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Math Help - Proof!

  1. #1
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    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:
    
    
    
    
    
    \text{Re} \left(\alpha+\mu\right) > 0, 
    
    
    
    
    \text{Re} \left(v-\alpha+2\right) > 0, 
    
    
    
    
    \text{Re} (1-\alpha-v) > 0.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Legendre View Post
    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:
    
    
    
    
    
    \text{Re} \left(\alpha+\mu\right) > 0, 
    
    
    
    
    \text{Re} \left(v-\alpha+2\right) > 0, 
    
    
    
    
    \text{Re} (1-\alpha-v) > 0.
    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?
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  3. #3
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    Quote Originally Posted by Drexel28 View Post
    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!
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  4. #4
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    Quote Originally Posted by Drexel28 View Post
    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
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