Complex Analysis Problems

Complex Analysis Problems transcribed from the originals by William J. DeMeo October 2, 2008 Contents 99 November 2 2 2 200 November 26 4 3 2006 November 3 6 4 2007 April 6 7 5 2007 November 6 8

99 NOVEMBER 2 99 November 2 INSTRUCTIONS: In each of sections A, B, and C, do all but one problem. TIME LIMIT: 2 hours SECTION A (Do 3 of the 4 problems.). Where does the function f(z) = zrez + zimz + z have a complex derivative? Compute the derivative wherever it exists. 2. (a) Prove that any nonconstant polynomial with complex coefficients has at least one root. (b) From (a) it follows that every nonconstant polynomial P has the factorization N P (z) = a (z λ n ), n= where a and each root λ n are complex constants. Prove that if P has only real coefficients, then P has a factorization K M P (z) = a (z r k ) (z 2 b m z + c m ), where a and each r k, b m, c m are real constants. k= m= 3. Use complex residue methods to compute the integral π 0 5 + 3 cos θ dθ. 4. (a) Explain how to map an infinite strip (i.e. the region strictly between two parallel lines) onto the unit disk by a one-to-one conformal mapping. (b) Two circles lie outside one another except for common point of tangency. Explain how to map the region exterior to both circles (including the point at infinity) onto an infinite strip by a one-to-one conformal mapping. SECTION B (Do 3 of the 4 problems.) 5. Suppose that f is analytic in the annulus < z < 2, and that there exists a sequence of polynomials converging to f uniformly on every compact subset of this annulus. Show that f has an analytic extension to all of the disk z < 2. See also: April 96 (8). 2

99 NOVEMBER 2 6. Let f be analytic in z < 2, with the only zeros of f being the distinct points a, a 2,..., a n, of multiplicities m, m 2,..., m n, respectively, and with each a j lying in the disk z <. Given that g is analytic in z < 2, what is f (z)g(z) dz? f(z) (Verify your answer.) z = 7. Let {f n } be a sequence of analytic functions in the unit disk D, and suppose there exists a positive constant M such that f n (z) dz M C for each f n and for every circle C lying in D. Prove that {f n } has a subsequence converging uniformly on compact subsets of D. 8. State and prove: (a) the mean value property for analytic functions (b) the maximum principle for analytic functions. SECTION C (Do 2 of the 3 problems.) 9. Let X be a Hausdorff topological space, let K be a compact subset of X, and let x be a point of X not in K. Show that there exist open sets U and V such that K U, x V, U V =. 0. A topological space X satisfies the second axiom of countability. Prove that every open cover of X has a countable subcover.. Let X be a topological space, and let U be a subset of X. (a) Show that if an open set intersects the closure of Y then it intersects Y. (b) Show that if Y is connected and if Y Z Ȳ, then Z is connected. 3

2 200 NOVEMBER 26 2 200 November 26 Instructions. Make a substantial effort on all parts of the following problems. If you cannot completely answer Part (a) of a problem, it is still possible to do Part (b). Partial credit is given for partial progress. Include as many details as time permits. Throughout the exam, z denotes a complex variable, and C denotes the complex plane.. (a) Suppose that f(z) = f(x + iy) = u(x, y) + iv(x, y) where u and v are C functions defined on a neighborhood of the closure of a bounded region G C with boundary which is parametrized by a properly oriented, piecewise C curve γ. If u and v obey the Cauchy-Riemann equations, show that Cauchy s theorem f(z) dz = 0 follows from Green s theorem, namely γ ( Q P dx + Q dy = x P ) dx dy for C functions P and Q. () y γ G (b) Suppose that we do not assume that u and v are C, but merely that u and v are continuous in G and f f(z) f(z 0 ) (z 0 ) = lim z z 0 z z 0 exists at some (possibly only one!) point z 0 G. Show that given any ɛ > 0, we can find a triangular region containing z 0, such that if T is the boundary curve of, then f(z) dz = 2 ɛl2, where L is the length of the perimeter of. T Hint for (b) Note that part (a) yields (az + b) dz = 0 for a, b C, which you can use here in (b), even T if you could not do Part (a). You may also use the fact that T g(z) dz L sup{ g(z) : z T } for g continuous on T. 2. Give two quite different proofs of the Fundamental Theorem of Algebra, that if a polynomial with complex coefficients has no complex zero, then it is constant. You may use independent, well-known theorems and principles such as Liouville s Theorem, the Argument Principle, the Maximum Principle, Rouché s Theorem, and/or the Open Mapping Theorem. 3. (a) State and prove the Casorati-Weierstrass Theorem concerning the image of any punctured disk about a certain type of isolated singularity of an analytic function. You may use the fact that if a function g is analytic and bounded in the neighborhood of a point z 0, then g has a removable singularity at z 0. (b) Verify the Casorati-Weierstrass Theorem directly for a specific analytic function of your choice, with a suitable singularity. 4. (a) Define γ : [0, 2π] C by γ(t) = sin(2t) + 2i sin(t). This is a parametrization of a figure 8 curve, traced out in a regular fashion. Find a meromorphic function f such that f(z) dz =. Be careful with γ minus signs and factors of 2πi. 4

2 200 NOVEMBER 26 (b) From the theory of Laurent expansions, it is known that there are constants a n < z < 4, z 2 5z + 4 = a n z n. Find a 0 and a 0 by the method of your choice. n= such that, for 5. (a) Suppose that f is analytic on a region G C and {z C : z a R} G. Show that if f(z) M for all z with z a = R, then for any w, w 2 {z C : z a 2R}, we have f(w ) f(w 2 ) 4M R w w 2 (b) Explain how Part (a) can be used with the Arzela-Ascoli Theorem to prove Montel s Theorem asserting the normality of any locally bounded family F of analytic functions on a region G. 5

3 2006 NOVEMBER 3 3 2006 November 3 Notation: C is the set of complex numbers, D = {z C : z < } is the open unit disk, Π + and Π are the upper and lower half-planes, respectively, and, given an open set G C, H(G) is the set of holomorphic functions on G.. (a) Suppose that f H(D \ {0}) and that f(z) < for all 0 < z <. Prove that there is F H(D) with F (z) = f(z) for all z D \ {0}. (b) State a general theorem about isolated singularities for holomorphic functions. 2. (a) Explicitly construct, through a sequence of mappings, a one-to-one holomorphic function mapping the disk D onto the half disk D Π +. (b) State a general theorem concerning one-to-one mappings of D onto domains Ω C. 3. (a) State the Schwarz Lemma. (b) Suppose that f H(Π + ) and that f(z) < for all z Π +. If f(i) = 0 how large can f (i) be? Find the extremal functions. 4. (a) State Cauchy s theorem and its converse. (b) Suppose that f is a continuous function defined on the entire complex plane. Assume that (i) f H(Π + Π ) (ii) f( z) = f(z) all z C. Prove that f is an entire function. 5. (a) Define what it means for a family F H(Ω) to be a normal family. State the fundamental theorem for normal families. (b) Suppose f H(Π + ) and f(z) < all z Π +. Suppose further that lim t 0+f(it) = 0. Prove that f(z n ) 0 whenever the sequence z n 0 and z n Γ where Γ = {z Π + : Rez Imz}. Hint. Consider the functions f t (z) = f(tz) where t > 0. 6

4 2007 APRIL 6 4 2007 April 6 Notation: C is the set of complex numbers, D = {z C : z < }, and, for any open set G C, H(G) is the set of holomorphic functions on G.. Give the Laurent series expansion of z(z ) in the region A {z C : 2 < z + 2 < 3}. 2. (i) Prove: Suppose that for all z D and all n N we have that f n is holomorphic in D and f n (z) <. Also suppose that lim n Imf n (x) = 0 for all x (, 0). Then lim n Imf n (/2) = 0. (ii) Give a complete statement of the convergence theorem that you use in part (2i). 3. Use the residue theorem to evaluate +x 4 dx. 4. Present a function f that has all of the following properties: (i) f is one-to-one and holomorphic on D. (ii) {f(z) : z D} = {w C : Rew > 0 and Imw > 0}. (iii) f(0) = + i. 5. (i) Prove: If f : D D is holomorphic and f(/2) = 0, then f(0) /2. (ii) Give a complete statement of the maximum modulus theorem that you use in part (i). 6. Prove: If G is a connected open subset of C, any two points of G can be connected by a parametric curve in G. 7

5 2007 NOVEMBER 6 5 2007 November 6 Do as many problems as you can. Complete solutions (except for minor flaws) to 5 problems would be considered an excellent performance. Fewer than 5 complete solutions may still be passing, depending on the quality.. Let G be a bounded open subset of the complex plane. Suppose f is continuous on the closure of G and analytic on G. Suppose further that there is a constant c 0 such that f = c for all z on the boundary of G. Show that either f is constant on G or f has a zero in G. 2. a. State the Residue Theorem. b. Use contour integration to evaluate 0 x 2 (x 2 + ) 2 dx. Important: You must carefully: specify your contours, prove the inequalities that provide your limiting arguments, and show how to evaluate all relevant residues. 3. a. State the Schwarz Lemma. b. Suppose f is holomorphic in D = {z : z < } with f(d) D. Let f n denote the composition of f with itself n times (n = 2, 3,... ). Show that if f(0) = 0 and f (0) <, then {f n } converges to 0 locally uniformly on D. 4. Exhibit a conformal mapping of the region common to the two disks z < and z < onto the region inside the unit circle z =. 5. Let {f n } be a sequence of functions analytic in the complex plane C, converging uniformly on compact subsets of Cto a polynomial p of positive degree m. Prove that, if n is sufficiently large, then f n has at least m zeros (counting multiplicities). Do not simply refer to Hurwitz s Theorem; prove this version of it. 6. Let (X, d) be a metric space. a. Define what it means for a subset K X to be compact. b. Prove (using your definition in a.) that K X is compact implies that K is both closed and bounded in X. c. Give an example that shows the converse of the statement in b. is false. 8