{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

MA530_JAN03 - of the domain then it is identically zero Can...

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

View Full Document Right Arrow Icon
QUALIFYING EXAMINATION JANUARY 2003 MATH 530 - Prof. Bell 1. (10 pts) Suppose P and Q are polynomials and that the degree of P is at least two less than the degree of Q . Prove that the sum of all the residues of P/Q in the complex plane is zero. 2. (20 pts) a) Prove that an uncountable subset of a domain in the complex plane must have a limit point in the domain. b) Prove that an analytic function on a domain can have at most countably many zeroes if it is not identically zero. c) Does there exist a non-constant analytic function on the unit disc with infin- itely many zeroes on the unit disc? Prove or give a counterexample. 3. (20 pts) Prove that if a harmonic function on a domain vanishes on an open subset
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: of the domain, then it is identically zero. Can it vanish on a set with a limit point in the domain and not be identically zero? 4. (20 pts) Suppose that f ( z ) has an essential singularity at z = a . Prove that there is a sequence of points z n tending to a such that ( z n-a ) n f ( z n ) tends to ∞ . 5. (10 pts) Does there exist a sequence of polynomials P n ( z ) which converges uni-formly to 1 /z on { z : | z | = 1 } ? Explain. 6. (20 pts) Suppose that f is a continuous complex valued function on a domain Ω. Prove that if f 2 and f 3 are analytic on Ω, then f itself must be analytic....
View Full Document

{[ snackBarMessage ]}