# Math Help - Complex anaylsis

1. ## Complex anaylsis

Use the definition to show that f'(z) does not exist when f(z)=Re(z).

Thanks!

2. Originally Posted by LCopper2010
Use the definition to show that f'(z) does not exist when f(z)=Re(z).

Thanks!

In case the limit exists we have $f'(z_0)=\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}=\lim_{z\to z_0}\frac{Re(z)-Re(z_0)}{z-z_0}$ , so put $z=x+yi\,,\,\,z_0=a+bi$:

$f'(z_0)=\lim_{z\to z_0}\frac{x-a}{(x-a)+(y-b)i}$ . Now, choose $y=b$ and make $x\rightarrow a$ , so:

$\lim_{x\to a}\frac{x-a}{x-a}=1$ , and now choose $x=a$ and make $y\rightarrow b$

$\lim_{y\to b}\frac{0}{(y-b)i}=0$.

Since the wanted limit depends on how the complex variable $z$ approaches $a+bi$ the limit, and thus the derivative, doesn't exist.

Tonio