Analysis Comprehensive Exam, January 2011 Instructions: Do as many problems as you can. You should attempt to answer completely some questions in both

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74 Analysis Comprehensive Exam, January 2011 Instructions: Do as many problems as you can. You should attempt to answer completely some questions in both real and complex analysis. You have 3 hours. Real 1. Let A be the subset of [0, 1] which consists of all numbers which do not have the digit 4 appearing in their decimal expansion. Find the Lebesgue measure of A. 2. Prove that {f : [0, 1] R f C 1, f 10, f 7} is a relatively compact subset of C([0, 1]) = {f : [0, 1] R f is continuous}, using the norm f sup = sup x f(x) to define the topology. 3. Suppose F : R 2 R is such that all partial derivatives exist and are continuous on R 2. Prove that F is differentiable on R Let f be Lebesgue integrable on R, (i.e., f L 1 (R, m)) and define F (x) = F is continuous on R. x f(t)dt. Show that Complex 5. Let For the contour compute the integral f(z) = (z2 + 1) z 2 (z 2 + 1/4). Γ = {z C z = 1/3} 1 f(z) dz. 2πi Γ 6. Suppose a sequence of functions {f n }, analytic in a open domain D C converges to f uniformly on compacts subsets of D. Show that f n f on D and the convergence is uniform on compact subsets of D. 7. Let D r = {z C : z < r}. Determine the largest r for which the following series defines an analytic function on D r : z p = z 2 + z 3 + z 5 + p prime 8. Using the multiple valued function f(z) = z i, describe completely all images of the positive imaginary axis under f. 1

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79 ANALYSIS COMPREHENSIVE EXAM, AUGUST 2012 INSTRUCTIONS: There are 8 problems (see back). Do as many problems as you can. You should attempt to answer completely some questions in both real and complex analysis. You have 3 hours. Real: 1. Let F be an equicontinuous family of real-valued functions on a compact interval [a, b]. Let L be the family of all pointwise limits of sequences in F. I.e., f L if and only if there is a sequence {f n } F such that f n (x) f(x) for every x. Prove L is also equicontinuous. 2. If A is an n n matrix, define Prove that and that e ta = k=0 t k k! Ak, t R. d dt eta = Ae ta = e ta A, t R, e (s+t)a = e sa e ta, s, t, R. 3. Let (X, B, µ) be a measure space, and let {f n } be a sequence of integrable functions on X such that f n f µ-a.e. with f an integrable function. Prove f f n dµ 0 if and only if f n dµ f du. 4. Let f : R R be differentiable. Assume there is no x R such that f and f both vanish at x. Show that S = {x [0, 1] : f(x) = 0} is finite. Complex: 5. For any fixed integer n 2, show that the equation z n + z + 1 = 0 has n distinct complex roots in {z C : z < 2}. 6. Let R C be the region inside the disc { z 1 < 1} but outside the disc { z 1/2 1/2}. Find a conformal transformation from R to the upper half plane. 1

80 2 7. Suppose that f is a bounded holomorphic function on the domain {z C : z > 1}. a) Prove that lim f(z) exists. z b) Let L denote the limit in (a). Show that 1 2πi ζ =R f(ζ) dζ = L f(z) for z > R > 1. ζ z 8. Let f be a holomorphic function which maps the unit disc into the unit disc with f(0) = 0. Show that f(z) + f( z) 2 z 2 for all z in the unit disc, and if equality holds for some z 0, then f(z) = e iθ z 2 for some real θ.

81 ANALYSIS COMPREHENSIVE EXAM, JANUARY 2013 INSTRUCTIONS: There are 8 problems (see back). Do as many problems as you can. You should attempt to answer completely some questions in both real and complex analysis. You have 3 hours. Real: 1. Show that if {x n } is a convergent sequence, then the sequence given by the averages y n = x 1 + x x n n also converges to the same limit. And give an example to show that it is possible for the sequence of averages {y n } to converge even if {x n } does not. 2. Assume f : [0, ) R is bounded and continuous, and g(s) := Show that f 0. 0 f(t)e st dt = 0, s > Enumerate Q [0, 1] as Let Q [0, 1] = {r 1, r 2,... }. f(x) = { 1 k, x = r k 0, x [0, 1]\Q. Prove that f is Riemann integrable on [0, 1] and compute 1 f(t) dt Let A R n be an open set. If f : A R is of class C r and Df(x 0 ) = 0, D 2 f(x 0 ) = 0,..., D r 1 f(x 0 ) = 0 but D r f(x 0 )(x,..., x) < 0 for all x R n, x 0, then prove that f has a local maximum at x 0. Complex: In the following problems, D(0, 1) denotes the unit disc centered at the origin, D(0, 1) = {z C : z < 1}. 5. Evaluate for z C. g(z) = e s2 +isz ds, 6. Suppose f(z) is a holomorphic polynomial and for j = 0, 1, 2,.... Prove that f(z) 0. D(0,1) f(z) z j dz = 0 1

82 7. Evaluate 1 z n 1 2πi D(0,1) 3z n 1 dz, where n is a positive integer and the orientation is counterclockwise. 8. Find all meromorphic functions f on C such that f(z) log(2 + z 2 ), z 0. z Give explicit formulas for the functions and give a proof for your answer.

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