# Show a complex function is differentiable at 0

• Feb 28th 2013, 10:32 AM
Ragnarok
Show a complex function is differentiable at 0
(Sorry for all the questions on this, I am really struggling with this and really appreciate all help.)

Show that the function $\displaystyle f(z)=|z|^2$ is differentiable at the origin.

So if $\displaystyle z=x+iy$, then $\displaystyle |z|^2=x^2+y^2$.

To show that it's differentiable you have to show that the limit of the difference quotient exists at 0:

$\displaystyle \lim_{\Delta z\to 0}\frac{f(z_0+\Delta z)-f(z_0)}{\Delta z}$

$\displaystyle =\lim_{\Delta z\to0}\frac{f(0+\Delta z)-f(0)}{\Delta z}$

$\displaystyle =\lim_{\Delta z\to0}\frac{(\Delta x)^2+(\Delta y)^2-0}{\Delta x+i\Delta y}$

Is this the right way to start?
• Feb 28th 2013, 03:38 PM
chiro
Re: Show a complex function is differentiable at 0
Hey Ragnarok.

Hint: You might want to try looking at the Cauchy-Riemann equations. Have you come across these in class?
• Feb 28th 2013, 06:23 PM
Ragnarok
Re: Show a complex function is differentiable at 0
Ah, we have covered the Cauchy-Riemann equations, though technically this problem is from a section before they are covered. I think it will work fine, though, thanks for the hint!