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Homework Help: Complex Variables

  1. Oct 20, 2009 #1
    1. The problem statement, all variables and given/known data

    Let z be a complex variable

    Suppose f is an entire function and [tex]Re(f(z))\leq c[/tex] for all z

    Show that f is constant.
    (Hint: Consider exp(f(z))

    2. Relevant equations
    possibly this: [tex]e^z=e^x(cos(y)+isin(y))[/tex] where [tex]z=x+iy[/tex]

    3. The attempt at a solution
    I had no idea how I would show this, so I just started off trying a few things:
    I first started off working with the hint to consider exp(f(z)), where exp((f(z))=ef(z)
    I set g(z) equal to exp((f(z)) and because f(z) is entire, g(z) would also have to be entire
    I first found a formula for the derivative of g(z) but that got me nowhere

    I also tried working off the fact that [tex]Re(g(z))\leq e^ccos(Im(f(z)))[/tex]
    but that got me nowhere as well...

    I have been thinking about this problem for so long now, and I couldn't think of a way to show that f is constant
  2. jcsd
  3. Oct 20, 2009 #2


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    Science Advisor
    Homework Helper

    Do you know Liouville's theorem? |exp(f(z))|<=exp(c).
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