# Complex Analysis

• Nov 6th 2011, 12:27 AM
sinichko
Complex Analysis
Hi! Help me please with this one.

Let's $\displaystyle f(z)$ be an entire function
and $\displaystyle Re f(z)\geq 0$for all $\displaystyle z$
Prove that $\displaystyle f$ is constant.

Thanks.
• Nov 6th 2011, 01:23 AM
Opalg
Re: Complex Analysis
Quote:

Originally Posted by sinichko
Hi! Help me please with this one.

Let's $\displaystyle f(z)$ be an entire function
and $\displaystyle Re f(z)\geq 0$ for all $\displaystyle z$
Prove that $\displaystyle f$ is constant.

Hint: What can you say about the function $\displaystyle e^{-f(z)}$ ?
• Nov 8th 2011, 07:42 AM
sinichko
Re: Complex Analysis
Quote:

Originally Posted by Opalg
Hint: What can you say about the function $\displaystyle e^{-f(z)}$ ?

I don't understand how it can help?
$\displaystyle e^{-f(z)}=\frac{1}{e^{u(x,y)}(\cos v(x,y)+i\sin v(x,y))}$
• Nov 8th 2011, 08:40 AM
Opalg
Re: Complex Analysis
Quote:

Originally Posted by sinichko
I don't understand how it can help?
$\displaystyle e^{-f(z)}=\frac{1}{e^{u(x,y)}(\cos v(x,y)+i\sin v(x,y))}$

$\displaystyle -f(z) = -u(x,y) - iv(x,y)$. So $\displaystyle e^{-f(z)}=e^{-u(x,y)}(\cos v(x,y)-i\sin v(x,y))$. You are told that $\displaystyle u(x,y)\geqslant0$ for all x,y. What does that tell you about $\displaystyle |e^{-f(z)}|$ ?
• Nov 8th 2011, 09:53 AM
sinichko
Re: Complex Analysis
Quote:

Originally Posted by Opalg
$\displaystyle -f(z) = -u(x,y) - iv(x,y)$. So $\displaystyle e^{-f(z)}=e^{-u(x,y)}(\cos v(x,y)-i\sin v(x,y))$. You are told that $\displaystyle u(x,y)\geqslant0$ for all x,y. What does that tell you about $\displaystyle |e^{-f(z)}|$ ?

$\displaystyle |e^{-f(z)}|$ is bounded?
• Nov 8th 2011, 10:36 AM
Opalg
Re: Complex Analysis
Quote:

Originally Posted by sinichko
$\displaystyle |e^{-f(z)}|$ is bounded?

Yes! So what do you deduce from that?
• Nov 8th 2011, 11:07 AM
sinichko
Re: Complex Analysis
Quote:

Originally Posted by Opalg
Yes! So what do you deduce from that?

Thanks.
Maybe I should apply Liouville's theorem
, but I have to prove that function $\displaystyle e^{-f(z)}$- an entire, given that $\displaystyle f(z)$ - entire function...how can i prove that $\displaystyle e^{-f(z)}$ entire function also?
• Nov 8th 2011, 11:46 AM
Opalg
Re: Complex Analysis
Quote:

Originally Posted by sinichko
Thanks.
Maybe I should apply Liouville's theorem
, but I have to prove that function $\displaystyle e^{-f(z)}$- an entire, given that $\displaystyle f(z)$ - entire function...how can i prove that $\displaystyle e^{-f(z)}$ entire function also?

Yes, that is exactly what you need to do. In fact, a differentiable function of a differentiable function is always differentiable (that is basically what the chain rule says).