Is the Function Analytic? Testing the Cauchy Riemann Equations

[Blanchdog]

Homework Statement

Use the definition of the complex derivative to find out if the function f(z) = z*
is analytic. (Hint: you may want to approach the point of interest from the real axis side
and from the imaginary axis side). What about the functions f(z) = z + z* and f(z) = z - z*

Relevant Equations

The definition of the derivative is df/dz = lim_h->0 of (f(z+h) - F(z))/h

The Cauchy-Riemann Equations are du/dx = dv/dy, and du/dy = -dv/dx

I tested the first function with the Cauchy Riemann equations and it seemed to fail that test, so I don't believe that function is analytic. However, I'm really not sure how to show that it is or is not analytic using the definition of the complex derivative.

 

[PeroK]

You need to show that the limit in the definition does not exist at some (or all) points.

 

[vela]

The definition of the derivative is df/dz = lim_h->0 of (f(z+h) - f(z))/h

It might be a little more illuminating if you define the derivative as
$$f'(z_0) = \lim_{z \to z_0} \frac{f(z)-f(z_0)}{z-z_0}.$$ (The definition you gave might mislead you into thinking $h$ is real.) For the limit to exist, it shouldn't matter from which direction $z$ approaches $z_0$. If you can show that the limit is different for two different paths, you've shown the derivative doesn't exist. The hint suggests two paths you should consider.