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Math Help - Elegant Trigonometric Integral

  1. #1
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    Elegant Trigonometric Integral

    I was looking at a book on Mathematical Physics today and found the following interesting identity involving sine and cosine integrals. But I am not going to tell you what the identity is, to make it more difficult to find it.

    Here it is below. In the special case where n,m are integers it solves nicely.

    But is also works for real numbers as well. Probably also for complex numbers but let us not go there.
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  2. #2
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    Hint1: let 2a=n and 2b=m.
    Hint2: Beta function.
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  3. #3
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    Okay here is the solution below.
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  4. #4
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    Applyin' Beta Function identity for x,y>0,

    \beta(x,y)=2\int_0^{\pi /2} {\sin ^{2x - 1} (u)\cos ^{2y - 1} (u)\,du} yields the result immediately:

    \int_0^{\pi /2} {\sin ^n x\cos ^m x\,dx}  = \frac{1}<br />
{2}\beta \left( {\frac{{n + 1}}<br />
{2},\frac{{m + 1}}<br />
{2}} \right) = \frac{1}<br />
{2} \cdot \frac{{\Gamma \left( {\dfrac{{n + 1}}<br />
{2}} \right)\Gamma \left( {\dfrac{{m + 1}}<br />
{2}} \right)}}<br />
{{\Gamma \left( {\dfrac{{n + m + 2}}<br />
{2}} \right)}}.

    ----

    I remember I saw this once, but my integration skills were low, so I couldn't get anything you wrote.

    Now with more knowledge this really makes sense to me
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Krizalid View Post
    I remember I saw this once, but my integration skills were low, so I couldn't get anything you wrote.
    this was less than a year ago. you have come far since then it seems... that gives little ol' me some hope...
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