- #1

- 901

- 3

## 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?