Why df/dz*=0 for analytic functions

After my recent post about quaternions, I was asked why we can characterize an analytic function of a complex variable by ∂f/∂z* = 0. Here's why this is true.

Let's write 

f(z) = f(x + i y) = u(x, y) + i v(x, y)

so that we have

z* = x – i y

x = (z + z*) / 2

y = -i (zz*) / 2

Now

f/∂z* = (∂f/∂x) (∂x/∂z*) + (∂f/∂y) (∂y/∂z*)

= (∂u/∂x + i ∂v/∂x) (1/2) + (∂u/∂y + i ∂v/∂y) (i/2)

= (∂u/∂x – ∂v/∂y) (1/2) + (∂v/∂x – ∂u/∂y) (i/2)

But the Cauchy-Riemann equations tell us that we have

u/∂x = ∂v/∂y

and

u/∂y = -∂v/∂x

so that just leaves us with

f/∂z* = (0)(1/2) + (0)(i/2) = 0

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