advertisement

Preliminary Examination 1999 Complex Analysis Do all problems. Notation: ¡ = {x : x is a real number} B ( a , r ) = { z ∈ £ :| z − a | < r} D = B(0,1) £ = { z : z is a complex number} ann( a , r1 ,r2 ) = { z ∈ £ : r1 < | z − a | < r2} For G ⊂ £ , let A (G ) denote the set of analytic functions on G (mapping G to £ ) and let Har (G ) denote the set of harmonic functions on G (mapping G to ¡ ) . ∞ 1. ∑a z Suppose the power series n= 0 n n converges on | z | < R where z and the an are complex numbers. If bn ∈ £ is such that | bn | < n | a n | for all n, prove that 2 ∞ ∑b z n= 0 n n converges for |z|<R. 2. Let f , g ∈A ( D ) . For 0 < r < 1, let Γ r = ∂B (0, r ) . a) Prove that the integral 1 2π ∫ Γr 1 z f ( w )g ( ) dw is independent of r provided w w that | z | < r < 1 and that it defines an analytic function h ( z ), | z | < 1 . b) 3. Prove or supply an counterexample: if f Let f ∈A ( D ) . Suppose that | f ( z ) | ≤ of f ) such that | f ′( z ) | ≤ 4. M (1− | z |) 3 0 and g 1 (1− | z |) 1 0, then h 0. . Prove that there exists a M (independent 2 . 2 Let f ∈A ( D ) such that f ( D) ⊂ D, f (0) = 0. Prove for z ∈ D that | f ( z ) + f ( −z ) | ≤ 2 | z |2 with equality if and only if f ( z ) = λ z 2 for some λ with | λ |= 1 . 5. In which quadrant do the roots of p ( z ) = z 4 + 2 z + 1 lie? 6. Let f ∈A ( D ) and suppose that f is not the identity map. How many fixed points can f have? 7. Let f be an entire function. Suppose that f has a root at z = + i and z = −i . Let M = max| f ( z ) | . Prove that | f ( z ) | ≤ | z |= 2 ∞ 8. Prove that ∏ (1 − n =1 ∞ 9. Evaluate ∫ 0 M 2 | z + 1| on B (0,2) . 3 z2 ) is an entire function and find its zeros, counting multiplicity. n2 x sin x dx . x2 +1 10. Let u ∈Har ( £) . Show that u can be positive on all of £ only if u is constant.