Purely Real-valued Analytic Functions?

[jtleafs33]

Homework Statement

Can a function which is purely real-valued be analytic? Describe the behavior of such functions?

Homework Equations

The Cauchy-Riemann conditions
ux=vy, vx=-uy

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?

 

[Dick]

Homework Statement

Can a function which is purely real-valued be analytic? Describe the behavior of such functions?

Homework Equations

The Cauchy-Riemann conditions
ux=vy, vx=-uy

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?

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

 

[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?

 

[Dick]

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?

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