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Are these functions analytic?

  1. Jan 25, 2012 #1
    1. The problem statement, all variables and given/known data
    Given that f0 = u(x,y) + i*v(x,y) is analytic in domain D, are these functions analytic in domain D?
    1. f1 = (u^2 - v^2) - 2i*u*v
    2. f2 = (e^u)cos(v) + i*(e^u)sin(v)
    3. f3 = u - i*v


    2. Relevant equations
    cauchy-riemann equations

    3. The attempt at a solution
    I kind of have an idea but I want to double check if my reasoning is right.

    If f0 is analytic then all of the partial derivatives of u(x,y) and v(x,y) wrt x and y exist and are continuous on domain D.

    1. f1 = f0^2. The product of differentiable and continuous function is also differentiable and continuous on domain D. Therefore f1 is analytic on domain D.

    2. f2 = e^u * 0.5*(e^-v + e^v) + i * e^u * 0.5(e^-v - e^v)
    Unsure. I can't explicity use Cauchy-Riemann to check. But I know that v and u are continuous. If a function g(x) is continuous and differentiable on some domain or set, is h(x) = e^g(x) also? If so, then f2 is analytic.

    3. f3 = conjugate(f0). Not differentiable because conjugate(z) is not differentiable everywhere. Therefore not analytic.
     
  2. jcsd
  3. Jan 25, 2012 #2

    Dick

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    Science Advisor
    Homework Helper

    I agree with all of those. h(x)=e^g(x) is analytic if g(x) is analytic. It's a composition of analytic functions. If f(z) is analytic and g(z) is analytic then f(g(z)) is analytic.
     
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