Results 1 to 3 of 3

Thread: question computing residues after a substitution

  1. #1
    Junior Member
    Joined
    Apr 2008
    Posts
    41

    question computing residues after a substitution

    Hi members,

    see attached PDF file for my question

    Thank you
    Attached Files Attached Files
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Mar 2010
    Posts
    1,050
    Thanks
    281

    Re: question computing residues after a substitution

    The residue of a function f(z) at z=0 is just $\displaystyle \oint_C f(z)\,dz$ where C is a contour around the origin.

    If we substitute 2u for z, the contour changes, but that doesn't matter because the integral is the same for all contours around the origin. But $\displaystyle \oint f(z)\,dz$ is twice the size of $\displaystyle \oint f(2u)\,du$ (think back to first-year calculus), so the residue with z is actually twice the residue with u.

    If you think of the simplest case, the residue of $\displaystyle \frac{1}{z}$ at $\displaystyle z=0$ is 1, but if you substitute 2u for z there, you get the residue of $\displaystyle \frac{1}{2u}$ at $\displaystyle u=0$, which is $\displaystyle \frac{1}{2}$. The residue is not the same when you change variables.

    In your case, you have $\displaystyle \frac{e^\frac{z-1}{2}}{z-1}$ with residue 1 at $\displaystyle z=1$, and $\displaystyle \frac{e^u}{2u}$ with residue $\displaystyle \frac{1}{2}$ at $\displaystyle u=0$.

    - Hollywood
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Apr 2008
    Posts
    41

    Re: question computing residues after a substitution

    Hi,

    Thank you very much for the answer<;

    Gilbert
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: Nov 1st 2014, 03:03 PM
  2. Replies: 1
    Last Post: Nov 27th 2012, 07:22 AM
  3. Replies: 1
    Last Post: Oct 12th 2012, 02:01 AM
  4. Question regarding covert multiple poles when calculating residues
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: Jun 12th 2011, 11:53 AM
  5. Replies: 1
    Last Post: Dec 2nd 2008, 03:51 PM

/mathhelpforum @mathhelpforum