(a) Use the polar form of the Cauchy-Riemann equations to show that:
g(z) = ln(r) + i(theta); r > 0 and 0 < (theta) < 2pi
is analytic in the given region and find its derivative.
(b) then show that the composite function G(z) = g(z^2 + 1) is analytic in the quadrant x > 0 and y > 0 and find its derivative.
The Attempt at a Solution
Ive done part (a) and got the correct answer, but I am having some trouble with (b). The main question I have is, how do I write this composite function? I can write:
z^2 + 1 as r^2(cos(2(theta)) + 1) + ir^2(sin(2(theta))) but I dont know if that helps me.
Thank you for your help!