Find where a complex function is differentiable

[boogiewithstu]

Using the definition of the derivative find at which points the function f(z) = Im(z)/z conjugate is complex differentiable.
I know that it is not complex differentiable anywhere but I need to show it using the definition and not the Cauchy Riemann equations.

 

[HallsofIvy]

Write z= x+ iy, so that f(z)= f(x+ iy)= \frac{y}{x+ iy}= \frac{xy- iy^2}{x^2+ y^2}[/tex] Apply the definition of "differentiable" to that. (It is sufficent to look at the real and complex parts separately.)