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Purely Real-valued Analytic Functions?

  1. Oct 24, 2012 #1
    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. jcsd
  3. Oct 24, 2012 #2

    Dick

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    No. If your function is real valued then v=0. What does that tell you about u?
     
  4. Oct 24, 2012 #3
    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?
     
  5. Oct 24, 2012 #4

    Dick

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