Complex integration

1. Apr 15, 2014

stripes

1. The problem statement, all variables and given/known data

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.

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2. Apr 16, 2014

Dick

Just substitute $e^{i \theta}$ for $z$ in the integral and turn it into an integral $d\theta$. Then use deMoivre.

3. Apr 16, 2014

stripes

Thanks Dick! Easy as pie now

4. Apr 16, 2014

Saitama

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}