Given the analytic function $f(z) = u(x,y) + iv(x,y)$, given that $$ u(x,0) = \sin^2x \ , \ v(x,0) = 0 $$ Find $f(z_0)$ for $z_0 = 5 + i8$.
From Cauchy integral representation I could say that
$$
f(z_0) = \frac{1}{2 \pi i} \int_{\gamma} \frac{f(z)}{z-z_0}dz
$$
with $\gamma$ = $\gamma_1$ + $\gamma_2$ and
$$
\gamma_1 = \mathbb{R} \ , \ \gamma_2 = Re^{i\theta} \ \text{for} \ -\pi \leq \theta \leq + \pi
$$
Thus I am able to evaluate $g(z) = \dfrac{f(z)}{z-z_0}$ over $\gamma_1$, but not over $\gamma_2$. How could I proceed?