1. The problem statement, all variables and given/known data determine if a) f(z) = e^z / (z^2 + 4) b) f(z) = conj(z) / |z|^2 c) f(z) = sum from 0 to inf. [ (e^z / 3^n) * (2z - 4)^n ] is analytic and sketch the domain where it is analytic. 2. Relevant equations 3. The attempt at a solution a) i don't know how to separate the function into a real and imaginary part. I have a feeling that the denominator needs to be manipulated but I have no clue how. f(z) = e^(x + yi) / ((x+yi)^2 + 4) f(z) = (e^x * cosy) / ((x+yi)^2 + 4) + (ie^x * siny) / ((x+yi)^2 + 4) (e^x * cosy)* ((x-yi)^2 + 4) / ((x+yi)^2 + 4)((x-yi)^2 + 4) I'm guessing it is analytic everywhere except z^2 = -4 b) f(z) = conj(z) / |z|^2 f(z) = 1 / z 1/z = 1 / ( x+yi) (x - yi) / (x^2 + y ^2) = x / (x^2 + y ^2) - yi / (x^2 + y ^2) since du/dx u(x,y) = dv/dy u(x,y) and dv/dx u(x,y) = - dv/dx u(x,y) the function is analytic everywhere except at the origin. c) I used the ratio test and got lim n-> inf |1/3 (2z - 4)| = |1/3 (2z - 4)| it's analytic on 0 < |2z - 4| < 3 ?