(a) If $f(z)$ is an analytic function with its real part is constant,…

Question

(a) If $f(z)$ is an analytic function with its real part is constant, then prove that it is a constant function.

(b) Determine the analytic function whose imaginary part is $\sinh x \cos y$.

(c) Find the image of the line $2x - y + 3 = 0$ under the transformation $w = \frac{1}{z}$.