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Solve integral

  1. Jan 8, 2012 #1
    1. The problem statement, all variables and given/known data

    Solve the integral annd express it through the gamma f

    2. Relevant equations

    cos(theta)^(2k+1)

    3. The attempt at a solution
     

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  2. jcsd
  3. Jan 8, 2012 #2

    Simon Bridge

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    You mean:
    [tex]\int_0^{\frac{\pi}{2}} \cos^{2k+1}(\theta)d\theta[/tex]... eg: evaluate the definite integral of an arbitrary odd-power of cosine.

    The standard approach is to start by integrating by parts.
    You'll end up with a reducing formula which you can turn into a ratio of factorials - apply the limits - after which it is a matter of relating that to the factorial form of the gamma function.

    eg. http://mathworld.wolfram.com/CosineIntegral.html
     
  4. Jan 8, 2012 #3
    Its difficult do see what is happening here.
     

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  5. Jan 8, 2012 #4

    Simon Bridge

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    If it was easy there'd be no point setting it as a problem.
    I'm not going to do it for you ...

    Do you know what a gamma function is? You can represent it as a factorial?
    Can you identify where you are having trouble seeing what is going on?
    Perhaps you should try to do the derivation for yourself?
     
    Last edited: Jan 8, 2012
  6. Jan 8, 2012 #5
    Thanks.
     

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  7. Jan 8, 2012 #6

    Simon Bridge

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    Really? And I thought I was being mean.....

    The trig-form of the beta function aye - yep, that's a tad more elegant that the path I was suggesting before (the more usual one)... but relies on a hand-wave: do you know how the beta function is derived?

    Also - you have [itex]\frac{1}{2}B(\frac{1}{2},k+1)[/itex] but you've spotted that.

    If you look at the cosine formula - you have to evaluate the limits ... at first it looks grim because it gives you a sum of terms like [itex]\sin\theta\cos^{2k}\theta[/itex] which is zero at both limits ... unless k=0 ... which is the first term in the sum, which is 1.

    After that it is a matter of subbing in the factorial representation of the gamma function.
    Which would be a concrete proof.

    Yours is shorter and if you have the beta function in class notes then you should be fine using it.
     
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