1. The problem statement, all variables and given/known data We have the conformal map w = f(z) = z + K/z. Prove this mapping is indeed conformal. 2. Relevant equations z = x + iy A map w = f(z) is conformal if it is analytic and df/dz is nonzero. f(z) = u(x,y) + iv(x,y) 3. The attempt at a solution df/dz = 1 - Kz^-2 =/= 0 for finite z, nonzero derivative condition met. Attempting to prove its analytic: f(z) = z + K/z = x+iy + K/(x+iy) = x+iy + K(x-iy)/(x+iy)(x-iy) = x+iy + K(x-iy)/(x^2+y^2) = [x + Kx/(x^2+y^2)] + i [y - Ky/(x^2+y^2)] = u(x,y) + iv(x,y) u(x,y) = x + Kx(x^2+y^2)^-1 v(x,y) = y - Ky(x^2+y^2)^-1 du/dx = 1 - Kx(x^2+y^2)^-2 * 2x dv/dy = 1 + Ky(x^2+y^2)^-2 * 2y they aren't equal therefore they do not satisfy Cauchy-Riemann relations, and the function is not analytic. However I'm directly told that it MUST be analytic because it is a conformal map. Did I make a mistake or is the problem mistyped?