this have a clever solution or did you just find the most complicated looking integral on the internet and post it up here?
December 16th 2009, 11:53 PM
NonCommAlg

Quote:

Originally Posted by Drexel28

Challenge problems should be something that requires you to be clever, nto something that takes two weeks to do because it is so dang laborious. Does this have a clever solution or did you just find the most complicated looking integral on the internet and post it up here?

dang laborious! (Rofl)
January 7th 2010, 02:20 PM
TWiX

Quote:

Originally Posted by Drexel28

Challenge problems should be something that requires you to be clever, nto something that takes two weeks to do because it is so dang laborious. Does this have a clever solution or did you just find the most complicated looking integral on the internet and post it up here?

xD
(Rofl)(Rofl)