math185f08-hw9 - MATH 185: COMPLEX ANALYSIS FALL 2008/09...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
MATH 185: COMPLEX ANALYSIS FALL 2008/09 PROBLEM SET 9 For f : Ω C and n N , recall that g = f n is the function defined by g ( z ) = [ f ( z )] 2 for all z Ω. A function on Ω C is said to be meromorphic if it has only removable singularities or poles in Ω (i.e. no essential singularities or non-isolated singularities). 1. Let Ω C be a region and let f : Ω C . Suppose g = f 2 and h = f 3 are both analytic on Ω. Show that f is analytic on Ω. 2. Is there a polynomial p ( z ) such that p ( z ) e 1 /z is an entire function? 3. Let f : C C be a nonconstant entire function. Prove that f ( C ) is dense in C . 4. Show that each of the following series define a meromorphic function on C and determine the set of poles and their orders. f ( z ) = X n =0 ( - 1) n n !( z + n ) , g ( z ) = X n =0 sin( nz
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 06/15/2009 for the course MATH 185 taught by Professor Lim during the Fall '07 term at University of California, Berkeley.

Ask a homework question - tutors are online