FREE PROBABILITY AND RANDOM MATRICES
|
|
- Lucas Barber
- 5 years ago
- Views:
Transcription
1 FEE POBABILITY AND ANDOM MATICES Lecture 4: Free Harmonic Analysis, October 4, 007 The Cauchy Transform. Let C + = {z C Im(z > 0} denote the complex upper half plane, and C = {z Im(z < 0} denote the lower half plane. Let ν be a probability measure on and for z let G(z = z t dν(t G is the Cauchy transform of the measure ν. Let us briefly check that the integral converges to an analytic function on C +. Lemma. G is an analytic function on C + with range contained in C. Since z t Im(z and ν is a probability measure the integral is always convergent. If Im(w 0 and z w < Im(w / then for t we have z w t w < Im(w Im(w = so the series n=0 ( z w t w n converges uniformly to t w on z w < t z Im(w /. Thus (z t = n=0 (t w (n+ (z w n on z w < Im(w /. Hence [ ] G(z = (t w (n+ dν(t (z w n n=0 is analytic on z w < Im(w /. Finally note that for Im(z > 0, we have for t, Im((z t < 0, and hence Im(G(z < 0. Thus G maps C + into C. Lemma. (i lim yg(iy = i and (ii sup y G(x + iy = y y 0,x Proof. (i y Im(G(iy = = ( y Im dν(t = iy t dν(t +(t/y y y + t dν(t dν(t where y and since ( + (t/y we may apply Lebesgue s Dominated Convergence Theorem.
2 LECTUE 4: FEE HAMONIC ANALYSIS We have y e(g(iy = yt dν(t. But for all y>0and for all t y + t yt y + t and yt/(y + t converges to 0 as y. Therefore y e(g(iy 0 as y, again by the Dominated Convergence Theorem. (ii Fory>0 and z = x + iy, y G(z y z t dν(t = y (x t + y dν(t Thus sup y G(x + iy. By (i however, the supremum is. y 0,x Theorem. Suppose a<b. Then lim y 0 + π b a Im(G(x + iy dx = ν((a, b + ν({a, b} If ν and ν are probability measures with G ν = G ν, then ν = ν. Proof. We have Im(G(x + iy = ( Im x t + iy Thus b Im(G(x + iy dx = a = = b a (b t/y dν(t = y dx dν(t (x t + y (a t/y tan ( b t y + x d xdν(t y (x t + y dν(t tan ( a t y where we have let x =(x t/y. So let f(y, t = tan ((b t/y tan ((a t/y and 0 t/ [a, b] f(t = π/ t {a, b} π t (a, b dν(t
3 LECTUE 4: FEE HAMONIC ANALYSIS 3 Then lim y 0 + f(y, t =f(t, and, for all y>0and for all t, we have f(y, t π. So by Lebesgue s Dominated Convergence Theorem b lim Im(G(x + iy dx = lim f(y, t dν(t y 0 + a y 0 + = f(t dν(t = π (ν((a, b + ν({a, b} This proves the first claim. The first claim shows that if G ν = G ν then ν and nu agree on all intervals and thus are equal. emark. The proof of the next theorem depends on a fundamental result of. Nevalinna which provides an integral representation for an analytic function from C + to C +. Suppose that φ : C + C + is analytic, then there is a unique Borel measure σ on and real numbers α and β, with β 0 such that for z C + φ(z =α + βz + t z dσ(t This integral representation is achieved by letting ξ = iz+, and thus iz z = i +ξ. Then one defines ψ by ψ(ξ =iφ(z and obtains an analytic function ψ mapping the open unit disc, D, into the complex right ξ half plane. Since ψ the real and imaginary parts of ψ are harmonic conjugates, it is enough to work with the real part. Now e(ψ isa positive harmonic function on D; when e(ψ is harmonic on an open set containing D one gets the measure from Cauchy s Integral Theorem; in general one dilates e(ψ to get a harmonic function on D, and then takes a limit as the dilation shrinks to e(ψ. This integral representation is usually attributed to G. Herglotz (9. The details can be found in the book of Akhiezer and Glazman. Theorem (. Nevanlinna. Suppose G : C + C is analytic and sup y G(x + iy =. Then there is a unique probability measure on such that G(z = dν(t. z t Proof. By the remark above there is a unique finite measure on such that G(z =α + βz + +tz dσ(t with α and β 0. z t Next we will use the condition ( sup y G(x + iy = y>0,x Considering first the imaginary part we get that β = 0 and for all N, N y ( + t dσ(t. Thus ( + N t +y t dσ(t. Let N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space (96, vol., Ch. VI, 58
4 4 LECTUE 4: FEE HAMONIC ANALYSIS ν(e = E +t dσ(t. From the real part of ( we get that for all y>0 y α + t(y t + y dν(t Therefore t( y α = lim y dσ(t t + y = lim t y y dσ(t = tdσ(t +(t/y Hence G(z = t + +tz tz + t z t dσ(t = ++tz dσ(t z t = z t ( + t dσ(t = z t dν(t emark. If {ν n } n is a sequence of finite Borel measures on we say that {ν n } n converges in distribution to the measure ν if for every f C b ( (the continuous bounded functions on we have lim n f(t dνn (t f(t dν(t. We say that {ν n } n converges vaguely to ν if for every f C 0 ( (the continuous functions on vanishing at infinity we have lim n f(t dνn (t f(t dν(t. Convergence in distribution implies vague convergence but not conversely. However if ν is a probability measure then {ν n } n converges vaguely to ν does imply that {ν n } n converges to ν in distribution. Theorem. Suppose that (ν n n is a sequence of probability measures on with G n the Cauchy transform of ν n. Suppose {G n } n converges pointwise to G on C +. If lim y yg(iy = i then there is a unique probability measure ν on such that ν n ν in distribution, and G(z = dν(t. z t Proof. {G n } n is uniformly bounded on compact subsets of C + (as G(z Im(z for the Cauchy transform of any probability measure, so by Montel s theorem {G n } n is relatively compact in the topology of uniform convergence on compact subsets of C +, thus, in particular, {G n } n has a subsequence which converges uniformly on compact subsets of C + to an analytic function, which must be G. ThusG is analytic, as it is the uniform limit of analytic functions. Now for z C +, G(z C. Also for each n N, x and y 0 y G n (x + iy. see Kai Lai Chung Probability, nd ed. ( and 3.4.
5 LECTUE 4: FEE HAMONIC ANALYSIS 5 Thus x, y 0, y G(x + iy. So in particular, G is nonconstant. If for some z C +, Im(G(z = 0 then by the minimum modulus principle G would be constant. Thus G maps C + into C. Hence by Nevanlinna s theorem there is a unique probability measure ν such that G(z = dν(t. z t Now by the Helley selection theorem 3 there is a subsequence {ν nk } k converging vaguely to some measure ν. For fixed z the function t /(z t isinc 0 (. Thus for Im(z > 0, G nk (z = dν z t n k (t d ν(t. Therefore G(z = dν(t i.e. ν = ν. Thus{ν z t z t n k } k converges vaguely to ν. Since ν is a probability measure {ν nk } k converges vaguely to ν. Thus{ν n } converges in distribution to ν. Corollary. If {ν n } n is a sequence of probability measures on with G n G pointwise on C +. Then {ν n } converges vaguely to a finite measure ν. The analytic relation between moments and free cumulants involves finding a functional inverse for the Cauchy transform of a probability measure, ν. One cannot do this in general without making some assumptions about ν. A fairly general condition was found by Hans Maassen in 99 4 Theorem (H. Maassen. Let ν be a probability measure on with σ = t dν(t < and tdν(t =0. Then G is a bijection between {z Im(z >σ} and { z Im(z > σ} i σ Compactly Supported Measures. When the probability measure ν has compact support then one can find a region in C + upon which the Cauchy transform is univalent. Suppose ν is a probability measure on and supp(ν [ r, r]. Then ν has moments m n = t n dν(t of all orders and m n r n. Let M(z =+m z + m z + be the moment generating function. If 3 E. Lukacs, Characteristic Functions, nd ed., Griffin, 97 4 Addition of Freely Independent andom Variables, J. Funct. Anal., 06 (99,
6 6 LECTUE 4: FEE HAMONIC ANALYSIS z <r the series converges. If z >rthen z <r and G(z = z t dν(t = z n t n dν(t z n 0 = z n t n dν(t = z n m n = ( z z z M z n 0 Let f(z =zm(z =G( =z + m z z + m z 3 +. Then f(0=0 and f (0 =. Suppose z, z (4r then f(z f(z ( f(z f(z z z e z z [ ] d f(z + t(z z = e dt dt z z = 0 0 n 0 e(f (z + t(z z dt Now e(f (z = e( + zm +3x m + z r 3 z r = (+( z r+3( z r + = So for z < (4r we have ( z r e(f (z ( /4 = 9 Hence for z, z < (4r we have f(z f(z z 9 z. In particular f is one-to-one on {z z < (4r }. Also f(z = z +m z + m z + ( z ( ( + r z + r z + = z ( z = /4 3 z r z Thus for z =(4r we have f(z (6r. Moreover, since f is one-to-one on {z z (4r }, the curve γ = {f(e it /(4r 0 t π} only wraps once around the origin. Thus f maps {z z < (4r } to the interior of the curve γ. Hence {z z < (6r } {f(z z < (4r }
7 LECTUE 4: FEE HAMONIC ANALYSIS 7 We will denote the inverse of f under composition by f.thusf is analytic on {z z < (6r } and has a simple zero at z =0. Thus K(z = f (z is meromorphic on {z z < (6r } and has a simple pole at z =0. Now K(G(z=/(f (G(z=/(f (f(z = z and G(K(z = f(f (z = z. Theorem. Suppose supp(ν [ r, r], then G ν is univalent on the open disc with centre 0 and radius (6r Let (z =K(z. Then is analytic on {z z < z (6r }, and ( G z + (z = z = (G(z + G(z which is the same relation as was found combinatorially. Bercovici and Voiculescu found a theorem that relates convergence in distribution of a sequence of probability measures with support in [ r, r] to the uniform convergence of the corresponding -transforms on a disc. Theorem (Bercovici & Voiculescu. Let {ν n } n be a sequence of probability measures on with compact support, let n be the corresponding sequence of -transforms. Then i r s.t. supp(ν n [ r, r] and ν with supp(ν [ r, r] such that ν D n ν if and only if ii s>0 such that { n } n converges uniformly on {z z <s} to an analytic function on {z z <s}. Example: The Semi-circle Law. As an example of Stieltjes inversion let us take a familiar example and calculate its Cauchy transform and then using only the Cauchy transform find the density by using Stieltjes inversion. The density of the semi-circle law is given by dν dt = the moments are given by 4 t sdt on [, ] π m n = { 0 n odd t n dν(t = n even c n/
8 8 LECTUE 4: FEE HAMONIC ANALYSIS where the c n s are the Catalan numbers: c n = ( n n + n Now let M(z be the moment generating function then M(z = M(z =+c z + c z 4 + c m c n z (m+n = m,n 0 k 0 Now we saw in lecture three that c m c n = c k+ so therefore m+n=k ( m+n=k c m c n z k M(z = c k+ z k = c z k+ z (k+ k 0 k 0 z M(z = M(z orm(z =+z M(z Therefore z M(z M(z + = 0. Solving the quadratic equation we have M(z = ± 4z and M(0= z Now + 4t as t 0, t whereas 4t = t + 4t ast 0 Therefore we choose the minus sign M(z = 4z z Now G(z = ( z M = z z 4z = z z 4 z Also + z M(z = M(z implies ( ( ( + z M = z z z M(z
9 LECTUE 4: FEE HAMONIC ANALYSIS 9 i.e. +G(z = zg(z orz = G(z+ G(z Thus K(z =z + and (z =z i.e. all so all cumulants of the z semi-circle law are 0 except κ, which equals. Now let us apply Stieltjes inversion to G(z. To do this we have to choose a branch of z 4. To achieve this we choose a branch for each of z and z +. Now z = z / e iθ/ where θ is the argument of z. Choosing a branch means connecting to with a curve along which θ will have a jump discontinuity.. z θ (z θ (z We choose θ, the argument of z, to be such that 0 θ < π. Similarly we choose θ, the argument of z +, such that 0 θ < π. Thus θ + θ is continuous on C \ [,. However e i(θ +θ / is continuous on C \ [, ] because e i(0+0/ ==e i(π+π/, so there is no jump as the half line [, is crossed. Now z 4= z z += z 4 / e iθ/ e iθ/ = z 4 / e i(θ +θ / Im( (x + iy 4 = (x + iy 4 / sin((θ + θ / lim Im( (x + iy 4 = x 4 / y 0 = { 0 x > x < { 0 x > 0 4 x x < lim y 0 ( x + iy (x + Im(G(x + iy = lim Im iy 4 + y 0 + { 0 x > = 4 x x <
10 0 LECTUE 4: FEE HAMONIC ANALYSIS Therefore { 0 x > lim Im(G(x + iy = y 0 + π 4 x π x < Hence we recover our original density. Stieltjes Inversion at an Atom. When the Cauchy transform has a pole at a the limit calculation using the dominated convergence theorem used above will not apply. A situation that frequently occurs is that G(z has a simple pole at a, i.e. G(z =f(a/(z a with f analytic on a neighbourhood of a, then ν has an atom at a with weight f(a. Theorem. Suppose G(z =f(z/(z a with a a real number and f analytic on a neighbourhood, B(a, r, ofa. Then a+r lim Im(G(x + iy dx = f(a y 0 + π a r Proof. Let C be the boundary of the rectangle with vertices at a+y+ir, a y + ir, a y ir, and a + y ir. We will parameterize the four sides as follows. C a + r + iy C a C 4 a r iy C 3 C : the top side; γ (t =t+iy where t decreases from a+r to a r; C : the left side; γ = a r + it where t decreases from y to y; C 3 : the bottom side; γ 3 (t =t iy where t increases from a r to a + r C 4 : the right side; γ 4 (t =a + r + it where t increases from y to y Then by Cauchy s integral theorem πi C f(z dz = f(a z a
11 LECTUE 4: FEE HAMONIC ANALYSIS So let us calculate separately each integral over C i.now a r a+r G(z dz = G(x + iy and G(z dz = G(x iy C a+r C 3 a r Thus a+r G(z dz = G(x iδ G(x + iy dx C +C 3 a r = a+r a r i Im(G(x + iy dx Thus a+r Im(G(x + iy dx = G(z dz a r π πi C +C 3 So we only need to prove that for i = or 4 lim G(z dz =0 y 0 + C i we shall only do the case i =. Since f is analytic on B(0,r there is M>0such that f(z M for z C.Thus y G(z dz M My dt C y r it r Thus for i = and 4 lim G(z dz =0 y 0 + C i
Solutions to practice problems for the final
Solutions to practice problems for the final Holomorphicity, Cauchy-Riemann equations, and Cauchy-Goursat theorem 1. (a) Show that there is a holomorphic function on Ω = {z z > 2} whose derivative is z
More informationFINAL EXAM MATH 220A, UCSD, AUTUMN 14. You have three hours.
FINAL EXAM MATH 220A, UCSD, AUTUMN 4 You have three hours. Problem Points Score There are 6 problems, and the total number of points is 00. Show all your work. Please make your work as clear and easy to
More informationOn the law of large numbers for free identically distributed random variables 1
General Mathematics Vol. 5, No. 4 (2007), 55-68 On the law of large numbers for free identically distributed random variables Bogdan Gheorghe Munteanu Abstract A version of law of large numbers for free
More informationFailure of the Raikov Theorem for Free Random Variables
Failure of the Raikov Theorem for Free Random Variables Florent Benaych-Georges DMA, École Normale Supérieure, 45 rue d Ulm, 75230 Paris Cedex 05 e-mail: benaych@dma.ens.fr http://www.dma.ens.fr/ benaych
More informationMORE CONSEQUENCES OF CAUCHY S THEOREM
MOE CONSEQUENCES OF CAUCHY S THEOEM Contents. The Mean Value Property and the Maximum-Modulus Principle 2. Morera s Theorem and some applications 3 3. The Schwarz eflection Principle 6 We have stated Cauchy
More informationPart IB. Further Analysis. Year
Year 2004 2003 2002 2001 10 2004 2/I/4E Let τ be the topology on N consisting of the empty set and all sets X N such that N \ X is finite. Let σ be the usual topology on R, and let ρ be the topology on
More information2 Write down the range of values of α (real) or β (complex) for which the following integrals converge. (i) e z2 dz where {γ : z = se iα, < s < }
Mathematical Tripos Part II Michaelmas term 2007 Further Complex Methods, Examples sheet Dr S.T.C. Siklos Comments and corrections: e-mail to stcs@cam. Sheet with commentary available for supervisors.
More informationAnalysis Comprehensive Exam, January 2011 Instructions: Do as many problems as you can. You should attempt to answer completely some questions in both
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
More informationPart IB Complex Analysis
Part IB Complex Analysis Theorems Based on lectures by I. Smith Notes taken by Dexter Chua Lent 2016 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after
More informationIII. Consequences of Cauchy s Theorem
MTH6 Complex Analysis 2009-0 Lecture Notes c Shaun Bullett 2009 III. Consequences of Cauchy s Theorem. Cauchy s formulae. Cauchy s Integral Formula Let f be holomorphic on and everywhere inside a simple
More informationINTEGRATION WORKSHOP 2004 COMPLEX ANALYSIS EXERCISES
INTEGRATION WORKSHOP 2004 COMPLEX ANALYSIS EXERCISES PHILIP FOTH 1. Cauchy s Formula and Cauchy s Theorem 1. Suppose that γ is a piecewise smooth positively ( counterclockwise ) oriented simple closed
More informationMATH FINAL SOLUTION
MATH 185-4 FINAL SOLUTION 1. (8 points) Determine whether the following statements are true of false, no justification is required. (1) (1 point) Let D be a domain and let u,v : D R be two harmonic functions,
More informationPart IB. Complex Analysis. Year
Part IB Complex Analysis Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 Paper 1, Section I 2A Complex Analysis or Complex Methods 7 (a) Show that w = log(z) is a conformal
More informationComplex Variables. Instructions Solve any eight of the following ten problems. Explain your reasoning in complete sentences to maximize credit.
Instructions Solve any eight of the following ten problems. Explain your reasoning in complete sentences to maximize credit. 1. The TI-89 calculator says, reasonably enough, that x 1) 1/3 1 ) 3 = 8. lim
More informationComplex 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
More informationProperties of Analytic Functions
Properties of Analytic Functions Generalizing Results to Analytic Functions In the last few sections, we completely described entire functions through the use of everywhere convergent power series. Our
More informationMA3111S COMPLEX ANALYSIS I
MA3111S COMPLEX ANALYSIS I 1. The Algebra of Complex Numbers A complex number is an expression of the form a + ib, where a and b are real numbers. a is called the real part of a + ib and b the imaginary
More informationComplex Analysis. Travis Dirle. December 4, 2016
Complex Analysis 2 Complex Analysis Travis Dirle December 4, 2016 2 Contents 1 Complex Numbers and Functions 1 2 Power Series 3 3 Analytic Functions 7 4 Logarithms and Branches 13 5 Complex Integration
More informationQualifying Exam Complex Analysis (Math 530) January 2019
Qualifying Exam Complex Analysis (Math 53) January 219 1. Let D be a domain. A function f : D C is antiholomorphic if for every z D the limit f(z + h) f(z) lim h h exists. Write f(z) = f(x + iy) = u(x,
More informationNATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part II
NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part II Chapter 2 Further properties of analytic functions 21 Local/Global behavior of analytic functions;
More information1. If 1, ω, ω 2, -----, ω 9 are the 10 th roots of unity, then (1 + ω) (1 + ω 2 ) (1 + ω 9 ) is A) 1 B) 1 C) 10 D) 0
4 INUTES. If, ω, ω, -----, ω 9 are the th roots of unity, then ( + ω) ( + ω ) ----- ( + ω 9 ) is B) D) 5. i If - i = a + ib, then a =, b = B) a =, b = a =, b = D) a =, b= 3. Find the integral values for
More informationFourth Week: Lectures 10-12
Fourth Week: Lectures 10-12 Lecture 10 The fact that a power series p of positive radius of convergence defines a function inside its disc of convergence via substitution is something that we cannot ignore
More informationComplex Analysis review notes for weeks 1-6
Complex Analysis review notes for weeks -6 Peter Milley Semester 2, 2007 In what follows, unless stated otherwise a domain is a connected open set. Generally we do not include the boundary of the set,
More informationGeneral Notation. Exercises and Problems
Exercises and Problems The text contains both Exercises and Problems. The exercises are incorporated into the development of the theory in each section. Additional Problems appear at the end of most sections.
More information11 COMPLEX ANALYSIS IN C. 1.1 Holomorphic Functions
11 COMPLEX ANALYSIS IN C 1.1 Holomorphic Functions A domain Ω in the complex plane C is a connected, open subset of C. Let z o Ω and f a map f : Ω C. We say that f is real differentiable at z o if there
More informationQuasi-conformal maps and Beltrami equation
Chapter 7 Quasi-conformal maps and Beltrami equation 7. Linear distortion Assume that f(x + iy) =u(x + iy)+iv(x + iy) be a (real) linear map from C C that is orientation preserving. Let z = x + iy and
More informationLECTURE-15 : LOGARITHMS AND COMPLEX POWERS
LECTURE-5 : LOGARITHMS AND COMPLEX POWERS VED V. DATAR The purpose of this lecture is twofold - first, to characterize domains on which a holomorphic logarithm can be defined, and second, to show that
More information3. 4. Uniformly normal families and generalisations
Summer School Normal Families in Complex Analysis Julius-Maximilians-Universität Würzburg May 22 29, 2015 3. 4. Uniformly normal families and generalisations Aimo Hinkkanen University of Illinois at Urbana
More informationComplex Analysis, Stein and Shakarchi The Fourier Transform
Complex Analysis, Stein and Shakarchi Chapter 4 The Fourier Transform Yung-Hsiang Huang 2017.11.05 1 Exercises 1. Suppose f L 1 (), and f 0. Show that f 0. emark 1. This proof is observed by Newmann (published
More informationIII.2. Analytic Functions
III.2. Analytic Functions 1 III.2. Analytic Functions Recall. When you hear analytic function, think power series representation! Definition. If G is an open set in C and f : G C, then f is differentiable
More informationIV. Conformal Maps. 1. Geometric interpretation of differentiability. 2. Automorphisms of the Riemann sphere: Möbius transformations
MTH6111 Complex Analysis 2009-10 Lecture Notes c Shaun Bullett 2009 IV. Conformal Maps 1. Geometric interpretation of differentiability We saw from the definition of complex differentiability that if f
More informationComplex Analysis, Stein and Shakarchi Meromorphic Functions and the Logarithm
Complex Analysis, Stein and Shakarchi Chapter 3 Meromorphic Functions and the Logarithm Yung-Hsiang Huang 217.11.5 Exercises 1. From the identity sin πz = eiπz e iπz 2i, it s easy to show its zeros are
More informationMATH 311: COMPLEX ANALYSIS CONTOUR INTEGRALS LECTURE
MATH 3: COMPLEX ANALYSIS CONTOUR INTEGRALS LECTURE Recall the Residue Theorem: Let be a simple closed loop, traversed counterclockwise. Let f be a function that is analytic on and meromorphic inside. Then
More informationComplex Analysis Qualifying Exam Solutions
Complex Analysis Qualifying Exam Solutions May, 04 Part.. Let log z be the principal branch of the logarithm defined on G = {z C z (, 0]}. Show that if t > 0, then the equation log z = t has exactly one
More informationMATH MEASURE THEORY AND FOURIER ANALYSIS. Contents
MATH 3969 - MEASURE THEORY AND FOURIER ANALYSIS ANDREW TULLOCH Contents 1. Measure Theory 2 1.1. Properties of Measures 3 1.2. Constructing σ-algebras and measures 3 1.3. Properties of the Lebesgue measure
More informationSolutions to Complex Analysis Prelims Ben Strasser
Solutions to Complex Analysis Prelims Ben Strasser In preparation for the complex analysis prelim, I typed up solutions to some old exams. This document includes complete solutions to both exams in 23,
More informationMATH 185: COMPLEX ANALYSIS FALL 2009/10 PROBLEM SET 9 SOLUTIONS. and g b (z) = eπz/2 1
MATH 85: COMPLEX ANALYSIS FALL 2009/0 PROBLEM SET 9 SOLUTIONS. Consider the functions defined y g a (z) = eiπz/2 e iπz/2 + Show that g a maps the set to D(0, ) while g maps the set and g (z) = eπz/2 e
More informationThe result above is known as the Riemann mapping theorem. We will prove it using basic theory of normal families. We start this lecture with that.
Lecture 15 The Riemann mapping theorem Variables MATH-GA 2451.1 Complex The point of this lecture is to prove that the unit disk can be mapped conformally onto any simply connected open set in the plane,
More informationThe Residue Theorem. Integration Methods over Closed Curves for Functions with Singularities
The Residue Theorem Integration Methods over losed urves for Functions with Singularities We have shown that if f(z) is analytic inside and on a closed curve, then f(z)dz = 0. We have also seen examples
More informationTheorem [Mean Value Theorem for Harmonic Functions] Let u be harmonic on D(z 0, R). Then for any r (0, R), u(z 0 ) = 1 z z 0 r
2. A harmonic conjugate always exists locally: if u is a harmonic function in an open set U, then for any disk D(z 0, r) U, there is f, which is analytic in D(z 0, r) and satisfies that Re f u. Since such
More informationMoment Inequalities for Equilibrium Measures in the Plane
Pure and Applied Mathematics Quarterly Volume 7, Number 1 (Special Issue: In honor of Frederick W. Gehring) 47 82, 2011 Moment Inequalities for Equilibrium Measures in the Plane A. Baernstein II, R. S.
More informationC4.8 Complex Analysis: conformal maps and geometry. D. Belyaev
C4.8 Complex Analysis: conformal maps and geometry D. Belyaev November 4, 2016 2 Contents 1 Introduction 5 1.1 About this text............................ 5 1.2 Preliminaries............................
More informationChapter 4: Open mapping theorem, removable singularities
Chapter 4: Open mapping theorem, removable singularities Course 44, 2003 04 February 9, 2004 Theorem 4. (Laurent expansion) Let f : G C be analytic on an open G C be open that contains a nonempty annulus
More informationPOWER SERIES AND ANALYTIC CONTINUATION
POWER SERIES AND ANALYTIC CONTINUATION 1. Analytic functions Definition 1.1. A function f : Ω C C is complex-analytic if for each z 0 Ω there exists a power series f z0 (z) := a n (z z 0 ) n which converges
More informationlim n C1/n n := ρ. [f(y) f(x)], y x =1 [f(x) f(y)] [g(x) g(y)]. (x,y) E A E(f, f),
1 Part I Exercise 1.1. Let C n denote the number of self-avoiding random walks starting at the origin in Z of length n. 1. Show that (Hint: Use C n+m C n C m.) lim n C1/n n = inf n C1/n n := ρ.. Show that
More informationHere are brief notes about topics covered in class on complex numbers, focusing on what is not covered in the textbook.
Phys374, Spring 2008, Prof. Ted Jacobson Department of Physics, University of Maryland Complex numbers version 5/21/08 Here are brief notes about topics covered in class on complex numbers, focusing on
More informationAn Introduction to Complex Analysis and Geometry John P. D Angelo, Pure and Applied Undergraduate Texts Volume 12, American Mathematical Society, 2010
An Introduction to Complex Analysis and Geometry John P. D Angelo, Pure and Applied Undergraduate Texts Volume 12, American Mathematical Society, 2010 John P. D Angelo, Univ. of Illinois, Urbana IL 61801.
More informationMORE NOTES FOR MATH 823, FALL 2007
MORE NOTES FOR MATH 83, FALL 007 Prop 1.1 Prop 1. Lemma 1.3 1. The Siegel upper half space 1.1. The Siegel upper half space and its Bergman kernel. The Siegel upper half space is the domain { U n+1 z C
More informationSummer Jump-Start Program for Analysis, 2012 Song-Ying Li
Summer Jump-Start Program for Analysis, 01 Song-Ying Li 1 Lecture 6: Uniformly continuity and sequence of functions 1.1 Uniform Continuity Definition 1.1 Let (X, d 1 ) and (Y, d ) are metric spaces and
More informationMetric Spaces and Topology
Chapter 2 Metric Spaces and Topology From an engineering perspective, the most important way to construct a topology on a set is to define the topology in terms of a metric on the set. This approach underlies
More informationMath 411, Complex Analysis Definitions, Formulas and Theorems Winter y = sinα
Math 411, Complex Analysis Definitions, Formulas and Theorems Winter 014 Trigonometric Functions of Special Angles α, degrees α, radians sin α cos α tan α 0 0 0 1 0 30 π 6 45 π 4 1 3 1 3 1 y = sinα π 90,
More informationPart IB Complex Analysis
Part IB Complex Analysis Theorems with proof Based on lectures by I. Smith Notes taken by Dexter Chua Lent 206 These notes are not endorsed by the lecturers, and I have modified them (often significantly)
More informationSolutions for Problem Set #5 due October 17, 2003 Dustin Cartwright and Dylan Thurston
Solutions for Problem Set #5 due October 17, 23 Dustin Cartwright and Dylan Thurston 1 (B&N 6.5) Suppose an analytic function f agrees with tan x, x 1. Show that f(z) = i has no solution. Could f be entire?
More informationWorked examples Conformal mappings and bilinear transformations
Worked examples Conformal mappings and bilinear transformations Example 1 Suppose we wish to find a bilinear transformation which maps the circle z i = 1 to the circle w =. Since w/ = 1, the linear transformation
More informationcarries the circle w 1 onto the circle z R and sends w = 0 to z = a. The function u(s(w)) is harmonic in the unit circle w 1 and we obtain
4. Poisson formula In fact we can write down a formula for the values of u in the interior using only the values on the boundary, in the case when E is a closed disk. First note that (3.5) determines the
More informationHomework 27. Homework 28. Homework 29. Homework 30. Prof. Girardi, Math 703, Fall 2012 Homework: Define f : C C and u, v : R 2 R by
Homework 27 Define f : C C and u, v : R 2 R by f(z) := xy where x := Re z, y := Im z u(x, y) = Re f(x + iy) v(x, y) = Im f(x + iy). Show that 1. u and v satisfies the Cauchy Riemann equations at (x, y)
More informationMATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5
MATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5.. The Arzela-Ascoli Theorem.. The Riemann mapping theorem Let X be a metric space, and let F be a family of continuous complex-valued functions on X. We have
More informationRIEMANN MAPPING THEOREM
RIEMANN MAPPING THEOREM VED V. DATAR Recall that two domains are called conformally equivalent if there exists a holomorphic bijection from one to the other. This automatically implies that there is an
More informationMATH SPRING UC BERKELEY
MATH 85 - SPRING 205 - UC BERKELEY JASON MURPHY Abstract. These are notes for Math 85 taught in the Spring of 205 at UC Berkeley. c 205 Jason Murphy - All Rights Reserved Contents. Course outline 2 2.
More informationComplex Analysis Important Concepts
Complex Analysis Important Concepts Travis Askham April 1, 2012 Contents 1 Complex Differentiation 2 1.1 Definition and Characterization.............................. 2 1.2 Examples..........................................
More informationConformal maps. Lent 2019 COMPLEX METHODS G. Taylor. A star means optional and not necessarily harder.
Lent 29 COMPLEX METHODS G. Taylor A star means optional and not necessarily harder. Conformal maps. (i) Let f(z) = az + b, with ad bc. Where in C is f conformal? cz + d (ii) Let f(z) = z +. What are the
More informationRudin Real and Complex Analysis - Harmonic Functions
Rudin Real and Complex Analysis - Harmonic Functions Aaron Lou December 2018 1 Notes 1.1 The Cauchy-Riemann Equations 11.1: The Operators and Suppose f is a complex function defined in a plane open set
More informationMATHS 730 FC Lecture Notes March 5, Introduction
1 INTRODUCTION MATHS 730 FC Lecture Notes March 5, 2014 1 Introduction Definition. If A, B are sets and there exists a bijection A B, they have the same cardinality, which we write as A, #A. If there exists
More information+ 2x sin x. f(b i ) f(a i ) < ɛ. i=1. i=1
Appendix To understand weak derivatives and distributional derivatives in the simplest context of functions of a single variable, we describe without proof some results from real analysis (see [7] and
More informationNotes on Complex Analysis
Michael Papadimitrakis Notes on Complex Analysis Department of Mathematics University of Crete Contents The complex plane.. The complex plane...................................2 Argument and polar representation.........................
More informationF (z) =f(z). f(z) = a n (z z 0 ) n. F (z) = a n (z z 0 ) n
6 Chapter 2. CAUCHY S THEOREM AND ITS APPLICATIONS Theorem 5.6 (Schwarz reflection principle) Suppose that f is a holomorphic function in Ω + that extends continuously to I and such that f is real-valued
More informationMATH COMPLEX ANALYSIS. Contents
MATH 3964 - OMPLEX ANALYSIS ANDREW TULLOH AND GILES GARDAM ontents 1. ontour Integration and auchy s Theorem 2 1.1. Analytic functions 2 1.2. ontour integration 3 1.3. auchy s theorem and extensions 3
More informationConformal Mappings. Chapter Schwarz Lemma
Chapter 5 Conformal Mappings In this chapter we study analytic isomorphisms. An analytic isomorphism is also called a conformal map. We say that f is an analytic isomorphism of U with V if f is an analytic
More information7.2 Conformal mappings
7.2 Conformal mappings Let f be an analytic function. At points where f (z) 0 such a map has the remarkable property that it is conformal. This means that angle is preserved (in the sense that any 2 smooth
More informationMULTIPLICATIVE MONOTONIC CONVOLUTION
Illinois Journal of Mathematics Volume 49, Number 3, Fall 25, Pages 929 95 S 9-282 MULTIPLICATIVE MONOTONIC CONVOLUTION HARI BERCOVICI Abstract. We show that the monotonic independence introduced by Muraki
More informationExercises for Part 1
MATH200 Complex Analysis. Exercises for Part Exercises for Part The following exercises are provided for you to revise complex numbers. Exercise. Write the following expressions in the form x + iy, x,y
More informationComplex Analysis. sin(z) = eiz e iz 2i
Complex Analysis. Preliminaries and Notation For a complex number z = x + iy, we set x = Re z and y = Im z, its real and imaginary components. Any nonzero complex number z = x + iy can be written uniquely
More informationComplex Variables, Summer 2016 Homework Assignments
Complex Variables, Summer 2016 Homework Assignments Homeworks 1-4, due Thursday July 14th Do twenty-four of the following problems. Question 1 Let a = 2 + i and b = 1 i. Sketch the complex numbers a, b,
More informationAnalysis Comprehensive Exam Questions Fall 2008
Analysis Comprehensive xam Questions Fall 28. (a) Let R be measurable with finite Lebesgue measure. Suppose that {f n } n N is a bounded sequence in L 2 () and there exists a function f such that f n (x)
More informationz b k P k p k (z), (z a) f (n 1) (a) 2 (n 1)! (z a)n 1 +f n (z)(z a) n, where f n (z) = 1 C
. Representations of Meromorphic Functions There are two natural ways to represent a rational function. One is to express it as a quotient of two polynomials, the other is to use partial fractions. The
More informationSolutions for Problem Set #4 due October 10, 2003 Dustin Cartwright
Solutions for Problem Set #4 due October 1, 3 Dustin Cartwright (B&N 4.3) Evaluate C f where f(z) 1/z as in Example, and C is given by z(t) sin t + i cos t, t π. Why is the result different from that of
More informationBochner s Theorem on the Fourier Transform on R
Bochner s heorem on the Fourier ransform on Yitao Lei October 203 Introduction ypically, the Fourier transformation sends suitable functions on to functions on. his can be defined on the space L ) + L
More informationSpring 2010 Exam 2. You may not use your books, notes, or any calculator on this exam.
MTH 282 final Spring 2010 Exam 2 Time Limit: 110 Minutes Name (Print): Instructor: Prof. Houhong Fan This exam contains 6 pages (including this cover page) and 5 problems. Check to see if any pages are
More informationComplex Variables & Integral Transforms
Complex Variables & Integral Transforms Notes taken by J.Pearson, from a S4 course at the U.Manchester. Lecture delivered by Dr.W.Parnell July 9, 007 Contents 1 Complex Variables 3 1.1 General Relations
More informationDepartment of Mathematics, University of California, Berkeley. GRADUATE PRELIMINARY EXAMINATION, Part A Fall Semester 2016
Department of Mathematics, University of California, Berkeley YOUR 1 OR 2 DIGIT EXAM NUMBER GRADUATE PRELIMINARY EXAMINATION, Part A Fall Semester 2016 1. Please write your 1- or 2-digit exam number on
More informationCHAPTER 6. Differentiation
CHPTER 6 Differentiation The generalization from elementary calculus of differentiation in measure theory is less obvious than that of integration, and the methods of treating it are somewhat involved.
More informationMath212a1413 The Lebesgue integral.
Math212a1413 The Lebesgue integral. October 28, 2014 Simple functions. In what follows, (X, F, m) is a space with a σ-field of sets, and m a measure on F. The purpose of today s lecture is to develop the
More information1. The COMPLEX PLANE AND ELEMENTARY FUNCTIONS: Complex numbers; stereographic projection; simple and multiple connectivity, elementary functions.
Complex Analysis Qualifying Examination 1 The COMPLEX PLANE AND ELEMENTARY FUNCTIONS: Complex numbers; stereographic projection; simple and multiple connectivity, elementary functions 2 ANALYTIC FUNCTIONS:
More informationANALYSIS QUALIFYING EXAM FALL 2016: SOLUTIONS. = lim. F n
ANALYSIS QUALIFYING EXAM FALL 206: SOLUTIONS Problem. Let m be Lebesgue measure on R. For a subset E R and r (0, ), define E r = { x R: dist(x, E) < r}. Let E R be compact. Prove that m(e) = lim m(e /n).
More informationMid Term-1 : Solutions to practice problems
Mid Term- : Solutions to practice problems 0 October, 06. Is the function fz = e z x iy holomorphic at z = 0? Give proper justification. Here we are using the notation z = x + iy. Solution: Method-. Use
More informationSelected Solutions To Problems in Complex Analysis
Selected Solutions To Problems in Complex Analysis E. Chernysh November 3, 6 Contents Page 8 Problem................................... Problem 4................................... Problem 5...................................
More informationBernstein s analytic continuation of complex powers
(April 3, 20) Bernstein s analytic continuation of complex powers Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/. Analytic continuation of distributions 2. Statement of the theorems
More informationMath Final Exam.
Math 106 - Final Exam. This is a closed book exam. No calculators are allowed. The exam consists of 8 questions worth 100 points. Good luck! Name: Acknowledgment and acceptance of honor code: Signature:
More informationChapter 6: The metric space M(G) and normal families
Chapter 6: The metric space MG) and normal families Course 414, 003 04 March 9, 004 Remark 6.1 For G C open, we recall the notation MG) for the set algebra) of all meromorphic functions on G. We now consider
More informationReal Analysis Problems
Real Analysis Problems Cristian E. Gutiérrez September 14, 29 1 1 CONTINUITY 1 Continuity Problem 1.1 Let r n be the sequence of rational numbers and Prove that f(x) = 1. f is continuous on the irrationals.
More informationProblem 1A. Suppose that f is a continuous real function on [0, 1]. Prove that
Problem 1A. Suppose that f is a continuous real function on [, 1]. Prove that lim α α + x α 1 f(x)dx = f(). Solution: This is obvious for f a constant, so by subtracting f() from both sides we can assume
More informationMTH4101 CALCULUS II REVISION NOTES. 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) ax 2 + bx + c = 0. x = b ± b 2 4ac 2a. i = 1.
MTH4101 CALCULUS II REVISION NOTES 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) 1.1 Introduction Types of numbers (natural, integers, rationals, reals) The need to solve quadratic equations:
More information4.6 Montel's Theorem. Robert Oeckl CA NOTES 7 17/11/2009 1
Robert Oeckl CA NOTES 7 17/11/2009 1 4.6 Montel's Theorem Let X be a topological space. We denote by C(X) the set of complex valued continuous functions on X. Denition 4.26. A topological space is called
More informationEconomics 204 Fall 2011 Problem Set 2 Suggested Solutions
Economics 24 Fall 211 Problem Set 2 Suggested Solutions 1. Determine whether the following sets are open, closed, both or neither under the topology induced by the usual metric. (Hint: think about limit
More information3 (Due ). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due ). Show that the open disk x 2 + y 2 < 1 is a countable union of planar elementary sets. Show that the closed disk x 2 + y 2 1 is a countable
More informationINTRODUCTION TO COMPLEX ANALYSIS W W L CHEN
INTRODUTION TO OMPLEX ANALYSIS W W L HEN c W W L hen, 986, 2008. This chapter originates from material used by the author at Imperial ollege, University of London, between 98 and 990. It is available free
More informationTopic 4 Notes Jeremy Orloff
Topic 4 Notes Jeremy Orloff 4 auchy s integral formula 4. Introduction auchy s theorem is a big theorem which we will use almost daily from here on out. Right away it will reveal a number of interesting
More informationOn rational approximation of algebraic functions. Julius Borcea. Rikard Bøgvad & Boris Shapiro
On rational approximation of algebraic functions http://arxiv.org/abs/math.ca/0409353 Julius Borcea joint work with Rikard Bøgvad & Boris Shapiro 1. Padé approximation: short overview 2. A scheme of rational
More informationMath 185 Fall 2015, Sample Final Exam Solutions
Math 185 Fall 2015, Sample Final Exam Solutions Nikhil Srivastava December 12, 2015 1. True or false: (a) If f is analytic in the annulus A = {z : 1 < z < 2} then there exist functions g and h such that
More informationCONSEQUENCES OF POWER SERIES REPRESENTATION
CONSEQUENCES OF POWER SERIES REPRESENTATION 1. The Uniqueness Theorem Theorem 1.1 (Uniqueness). Let Ω C be a region, and consider two analytic functions f, g : Ω C. Suppose that S is a subset of Ω that
More information