1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Complex trigonometric integral

  1. Nov 21, 2014 #1
    1. The problem statement, all variables and given/known data
    Calculate the complex integral along the closed path indicated:
    $$ \oint_C\frac{\sin{z}}{z^2+\pi^2}dz,\,\,|z-2i|=2.$$
    2. Relevant equations
    $$ \sin{z}=\frac{e^{iz}-e^{-iz}}{2i} $$
    $$ e^{iz}=e^{i(x+iy)}=e^{-y+ix}=e^{-y}(\cos{x}+i\sin{x}) $$
    3. The attempt at a solution
    I really don't know what to do here.. Everything I tried led me to a dead end. Is there a clever substitution to be made? I tried substituting ##z=x+iy##, I tried ##z=e^{it}+2i## and even tried expanding ##\sin{z}##, but it got me nowhere. Any help is appreciated. Thanks!
     
  2. jcsd
  3. Nov 21, 2014 #2

    ShayanJ

    User Avatar
    Gold Member

    You should use calculus of residues!
     
  4. Nov 21, 2014 #3

    Zondrina

    User Avatar
    Homework Helper

    The denominator ##z^2 + \pi^2## has singularities at ##z = \pm i \pi##.

    Do these lie within the positively oriented contour ##|z - 2i| = 2##?
     
  5. Nov 21, 2014 #4
    Yes, ##i\pi## lies within the contour, which means that the integral for any closed path around ##i\pi## would wield the right answer, but that didn't help me much. I still don't know how to calculate the integral.
     
  6. Nov 21, 2014 #5

    ShayanJ

    User Avatar
    Gold Member

    Check here!
     
  7. Nov 21, 2014 #6

    Zondrina

    User Avatar
    Homework Helper

    $$\oint_C f(z) \space dz = (2 \pi i) \times \space \text{Res}[f(z), i \pi]$$
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Complex trigonometric integral
  1. Trigonometric integral (Replies: 3)

Loading...