# Derivative of a constant complex function

• Dec 5th 2012, 07:48 AM
Ant
Derivative of a constant complex function
Let $\displaystyle f$ be an analytic function on a disc D.

Is it true that:

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

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, 12:02 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, 02:41 PM
Ant
Re: Derivative of a constant complex function
Thanks.

Do we also have the following:

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

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

Thanks!
• Dec 5th 2012, 05:02 PM
chiro
Re: Derivative of a constant complex function
Yes we should have and you could use the Cauchy-Riemann equations to show this.