Hi all, i have a couple of questions on my complex analysis course that i cannot do and was wondering if you could help at all.

QUESTION 1

a) Suppose that the functions f(z) = u(x,y) + iv(x,y) and g(x) = v(x,y) + iu(x,y) are analytic in some domain D. Show that both u and v are constant functions

b) Let f be a holomorphic function on the punctured disk D'(0,R) = {z in complex plane : 0 < |z| < R}, where R > 0 is fixed. Write down the formulae for c(subscript)n in the Laurent expansion

f(z) = sum from n = minus infinity to infinity c(subscript)n . z^n (by this i mean cn multiplied by z^n)

Using these formulae, prove that if f is bounded on D'(0,R), it has a removable singularity at 0.

c) FInd the maximal radius R > 0 for which the function f(z) = (sin z)^-1 is holomorphic in D'(0,R), and find the principal part of its Laurent expansion about z(subscript)0 = 0 (centred at 0)

(the z(subscript)0 refers to: sum from k = minus infinity to infinity c(subscript)k(z - z(subscript)0)^k the Laurent expansion)

QUESTION 2

Find the Laurent exansions of the function f(z) = z / (z^2 - 1) valid for:

i) 0 < |z - 1| < 2

ii) |z + 1| > 2

iii) |z| > 1

Thank you

Doug