(a) Evaluate $\int_{0}^{1+i} (x^2 - iy) dz$ along the line $x = y$. (…

Question

(a) Evaluate $\int_{0}^{1+i} (x^2 - iy) dz$ along the line $x = y$.

(b) Evaluate $\int_{a}^{b} f(z) dz$ along the curve given by $x = t + 1$, $y = 2t^2 - 1$, where $a = 1 - i, b = 2 + i$, and $f(z) = 2x + iy + 1$.

(c) Find the nature of singularities and residue at these points of the function $f(z) = \frac{e^z}{(z^2 + \pi^2)^2}$.