(a) Show that the function : $u(r, \theta) = r^3 \cos 3\theta - r \si…

Question

(a) Show that the function : $u(r, \theta) = r^3 \cos 3\theta - r \sin \theta$ is harmonic and find its conjugate function.

(b) Take a rectangular region C with vertices $2 \pm i$ and $-2 \pm i$ and let $g(a) = \oint_C \frac{(3z^2 - 2z + 5)}{z - a} dz$. Then find the values of $g(2)$, $g(-i)$, $g'(i)$ and $g'(2 + 2i)$.

(c) Explain four standard transformations with example in each case.