(a) Consider a function $f(z)$ given by $f(z) = \frac{(1+i)x^3 + (1-i…

Question

(a) Consider a function $f(z)$ given by $f(z) = \frac{(1+i)x^3 + (1-i)y^3}{|z|^2}$ for $z \neq 0$ and $f(0) = 0$. Examine the continuity of $f(z)$ at $z = 0$. Is it differentiable at the origin ?

(b) Evaluate : $\oint_C \frac{(\sin \pi z + \cos \pi z)dz}{(z-1)(z^2 - 4)}$ where C is the circle $|z| = \sqrt{5}$.

(c) Evaluate $\int_a^b f(z) dz$ along the straight line joining $a$ to $b$, where $a = 2 - i$, $b = 3 + 2i$ and $f(z) = 2x + 3iy + 1$.