# Thread: 2 complex analysis questions

1. ## 2 complex analysis questions

1.) Let G be a region and let f and g be analytic functions on G such that
f(z)g(z) = 0 on G. Show that either f(z) = 0 or g(z) = 0 on G.

2.) Let f be an entire function with constants 0<M, R>0 and positive integer n>1 such that |f(z)|<M|z|^n for all |z|<R. Prove that f is a polynomial of degree less than or equal to n.

2. Hey

1.) Let G be a region and let f and g be analytic functions on G such that
f(z)g(z) = 0 on G. Show that either f(z) = 0 or g(z) = 0 on G.
I dont know

2.) Let f be an entire function with constants 0<M, R>0 and positive integer n>1 such that |f(z)|<M|z|^n for all |z|<R. Prove that f is a polynomial of degree less than or equal to n.
f is an entire function, a holomorphic function on $\displaystyle D_\infty(0) = D_\infty(z_0)$

=> $\displaystyle f(x) = \sum^\infty_{n=0} a_n(z-z_0)^n = \sum^\infty_{n=0} a_n*z^n$

show that a_m = 0 if m > N

$\displaystyle |a_m| \le max_{\eta \in \partial D_R(0)} \frac{|f(\eta)|}{r^m}$

For $\displaystyle | \eta | = r \ge R$ is $\displaystyle |f(\eta )| \le M * r^N$

That means $\displaystyle |a_m| \le M*r^N / r^m = M * \frac{1}{r^{m-N}}$

$\displaystyle -> 0, r -> \infty$

$\displaystyle => |a_m| = 0$

$\displaystyle => f(z) = \sum^N_{n=0} a_n z^n$

=> f is a polynomial of degree less than or equal to N.

Hint: If $\displaystyle \{ z_n \}$ is a sequence of convergent distinct points in a region with limit point in the region and $\displaystyle h(z_n) = 0$ where $\displaystyle h$ is analytic on the region then $\displaystyle h$ is identically zero.