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Find all the holomorphic functions that satisfy certain condition

  1. Apr 25, 2014 #1
    The problem statement, all variables and given/known data

    Find all the holomorphic functions ##f: \mathbb C \to \mathbb C## such that ##f'(0)=1## and for all ##x,y \in \mathbb R##,

    ##f(x+iy)=e^xf(iy)##

    I am completely stuck with this exercise, for the second condition, I know that ##f(iy)=g(y)+ih(y)##, but is there something that ##g## and ##h## must satisfy? I would appreciate any hints.
     
  2. jcsd
  3. Apr 25, 2014 #2

    Dick

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    ##e^x (g(y)+ih(y))## needs to satisfy Cauchy-Riemann. See what you get out of that.
     
  4. Apr 28, 2014 #3
    As you've said, ##f## must satisfy the C-R equations, so ##f(x+iy)=e^x(g(y)+ih(y))##, with real part ##u(x,y)=e^xg(y)## and imaginary part ##v(x,y)=e^xh(y)##. By Cauchy Riemann we have

    (1) ##e^xg(y)=u_x=v_y=e^xh'(y)##
    (2) ##e^xg'(y)=u_y=-v_x=-e^xh(y)##

    (1) and (2) reduce to

    (3)##g(y)=h'(y)##
    (4)##g'(y)=-h(y)##

    Two functions that satisfy (3) and (4) are ##g(y)=k_1\cos(y)+c_1## and ##h(y)=k_2\sin(y)+c_2##, with ##k_1,k_2,c_1,c_2 \in \mathbb R##

    Now, my doubt is: how can I assure that sine and cosine are the only functions that satisfy equations (3) and (4)?
     
  5. Apr 28, 2014 #4

    micromass

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    Do you know existence and uniqueness theorems for differential equations?
     
  6. Apr 28, 2014 #5
    I know very little about differential equations, which hypothesis one needs to assure existence and uniqueness (or which textbook could you recommend me to read about this?)?
     
    Last edited: Apr 28, 2014
  7. Apr 28, 2014 #6

    Dick

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    No, you defined g(y) and h(y) to be the real and imaginary parts of f(iy), remember?
     
  8. Apr 28, 2014 #7
    Yes, I see and understand it now, I don't know why I got confused. Thanks
     
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