Evaluate $ \int_{C(0,5)} \frac{1}{i-\cos z}dz $.

Question

How do I evaluate $$ \int_{C(0,5)} \frac{1}{i-\cos z}dz? $$

Do I have to find the poles first, and then use residue theorem, or find where the function is holomorphic and then integrate using Cauchy's theorem? Could anyone give me some hints?

Answer 1

Following my comment:

$$e^{2iz}+1=2ie^{iz}\iff e^{2iz}-2ie^{iz}+1=0$$

The above quadratic's discriminant:

$$\Delta=-4-4=-8\implies e^{iz}_{1,2}=\frac{2i\pm2\sqrt2i}{2}=\begin{cases}(1+\sqrt2)i\\{}\\(1-\sqrt2)i\end{cases}$$

so taking the principal branch of the complex logarithm:

$$iz_{1,2}=\text{Log}\,(1\pm\sqrt2)i=\frac12\left(\log(3\pm2\sqrt2)+\pi i\right)\;$$

and from here

$$z_{1,2}=-\frac12i\left(\log(3\pm2\sqrt2)+\pi i\right)=\frac\pi2-\frac12\log(3\pm2\sqrt2)i$$

If you check the above numbers' modulus, one gets

$$|z_i|^2=\frac{\pi^2}4+\frac14\log^2(3\pm2\sqrt2)$$

and I think these two are within the integration path...

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