Conformal Mapping - Can't Prove Analyticity

[chill_factor]

Homework Statement

We have the conformal map w = f(z) = z + K/z.

Prove this mapping is indeed conformal.

Homework 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)

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?

 

[chill_factor]

Never mind I made a dumb mistake, it works.

 

[pasmith]

The Cauchy-Riemann equations are conditions for two functions \mathbb{R}^2 \to \mathbb{R} to be the real and imaginary parts of a differentiable function \mathbb{C} \to \mathbb{C}. But to prove analyticity you didn't need to show that the real and imaginary parts of f(z) = z + K/z satisfy the Cauchy-Riemann equations, because you already knew that f'(z) exists for all z \neq 0 and is equal to 1 - K/z^2.

Also, doesn't f'(\sqrt K) = f'(-\sqrt K) = 0? Your function is then conformal on the open set \mathbb{C} \setminus \{\sqrt K, 0, -\sqrt K\}.