Find all the holomorphic functions that satisfy certain condition

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Homework Statement

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.
 

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  • #2
Dick
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Homework Statement

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.
##e^x (g(y)+ih(y))## needs to satisfy Cauchy-Riemann. See what you get out of that.
 
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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)?
 
  • #4
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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)?
Do you know existence and uniqueness theorems for differential equations?
 
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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?)?
 
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  • #6
Dick
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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?)?

I have one more basic question: I've described ##f(iy)=g(y)+ih(y)##, but now that I think about, is this correctly expressed or is it instead ##f(iy)=g(iy)+ih(iy)##?

The problem with this last expression is that it doesn't satisfy the Cauchy Riemann equations because, for example, I would get

##g(iy)=u_x=v_y=ih'(iy)##, which is absurd
No, you defined g(y) and h(y) to be the real and imaginary parts of f(iy), remember?
 
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  • #7
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No, you defined g(y) and h(y) to be the real and imaginary parts of f(iy), remember?
Yes, I see and understand it now, I don't know why I got confused. Thanks
 

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