Find all the holomorphic functions that satisfy certain condition

[mahler1]

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.

 

[Dick]

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.

 

[mahler1]

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)?

 

[micromass]

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?

 

[mahler1]

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?)?

 

[Dick]

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?

 

[mahler1]

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