2. (a) Write the function f: ā ā ā, f(z) = exp(ų) in the form f(z) = u(x,y) + iv(x,y) where u and v are real valued functions of the real variables x and y.
(b) Is the function f: ā ā ā, f(z) = exp(ų)
(i) continuous at 0,
(ii) differentiable at 0?
Justify your answer in both cases.
(c) Define what is understood for a function u: ā" ā ā to be harmonic.
(d) Show that the function u(x,y) = xeĖ£ cos y - yeĖ£ sin y is harmonic and find the most general holomorphic function, f, with real part u(x,y). Write f as a function of z.