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Complex integration

  1. Apr 15, 2014 #1
    1. The problem statement, all variables and given/known data

    attachment.php?attachmentid=68696&stc=1&d=1397618882.jpg

    2. Relevant equations



    3. The attempt at a solution

    I did part (a) which is pretty easy. Use the Cauchy integral formula. Part (b) says "hence", which leads me to believe that part (a) can be used for part (b), but I cannot see anything remotely related between the two. I don't want to do weird manipulations and substitutions if part (a) can help.

    If someone could help me out with this I would really appreciate it.
     

    Attached Files:

  2. jcsd
  3. Apr 16, 2014 #2

    Dick

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    Science Advisor
    Homework Helper

    Just substitute ##e^{i \theta}## for ##z## in the integral and turn it into an integral ##d\theta##. Then use deMoivre.
     
  4. Apr 16, 2014 #3
    Thanks Dick! Easy as pie now
     
  5. Apr 16, 2014 #4
    Here is an elementary approach for part b) if you like:

    Notice that the integral you seek is
    $$\begin{aligned}
    \Re\left(\int_0^{\pi} e^{k\cos t}e^{ik\sin t}\,dt\right) &= \Re\left(\int_0^{\pi}e^{ke^{it}}\,dt\right)\\
    &=\Re\left(\int_0^{\pi} \left(1+ke^{it}+k^2e^{2it}+\cdots \right)\,dt\right)\\
    &=\boxed{\pi}\\
    \end{aligned}$$
     
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