We have the conformal map w = f(z) = z + K/z.
Prove this mapping is indeed conformal.
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?