# Homework Help: Purely Real-valued Analytic Functions?

1. Oct 24, 2012

### jtleafs33

1. The problem statement, all variables and given/known data
Can a function which is purely real-valued be analytic? Describe the behavior of such functions?

2. Relevant equations
The Cauchy-Riemann conditions
ux=vy, vx=-uy

3. The attempt at a solution

I can't think of any pure real-valued equations off the top of my head which satisfy the CR conditions. However, could any function like sin(x), cos(x), e^x, which is infinitely differentiable be considered analytic? Doesn't infinite differentiability imply that the CR conditions are already satisfied?

2. Oct 24, 2012

### Dick

No. If your function is real valued then v=0. What does that tell you about u?

3. Oct 24, 2012

### jtleafs33

If v=0, then vx=vy=0. So then, u must be constant, so that it's derivative will also be zero, and thus satisfy the CR conditions?

4. Oct 24, 2012

### Dick

Yes, that's it. The only purely real analytic functions are constant.