Complex Analysis, Contour Integral Formula

Question

I have difficulties with the following question: Let a be a given complex number satisfying $0 < \left\vert a\right\vert < 2$ and let $\gamma$ be the unit circle of radius $3$ oriented in the positive sense. $$ \mbox{Determine the value of the contour integral}\quad \oint_{\gamma}{\Re\left(z\right) \over z - a}\,\mathrm{d}z\quad \mbox{in terms of}\ a. $$

I have tried expressed $\Re\left(z\right)$ as $\left(z + \bar{z}\right)/2$, and I am stuck at this question. Someone please help give some inspiration, thank you so much !.

Answer 1

Hint: When $|z|=3$ then $2{\rm Re\ } z = z+\bar{z} = z + \frac{9}{z}$ and you are then reduced to a calculation of $$ \frac12 \oint_{\partial B_3} (z+\frac{9}{z}) \frac{1}{z-a} dz=\oint \frac{z/2}{z-a} + \frac{9}{2a} \left( \frac{1}{z-a} - \frac{1}{z}\right) dz.$$ If you know C.I.F then this should be feasible.

Answer 2

On the "unit circle of radius 3 oriented in the positive sense" we can write $3cos(\theta)+ 3i sin(\theta)$ with $\theta$ going from 0 to $2\pi$. The real part of z is $3cos(\theta)$ so this integral is $\int_0^{2\pi} \frac{cos(\theta)}{cos(\theta)+ isin(\theta)- a} d\theta$