# Derivative of a constant complex function

• Dec 5th 2012, 08:48 AM
Ant
Derivative of a constant complex function
Let $f$ be an analytic function on a disc D.

Is it true that:

$f$ is constant $\iff$ $f'(z) =0$ $\forall z \in D$

with ' denoting the derivative wrt to complex variable z. Or do we have to include some conditions on the partial derivatives wrt x,y (where Re(z)=x and Im(z)=y) ?

Thanks for anyhelp!
• Dec 5th 2012, 01:02 PM
chiro
Re: Derivative of a constant complex function
Hey Ant.

This is true and you can use the fact that all analytic functions in a region are infinitely differentiable with the caveat that differentiating a constant gives 0.
• Dec 5th 2012, 03:41 PM
Ant
Re: Derivative of a constant complex function
Thanks.

Do we also have the following:

$f'(z)=0 \iff$ both partial derivatives with respect to $x$ and $y$ are also zero.

(I think perhaps the above is false and only the $\Rightarrow$ direction holds??)

Thanks!
• Dec 5th 2012, 06:02 PM
chiro
Re: Derivative of a constant complex function
Yes we should have and you could use the Cauchy-Riemann equations to show this.