(a) Compute the limit of the following functions:
(i) \( \lim _{z \rightarrow 0} \frac{e^{z^{2}}-1}{z} \)
(2 marks)
(ii) \( \lim _{z \rightarrow 0} \frac{z^{2}+z-i}{2 z+3} \)
(2 marks)
(b) Show that the function \( e^{x}(\cos y+i \sin y) \) is analytic function, and find its derivative.
(3 marks)
(c) Evaluate \( \int_{1-i}^{2+i}(2 x+i y+1) d z \) along the following paths:
(i) \( x=t+1, y=2 t^{2}-1 \)
(2 marks)
(ii) THe straight line joining \( 1-i \) and \( 2+i \).
(3 marks)
(d) Use the knowledge of line integral to compute \( \int_{C} z^{2} d z \) where curve \( C \) is boundary of a triangle with vertices \( 0,1+i,-1+i \) clockwise.
(3 marks)
(e) Use Cauchy integral formula to calculate \( \int_{C} \frac{2 z+1}{z^{2}+z} d z \) where \( C \) is \( z=\frac{1}{2} \).
(3 marks)
