Take-home Examination 3 for Math 618 - Prof. Alexei Poltoratski, Exams of Mathematics

Download Take-home Examination 3 for Math 618 - Prof. Alexei Poltoratski and more Exams Mathematics in PDF only on Docsity! Math 618 Take-home Examination 3 April 20, 2007 Instructions Do any five of the following six problems. You may consult the textbook but not other sources; in particular, you may not ask another person for help solving the problems. You may cite and use results proved in the textbook or in class. Please submit your solutions to me in my office (Milner 202) before 4:00 p.m. on Friday, April 27. 1. In this problem, you will show how to compute the order of an entire function directly from the Maclaurin series coefficients. Recall that if f is an entire function, then M(r) denotes max{ |f(z)| : |z| = r }, and the order λ of f is defined as follows: λ = lim sup r→∞ log log M(r) log r . Suppose that f(z) = ∑∞ n=0 cnz n, and define a number β as follows: β = lim sup n→∞ n log n log 1 |cn| (interpret the fraction as 0 if cn = 0). Your task is to prove that in fact β = λ. Suggestion Prove separately (a) that β ≤ λ and (b) that λ ≤ β. (a) Fix a positive ǫ. By the definition of order, M(r) < er λ+ǫ for sufficiently large r. Bound |cn| for large n by applying Cauchy’s estimate with r = n1/(λ+ǫ), and deduce that β ≤ λ + ǫ. Let ǫ ↓ 0. (b) Fix a positive ǫ. Deduce from the definition of β that |cn| < n−n/(β+ǫ) for sufficiently large n, and use the triangle inequality to bound M(r) by ∑∞ n=0 |cn| r n. By splitting the sum at an integer n close to (2r)β+ǫ, show that λ < β + 2ǫ. Let ǫ ↓ 0. Remark The expression for the order in terms of the series coeffi- cients makes it easy to write down an entire function of any prescribed positive order a: namely, the series ∑∞ n=1 z n/nn/a does the job. More- over, the series ∑∞ n=1 z n/nn 2 represents a non-polynomial entire func- tion of order 0. Page 1 of 6 Dr. Boas Math 618 Take-home Examination 3 April 20, 2007 2. You know that there exists an entire function with prescribed zeroes of prescribed orders, subject to the necessary condition that the zeroes do not accumulate [Weierstrass]. You can immediately deduce by taking the reciprocal that there exists a meromorphic function in the plane with prescribed poles of prescribed orders. More is true: it is possible to prescribe not only the locations and the orders of the poles of the meromorphic function but also the principal part of the Laurent series at each pole. More precisely, suppose that {zj} ∞ j=1 is a sequence of distinct points having no accumulation point in the finite plane, and {pj} ∞ j=1 is a sequence of polynomials. You can think of pj( 1 z−zj ) as being the principal part of a Laurent series based at the point zj . Mittag-Leffler’s theorem says that there exists a function f meromorphic in the plane, with poles only at the points zj , such that for each j, the function f(z) − pj( 1 z−zj ) has a removable singularity at zj. In other words, the function f has the specified principal parts at the specified poles. Your task in this problem is to prove Mittag-Leffler’s theorem. Hint The first try is ∑∞ j=1 pj( 1 z−zj ), but this series need not converge. Show, however, that there exist polynomials qj such that the series ∑∞ j=1 ( pj( 1 z−zj ) − qj(z) ) does converge (uniformly on compact sets that avoid the sequence of poles). In fact, qj(z) can be taken to be a suitable partial sum of the Maclaurin series of pj( 1 z−zj ). Remarks Notice that the theorem of Mittag-Leffler uses infinite se- ries and additive convergence factors, while the theorem of Weierstrass uses infinite products and multiplicative convergence factors. Mittag-Leffler proved much more than is stated above.1 First of all, an analogous theorem holds on a general planar region. Secondly, one can simultaneously prescribe the locations of both the zeroes and the poles of a meromorphic function. Thirdly, one can prescribe any finite number of terms of the Laurent series (not just the terms containing negative powers). 1G. Mittag-Leffler, “Sur la représentation analytique des fonctions monogènes uni- formes d’une variable indépendante,” Acta Math. 4 (1884), 1–79. Page 2 of 6 Dr. Boas Math 618 Take-home Examination 3 April 20, 2007 5. In this problem, you are to supply proofs (modulo one ingredient) of Picard’s theorems about the range of analytic functions. The ingredient that you may assume without proof is the following result of Montel. Montel’s fundamental normality criterion Suppose a and b are two distinct complex numbers. The family of analytic functions on a region that take neither the value a nor the value b is a normal family in the region. Here normality is understood in the extended sense that the constant function ∞ is admitted as a possible limit function. (i) Picard’s little theorem The range of a nonconstant entire func- tion cannot omit two values. For the proof, suppose that f is an entire function whose range does omit two values, and consider the family of entire functions {fn} ∞ n=1 defined by fn(z) = f(nz). Apply Montel’s fundamental normality cri- terion to deduce that f must be constant. (ii) Picard’s great theorem In every punctured neighborhood of an essential singularity, an analytic function assumes every complex value—with one possible exception—infinitely often. For the proof, assume without loss of generality that the essential sin- gularity is at the origin, and suppose there are two values each taken only a finite number of times. After shrinking the neighborhood, one can assume that there are two omitted values. Consider the family of functions {fn} ∞ n=1 defined by fn(z) = f(z/n), and apply Montel’s cri- terion, remembering to account for the possibility of a limit function equal to the constant ∞. Remark Picard’s great theorem subsumes the little theorem, for an entire function either has an essential singularity at infinity or is a poly- nomial (and a nonconstant polynomial assumes every complex value). Thus one can strengthen the little theorem to say that a non-polynomial entire function takes every complex value infinitely often, with one pos- sible exception. Page 5 of 6 Dr. Boas Math 618 Take-home Examination 3 April 20, 2007 6. Suppose D is a bounded, simply connected planar region, and let p be a point in D. Consider the family of analytic functions f on D such that f maps D into (not necessarily onto) the unit disk, f(p) = 0 = f ′(p), and f ′′(p) is a positive real number. (a) Show that there is an extremal function that maximizes the value of f ′′(p) within the indicated class. (b) Show that the extremal function is unique by showing that it is equal to the square of the Riemann mapping function. Page 6 of 6 Dr. Boas