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?