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Homework Help: Analytic functions

  1. Jun 3, 2010 #1
    1. The problem statement, all variables and given/known data
    Let g(z) be an analytic function. I have to show that g'(z) is also analytic, using only the CR eqns. I am given that the 2nd partials are continuous

    2. Relevant equations
    let f(x,y)=u(x,y)+iv(x,y)
    CR: du/dx=dv/dy and du/dy=-dv/dx
    continuous 2nd partials: d/dx(du/dy)=d/dy(du/dx) and d/dx(dv/dy)=d/dy(dv/dx)

    3. The attempt at a solution
    I am confused as to how to express the derivative df/dz. would f'=du/dx+du/dy+idv/dx+idv/dy?
  2. jcsd
  3. Jun 3, 2010 #2


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    Staff Emeritus
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    Gold Member

    I don't think you want df/dz. If a function satisfies the CR equations and has continuous partials, what can you say about it?
  4. Jun 3, 2010 #3
    I have to solve this problem using only calculus. I can't use any theorems like using Taylor series. Maybe I can say that it is harmonic?
  5. Jun 4, 2010 #4
    nvm- got it! f'(z)=du/dx+idv/dx. Taking these and the real and imaginary parts, we can get the CR eqns for f' using the CR eqns for f and the fact that the mixed partials are equal.
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