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Homework Help: Beta Function, Solve the

  1. Aug 25, 2010 #1
    Beta Function Demonstration Problem.

    I pushed the "Enter" key by accident and the topics name got ruined.

    Good Day to everyone, I have this problem, (In the context of the Gamma and Beta functions) I have to demonstrate that:


    1. The problem statement, all variables and given/known data

    Demonstrate that http://img576.imageshack.us/i/integralsola.jpg/ [Broken]

    Uploaded with ImageShack.us

    2. Relevant equations

    Can I state that x=y so that tan^v(theta) can be expressed as the Trigonometric representation Beta Function?

    3. The attempt at a solution

    http://img202.imageshack.us/img202/1423/intento.jpg [Broken]

    Uploaded with ImageShack.us

    So once i make the substitution, i try to solve the integral, if've use both variable change and integration by parts, but always get everything messy and far away from the desired result, if someone could please give a direction, is there some essential substition that I'm not making?, or some property of the function I'm not applying?

    Any little direction you give i would gladly apreciatte

    Note: I drew these equations on M.Word

    Thank you all.
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Aug 26, 2010 #2
    I think that should be:

    [tex]\int_0^{\pi/2} \tan^v(x)dx=\pi/2 \sec(\pi v/2)[/tex]

    Now suppose we have the definition:

    [tex]\beta(x,y)=2\int_0^{\pi/2} \sin^{2x-1}(t)\cos^{2y-1}(t)dt[/tex]

    then representing tan in terms of sin and cos, can you figure what x and y are in terms of v, then once you get it into the beta format and then represent the beta function in terms of the gamma quotient (see mathworld on Beta), then represent that gamma quotient in terms of the secant function.
     
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