# Proving that an analytic function on a region is constant.

• Mar 18th 2011, 05:32 PM
jimmehhh
Proving that an analytic function on a region is constant.
Let f=u+iv be an analytic function defined on a region D. Suppose
a*u(x,y) + b*v(x,y) = c for all x+iy in D, where a,b,c (in C) are constants, not all 0.
Prove that f is constant on D.

My guess is that I'm meant to show df/dx and df/dy = 0? Because it's constant. But I'm not sure how to go about it. I've used the Cauchy-Riemann equations to derive expressions for df/dx and df/dy with u and v, but I'm not really sure what to do with them, nor the terms of a, b and c.
• Mar 18th 2011, 05:42 PM
tonio
Quote:

Originally Posted by jimmehhh
Let f=u+iv be an analytic function defined on a region D. Suppose
a*u(x,y) + b*v(x,y) = c for all x+iy in D, where a,b,c (in C) are constants, not all 0.
Prove that f is constant on D.

My guess is that I'm meant to show df/dx and df/dy = 0? Because it's constant. But I'm not sure how to go about it. I've used the Cauchy-Riemann equations to derive expressions for df/dx and df/dy with u and v, but I'm not really sure what to do with them, nor the terms of a, b and c.

The given condition is equivalent to f being bounded on D, and since it is entire

there Liouville Theorem states that it must be constant there.

Tonio