Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Integral of x^(m-1)/(1+x) wrt x?

  1. Jan 15, 2014 #1
    How does this:
    [itex]\int_0^\infty\frac{x^{m-1}}{1+x} dx[/itex]
    equal
    [itex]\frac{\pi}{sin(m\pi)}[/itex]
    ?
    It has been simply stated as a fact in a proof of the so called "Euler reflection formula" in a textbook.
    I have tried the usual ways, substitution, integration by parts and even series expansion of 1/1+x but I can't find how the above equality is true.
     
  2. jcsd
  3. Jan 15, 2014 #2

    maajdl

    User Avatar
    Gold Member

    I suggest you to first try on some special cases.
    For example: m=1/2, m=3/2, m=1/4.
    Then, also, analyze the domain of convergence of this integral.
    Besides, I think there is no elementary primitive.
    Have a look on the wiki article: http://en.wikipedia.org/wiki/Reflection_formula .
    Maybe you might prove the relation of your integral to the product [itex]\Gamma[/itex](m)[itex]\Gamma[/itex](1-m)
     
  4. Jan 15, 2014 #3

    maajdl

    User Avatar
    Gold Member

    Far from obvious!
    See pages 13-17

    http://warwickmaths.org/files/gamma.pdf [Broken]
     
    Last edited by a moderator: May 6, 2017
  5. Jan 15, 2014 #4
    Thanks. So I guess I will just have to accept that as a fact. No easy and 'real' way to obtain the result without resorting to other forms of Gamma function.
     
  6. Jan 15, 2014 #5

    maajdl

    User Avatar
    Gold Member

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Integral of x^(m-1)/(1+x) wrt x?
  1. Integral: x/1+sqrt(x) (Replies: 7)

  2. Integrating (x^2+x)^-1 (Replies: 6)

Loading...