NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India

Size: px
Start display at page:

Download "NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India"

Transcription

1 NPTEL web course on Complex Analysis A. Swaminathan I.I.T. Roorkee, India and V.K. Katiyar I.I.T. Roorkee, India A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 1 / 27

2 Complex Analysis Module: 2: Functions of a Complex Variable Lecture: 4: Analytic functions A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 2 / 27

3 Derivatives A small note So far, we have seen the mapping of functions that takes values from Z -plane (xy-plane) to the Ω-plane(uv-plane). In the Ω-plane, u and v are functions of x and y. w = u + iv Ω, u = u(x, y), v = v(x, y), x + iy Z (xy plane). A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 3 / 27

4 Derivatives A small note So far, we have seen the mapping of functions that takes values from Z -plane (xy-plane) to the Ω-plane(uv-plane). In the Ω-plane, u and v are functions of x and y. w = u + iv Ω, u = u(x, y), v = v(x, y), x + iy Z (xy plane). In the one-variable real case, the concept that follows continuity is derivative. Here, before the concept of differentiability, we consider the concept of partial derivatives which is similar to the two-variable real case. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 3 / 27

5 Partial Derivatives Definition Let u(x, y) be defined at all points in the neighbourhood of (x 0, y 0 ). Then we define the partial derivative of u with respect to x and y, respectively, as u x (x 0, y 0 ) = u y (x 0, y 0 ) = u(x 0 + x, y 0 ) u(x 0, y 0 ) lim x 0 x lim y 0 provided the limit exist, in each case. u(x 0, y 0 + y) u(x 0, y 0 ) y A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 4 / 27

6 Partial Derivatives Definition Let u(x, y) be defined at all points in the neighbourhood of (x 0, y 0 ). Then we define the partial derivative of u with respect to x and y, respectively, as u x (x 0, y 0 ) = u y (x 0, y 0 ) = u(x 0 + x, y 0 ) u(x 0, y 0 ) lim x 0 x lim y 0 provided the limit exist, in each case. Notation u(x 0, y 0 + y) u(x 0, y 0 ) y Some authors prefer to write the notation u x = u x. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 4 / 27

7 Partial Derivatives Second order partial derivatives u xx = 2 u x 2, u xy = u yy = 2 u y 2, 2 u x y, u yx = 2 u y x, where u xy and u yx are called second order mixed partial derivatives. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 5 / 27

8 Partial Derivatives Note Assume that, for the function u(x, y), the first order partial derivatives u x, u y and the second order partial derivatives u xx, u yy, u xy and u yx exist at (x 0, y 0 ). Further if all these partial derivatives are continuous in the neighbourhood (x 0, y 0 ), then the mixed partial derivatives are equal at (x 0, y 0 ); i.e., u xy (x 0, y 0 ) = u yx (x 0, y 0 ). A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 6 / 27

9 Partial Derivatives Example Let xy(x 2 y 2 ) u(x, y) = x 2 + y 2 for (x, y) (0, 0) 0 for (x, y) = (0, 0). For this function, u xy (0, 0) = 1 and u yx (0, 0) = 1, because the mixed partial derivatives are not continuous at (0, 0). A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 7 / 27

10 Derivatives Definition Let f : D 1 D 2. Let z 0 D. f is differentiable at z 0 if f (z 0 + z) f (z 0 ) lim 0 z exists. This derivative is denoted as f (z 0 ). A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 8 / 27

11 Derivatives Note. If f (z 0 ) exists for each z 0 D, then f is said to be differentiable in D. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 9 / 27

12 Derivatives Example Question.Find the derivative of f (z) = z n. Answer. Using binomial theorem, we write Thus (z + z) n z n f (z) = δz ( lim nz n 1 + z 0 nz n 1 z + = = nz n 1 + n(n 1) z n 2 ( z) ( z) n 2 z n(n 1) z n 2 ( z) + + ( z) n 1. 2 ) n(n 1) z n 2 ( z) + + ( z) n 1 = nz n 1. 2 A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 10 / 27

13 Derivatives Note. The above example is similar to the real-variable case. Hence the following results can be adapted from the real variable calculus. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 11 / 27

14 Derivatives Results Theorem If f and g are differentiable then (f ± g) (z) = f (z) ± g (z), (cf ) (z) = cf (z) for some non-zero real constant c, (fg) (z) = f (z)g (z) + f (z)g(z), ( ) f (z) = g(z)f (z) f (z)g (z) g (g(z)) 2, if g(z) 0. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 12 / 27

15 Derivatives Chain Rule Theorem Let f (z) be a differentiable function of z and w = f (z). If g(w) is a differentiable function of w, then the composition function g(f (z)) is also differentiable and g (w) = (g(f (z))) = g (f (z))f (z). A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 13 / 27

16 Derivatives Analytic functions A topological behaviour of a complex valued function, obtained from the concept of its derivative, is called Analytic function. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 14 / 27

17 Analytic function Definition Let f : D 1 D 2. Let f be differentiable at z 0 and also in the neighborhood of z 0. Then f is said to be analytic at z 0. If this is true for each z 0 D 1 then f is analytic in D 1. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 15 / 27

18 Analytic function Example (i). f (z) = z is analytic at everywhere. (ii). All polynomials are analytic in C. (iii). f (z) = z is not analytic at z = 1. 1 z A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 16 / 27

19 Non-Analytic function Example Question. To check if f (z) = z is analytic at origin. Answer. First we find the derivative of f (z) = z at origin. For this, we need to find f (0 + z) f (0) lim exists. z 0 z This implies f (0 + z) f (0) f ( z) f (0) lim = lim z 0 z z 0 z z = lim z 0 z = Now we consider two different cases. lim ( x, y) 0 z 0 = lim z 0 z x i y x + i y. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 17 / 27

20 non-analytic function Example case(i): Let x 0 first. Then, ( ) x i y lim lim y 0 x 0 x + i y case(ii): Let y 0 first. Then, ( lim x 0 lim y 0 x i y x + i y = lim ( 1) = 1. y 0 ) = lim (1) = 1. x 0 Since taking the limit in different ways leads to different values, f (z) = z is not differentiable at origin and hence not analytic. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 18 / 27

21 non-analytic function Example In general f (z) = z is nowhere differentiable and hence is not analytic anywhere in the complex plane C. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 19 / 27

22 non-analytic function Example In general f (z) = z is nowhere differentiable and hence is not analytic anywhere in the complex plane C. Remark Since f (z) = z is not analytic, any f (z) that can be written in terms of z is not analytic. Since 2Re z = z + z, 2iIm z = z z, f (z) is not analytic if it can be written in the form consisting of z, Re z, Im z. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 19 / 27

23 Analytic function A consequence One of the important consequences of analytic functions is the following result. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 20 / 27

24 Analytic function A consequence One of the important consequences of analytic functions is the following result. Theorem Let f be analytic in a open connected set (domain) D and f (z) = 0 everywhere in D. Then f is constant throughout D. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 20 / 27

25 Analytic function Remark Note that in the previous result, the connectedness of the domain is essential. For example, define f (z) by { α if z < r1 f (z) = β if z > r 1 + 2, for some constant α and r 1 > 0. Then also f (z) = 0. But f (z) is not constant. Here the domain where f (z) is defined is not connected. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 21 / 27

26 Analytic function Theorem Let f be analytic in a domain D 1 and g be analytic in a domain D 2, then their sum, difference and product are analytic in the domain D 1 D 2. Similarly their quotient f /g is analytic in D 1 D 2, if g(z) 0 for z D 1 D 2. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 22 / 27

27 Analytic function Choice of domain for resultant function Remark The domain for the resultant function of two analytic function should be chosen very carefully. For the composition of two analytic functions which follows the chain rule, an example is given below to illustrate the choice of the domain. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 23 / 27

28 Analytic function Choice of domain for resultant function Example Define f (z) = z 2. Clearly this is an entire function (analytic in the whole complex plane). Define g(z) = z 1/2. This is a multi-valued function. Converting this function into polar form and restricting the domain of definition as g(z) = z 1/2 = ρe iφ/2, (ρ > 0, π < φ < π) it can be seen that this function has a derivative at each point of z in the domain of definition and g (z) = 1/2g(z). A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 24 / 27

29 Analytic function Choice of domain for resultant function Example Hence this function is analytic everywhere in the given domain of definition. For the composition g(f (z)), writing w = f (z) = re iθ, the function is defined as g(f (z)) = g(w) = w 1/2 = re iθ/2, (r > 0, π < θ < π). Even though, the domain of f is C, we should restrict its domain such that the range of f is contained in the domain on which g is defined. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 25 / 27

30 Analytic function Choice of domain for resultant function Example Hence the largest possible domain in which f can be defined so that the range of f should lie in the domain r > 0, π < θ < π. For this purpose, we write w = ρ 2 e i2φ. This gives π < φ < π when π/2 < φ < π/2. Thus r = ρ 2 and θ = 2φ, where r > 0 and π < θ < π. Hence the plane ρ > 0, π/2 < φ < π/2 is the largest possible domain that can be taken for the definition of f. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 26 / 27

31 Analytic function Choice of domain for resultant function Example This domain gives the analyticity of f (g(z)) from the analyticity of f and g in the respective domain. Hence g(f (z)) = g(w) = ρ 2 e i2φ/2 = ρe iφ = z, for any point z in the domain ρ > 0, π/2 < φ < π/2. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 27 / 27

NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India

NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India NPTEL web course on Complex Analysis A. Swaminathan I.I.T. Roorkee, India and V.K. Katiyar I.I.T. Roorkee, India A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 1 / 20 Complex Analysis Module: 2:

More information

NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India

NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India NPTEL web course on Complex Analysis A. Swaminathan I.I.T. Roorkee, India and V.K. Katiyar I.I.T. Roorkee, India A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 1 / 16 Complex Analysis Module: 2:

More information

NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India

NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India NPTEL web course on Complex Analysis A. Swaminathan I.I.T. Roorkee, India and V.K. Katiyar I.I.T. Roorkee, India A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 1 / 18 Complex Analysis Module: 5:

More information

NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India

NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India NPTEL web course on Complex Analysis A. Swaminathan I.I.T. Roorkee, India and V.K. Katiyar I.I.T. Roorkee, India A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 1 / 14 Complex Analysis Module: 1:

More information

NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India

NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India NPTEL web course on Complex Analysis A. Swaminathan I.I.T. Roorkee, India and V.K. Katiyar I.I.T. Roorkee, India A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 1 / 20 Complex Analysis Module: 10:

More information

NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India

NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India NPTEL web course on Complex Analysis A. Swaminathan I.I.T. Roorkee, India and V.K. Katiyar I.I.T. Roorkee, India A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 1 / 17 Complex Analysis Module: 5:

More information

NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India

NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India NPTEL web course on Complex Analysis A. Swaminathan I.I.T. Roorkee, India and V.K. Katiyar I.I.T. Roorkee, India A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 1 / 28 Complex Analysis Module: 6:

More information

NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India

NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India NPTEL web course on Complex Analysis A. Swaminathan I.I.T. Roorkee, India and V.K. Katiyar I.I.T. Roorkee, India A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 1 / 19 Complex Analysis Module: 8:

More information

NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India

NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India NPTEL web course on omplex Analysis A. Swaminathan I.I.T. Roorkee, India and V.K. Katiyar I.I.T. Roorkee, India A.Swaminathan and V.K.Katiyar (NPTEL) omplex Analysis 1 / 18 omplex Analysis Module: 6: Residue

More information

NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India

NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India NPTEL web course on Complex Analysis A. Swaminathan I.I.T. Roorkee, India and V.K. Katiyar I.I.T. Roorkee, India A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 1 / 36 Complex Analysis Module: 7:

More information

NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India

NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India NPTEL web course on Complex Analysis A. Swaminathan I.I.T. Roorkee, India and V.K. Katiyar I.I.T. Roorkee, India A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 1 / 29 Complex Analysis Module: 4:

More information

CHAPTER 3. Analytic Functions. Dr. Pulak Sahoo

CHAPTER 3. Analytic Functions. Dr. Pulak Sahoo CHAPTER 3 Analytic Functions BY Dr. Pulak Sahoo Assistant Professor Department of Mathematics University Of Kalyani West Bengal, India E-mail : sahoopulak1@gmail.com 1 Module-4: Harmonic Functions 1 Introduction

More information

Complex Variables. Chapter 2. Analytic Functions Section Harmonic Functions Proofs of Theorems. March 19, 2017

Complex Variables. Chapter 2. Analytic Functions Section Harmonic Functions Proofs of Theorems. March 19, 2017 Complex Variables Chapter 2. Analytic Functions Section 2.26. Harmonic Functions Proofs of Theorems March 19, 2017 () Complex Variables March 19, 2017 1 / 5 Table of contents 1 Theorem 2.26.1. 2 Theorem

More information

Theorem [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

Theorem [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 information

Math 185 Fall 2015, Sample Final Exam Solutions

Math 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 information

Math 185 Homework Exercises II

Math 185 Homework Exercises II Math 185 Homework Exercises II Instructor: Andrés E. Caicedo Due: July 10, 2002 1. Verify that if f H(Ω) C 2 (Ω) is never zero, then ln f is harmonic in Ω. 2. Let f = u+iv H(Ω) C 2 (Ω). Let p 2 be an integer.

More information

Lecture Notes Complex Analysis. Complex Variables and Applications 7th Edition Brown and Churchhill

Lecture Notes Complex Analysis. Complex Variables and Applications 7th Edition Brown and Churchhill Lecture Notes omplex Analysis based on omplex Variables and Applications 7th Edition Brown and hurchhill Yvette Fajardo-Lim, Ph.D. Department of Mathematics De La Salle University - Manila 2 ontents THE

More information

Chapter 13: Complex Numbers

Chapter 13: Complex Numbers Sections 13.3 & 13.4 1. A (single-valued) function f of a complex variable z is such that for every z in the domain of definition D of f, there is a unique complex number w such that w = f (z). The real

More information

Math Homework 2

Math Homework 2 Math 73 Homework Due: September 8, 6 Suppose that f is holomorphic in a region Ω, ie an open connected set Prove that in any of the following cases (a) R(f) is constant; (b) I(f) is constant; (c) f is

More information

MATH 452. SAMPLE 3 SOLUTIONS May 3, (10 pts) Let f(x + iy) = u(x, y) + iv(x, y) be an analytic function. Show that u(x, y) is harmonic.

MATH 452. SAMPLE 3 SOLUTIONS May 3, (10 pts) Let f(x + iy) = u(x, y) + iv(x, y) be an analytic function. Show that u(x, y) is harmonic. MATH 45 SAMPLE 3 SOLUTIONS May 3, 06. (0 pts) Let f(x + iy) = u(x, y) + iv(x, y) be an analytic function. Show that u(x, y) is harmonic. Because f is holomorphic, u and v satisfy the Cauchy-Riemann equations:

More information

Assignment 2 - Complex Analysis

Assignment 2 - Complex Analysis Assignment 2 - Complex Analysis MATH 440/508 M.P. Lamoureux Sketch of solutions. Γ z dz = Γ (x iy)(dx + idy) = (xdx + ydy) + i Γ Γ ( ydx + xdy) = (/2)(x 2 + y 2 ) endpoints + i [( / y) y ( / x)x]dxdy interiorγ

More information

University of Regina. Lecture Notes. Michael Kozdron

University of Regina. Lecture Notes. Michael Kozdron University of Regina Mathematics 32 omplex Analysis I Lecture Notes Fall 22 Michael Kozdron kozdron@stat.math.uregina.ca http://stat.math.uregina.ca/ kozdron List of Lectures Lecture #: Introduction to

More information

EE2007 Tutorial 7 Complex Numbers, Complex Functions, Limits and Continuity

EE2007 Tutorial 7 Complex Numbers, Complex Functions, Limits and Continuity EE27 Tutorial 7 omplex Numbers, omplex Functions, Limits and ontinuity Exercise 1. These are elementary exercises designed as a self-test for you to determine if you if have the necessary pre-requisite

More information

2 Complex Functions and the Cauchy-Riemann Equations

2 Complex Functions and the Cauchy-Riemann Equations 2 Complex Functions and the Cauchy-Riemann Equations 2.1 Complex functions In one-variable calculus, we study functions f(x) of a real variable x. Likewise, in complex analysis, we study functions f(z)

More information

R- and C-Differentiability

R- and C-Differentiability Lecture 2 R- and C-Differentiability Let z = x + iy = (x,y ) be a point in C and f a function defined on a neighbourhood of z (e.g., on an open disk (z,r) for some r > ) with values in C. Write f (z) =

More information

MTH 3102 Complex Variables Final Exam May 1, :30pm-5:30pm, Skurla Hall, Room 106

MTH 3102 Complex Variables Final Exam May 1, :30pm-5:30pm, Skurla Hall, Room 106 Name (Last name, First name): MTH 02 omplex Variables Final Exam May, 207 :0pm-5:0pm, Skurla Hall, Room 06 Exam Instructions: You have hour & 50 minutes to complete the exam. There are a total of problems.

More information

Complex Analysis Homework 4: Solutions

Complex Analysis Homework 4: Solutions Complex Analysis Fall 2007 Homework 4: Solutions 1.5.2. a The function fz 3z 2 +7z+5 is a polynomial so is analytic everywhere with derivative f z 6z + 7. b The function fz 2z + 3 4 is a composition of

More information

Series Solution of Linear Ordinary Differential Equations

Series Solution of Linear Ordinary Differential Equations Series Solution of Linear Ordinary Differential Equations Department of Mathematics IIT Guwahati Aim: To study methods for determining series expansions for solutions to linear ODE with variable coefficients.

More information

CHAPTER 9. Conformal Mapping and Bilinear Transformation. Dr. Pulak Sahoo

CHAPTER 9. Conformal Mapping and Bilinear Transformation. Dr. Pulak Sahoo CHAPTER 9 Conformal Mapping and Bilinear Transformation BY Dr. Pulak Sahoo Assistant Professor Department of Mathematics University of Kalyani West Bengal, India E-mail : sahoopulak1@gmail.com 1 Module-4:

More information

MA102: Multivariable Calculus

MA102: Multivariable Calculus MA102: Multivariable Calculus Rupam Barman and Shreemayee Bora Department of Mathematics IIT Guwahati Differentiability of f : U R n R m Definition: Let U R n be open. Then f : U R n R m is differentiable

More information

Math 417 Midterm Exam Solutions Friday, July 9, 2010

Math 417 Midterm Exam Solutions Friday, July 9, 2010 Math 417 Midterm Exam Solutions Friday, July 9, 010 Solve any 4 of Problems 1 6 and 1 of Problems 7 8. Write your solutions in the booklet provided. If you attempt more than 5 problems, you must clearly

More information

MTH4101 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) 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 information

MTH 3102 Complex Variables Final Exam May 1, :30pm-5:30pm, Skurla Hall, Room 106

MTH 3102 Complex Variables Final Exam May 1, :30pm-5:30pm, Skurla Hall, Room 106 Name (Last name, First name): MTH 32 Complex Variables Final Exam May, 27 3:3pm-5:3pm, Skurla Hall, Room 6 Exam Instructions: You have hour & 5 minutes to complete the exam. There are a total of problems.

More information

7.2 Conformal mappings

7.2 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 information

MA3111S COMPLEX ANALYSIS I

MA3111S 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 information

2. Complex Analytic Functions

2. Complex Analytic Functions 2. Complex Analytic Functions John Douglas Moore July 6, 2011 Recall that if A and B are sets, a function f : A B is a rule which assigns to each element a A a unique element f(a) B. In this course, we

More information

Complex Function. Chapter Complex Number. Contents

Complex Function. Chapter Complex Number. Contents Chapter 6 Complex Function Contents 6. Complex Number 3 6.2 Elementary Functions 6.3 Function of Complex Variables, Limit and Derivatives 3 6.4 Analytic Functions and Their Derivatives 8 6.5 Line Integral

More information

Complex Analysis. Chapter V. Singularities V.3. The Argument Principle Proofs of Theorems. August 8, () Complex Analysis August 8, / 7

Complex Analysis. Chapter V. Singularities V.3. The Argument Principle Proofs of Theorems. August 8, () Complex Analysis August 8, / 7 Complex Analysis Chapter V. Singularities V.3. The Argument Principle Proofs of Theorems August 8, 2017 () Complex Analysis August 8, 2017 1 / 7 Table of contents 1 Theorem V.3.4. Argument Principle 2

More information

Lecture 14 Conformal Mapping. 1 Conformality. 1.1 Preservation of angle. 1.2 Length and area. MATH-GA Complex Variables

Lecture 14 Conformal Mapping. 1 Conformality. 1.1 Preservation of angle. 1.2 Length and area. MATH-GA Complex Variables Lecture 14 Conformal Mapping MATH-GA 2451.001 Complex Variables 1 Conformality 1.1 Preservation of angle The open mapping theorem tells us that an analytic function such that f (z 0 ) 0 maps a small neighborhood

More information

MATH 31BH Homework 5 Solutions

MATH 31BH Homework 5 Solutions MATH 3BH Homework 5 Solutions February 4, 204 Problem.8.2 (a) Let x t f y = x 2 + y 2 + 2z 2 and g(t) = t 2. z t 3 Then by the chain rule a a a D(g f) b = Dg f b Df b c c c = [Dg(a 2 + b 2 + 2c 2 )] [

More information

Class test: week 10, 75 minutes. (30%) Final exam: April/May exam period, 3 hours (70%).

Class test: week 10, 75 minutes. (30%) Final exam: April/May exam period, 3 hours (70%). 17-4-2013 12:55 c M. K. Warby MA3914 Complex variable methods and applications 0 1 MA3914 Complex variable methods and applications Lecture Notes by M.K. Warby in 2012/3 Department of Mathematical Sciences

More information

EE2007: Engineering Mathematics II Complex Analysis

EE2007: Engineering Mathematics II Complex Analysis EE2007: Engineering Mathematics II omplex Analysis Ling KV School of EEE, NTU ekvling@ntu.edu.sg V4.2: Ling KV, August 6, 2006 V4.1: Ling KV, Jul 2005 EE2007 V4.0: Ling KV, Jan 2005, EE2007 V3.1: Ling

More information

Students need encouragement. So if a student gets an answer right, tell them it was a lucky guess. That way, they develop a good, lucky feeling.

Students need encouragement. So if a student gets an answer right, tell them it was a lucky guess. That way, they develop a good, lucky feeling. Chapter 8 Analytic Functions Stuents nee encouragement. So if a stuent gets an answer right, tell them it was a lucky guess. That way, they evelop a goo, lucky feeling. 1 8.1 Complex Derivatives -Jack

More information

Part IB. Further Analysis. Year

Part 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 information

MTH 362: Advanced Engineering Mathematics

MTH 362: Advanced Engineering Mathematics MTH 362: Advanced Engineering Mathematics Lecture 1 Jonathan A. Chávez Casillas 1 1 University of Rhode Island Department of Mathematics September 7, 2017 Course Name and number: MTH 362: Advanced Engineering

More information

(z 0 ) = lim. = lim. = f. Similarly along a vertical line, we fix x = x 0 and vary y. Setting z = x 0 + iy, we get. = lim. = i f

(z 0 ) = lim. = lim. = f. Similarly along a vertical line, we fix x = x 0 and vary y. Setting z = x 0 + iy, we get. = lim. = i f . Holomorphic Harmonic Functions Basic notation. Considering C as R, with coordinates x y, z = x + iy denotes the stard complex coordinate, in the usual way. Definition.1. Let f : U C be a complex valued

More information

Midterm 1 Solutions Thursday, February 26

Midterm 1 Solutions Thursday, February 26 Math 59 Dr. DeTurck Midterm 1 Solutions Thursday, February 26 1. First note that since f() = f( + ) = f()f(), we have either f() = (in which case f(x) = f(x + ) = f(x)f() = for all x, so c = ) or else

More information

Spring Abstract Rigorous development of theory of functions. Topology of plane, complex integration, singularities, conformal mapping.

Spring Abstract Rigorous development of theory of functions. Topology of plane, complex integration, singularities, conformal mapping. MTH 562 Complex Analysis Spring 2007 Abstract Rigorous development of theory of functions. Topology of plane, complex integration, singularities, conformal mapping. Complex Numbers Definition. We define

More information

Sec. 1.1: Basics of Vectors

Sec. 1.1: Basics of Vectors Sec. 1.1: Basics of Vectors Notation for Euclidean space R n : all points (x 1, x 2,..., x n ) in n-dimensional space. Examples: 1. R 1 : all points on the real number line. 2. R 2 : all points (x 1, x

More information

Complex Analysis review notes for weeks 1-6

Complex 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 information

or E ( U(X) ) e zx = e ux e ivx = e ux( cos(vx) + i sin(vx) ), B X := { u R : M X (u) < } (4)

or E ( U(X) ) e zx = e ux e ivx = e ux( cos(vx) + i sin(vx) ), B X := { u R : M X (u) < } (4) :23 /4/2000 TOPIC Characteristic functions This lecture begins our study of the characteristic function φ X (t) := Ee itx = E cos(tx)+ie sin(tx) (t R) of a real random variable X Characteristic functions

More information

Complex Analysis. Travis Dirle. December 4, 2016

Complex 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 information

Leplace s Equations. Analyzing the Analyticity of Analytic Analysis DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING. Engineering Math EECE

Leplace s Equations. Analyzing the Analyticity of Analytic Analysis DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING. Engineering Math EECE Leplace s Analyzing the Analyticity of Analytic Analysis Engineering Math EECE 3640 1 The Laplace equations are built on the Cauchy- Riemann equations. They are used in many branches of physics such as

More information

MSO202: Introduction To Complex Analysis

MSO202: Introduction To Complex Analysis 1 MSO0: Introduction To Complex Analysis Lecture z a Geometrical Interpretation of Ha { z:im( ) 0}, b 0 b z We first give the geometrical of H0 { z:im( ) 0, b 1}, i.e. b when a = 0 and b 1. In this case,

More information

EE2 Mathematics : Complex Variables

EE2 Mathematics : Complex Variables EE Mathematics : omplex Variables J. D. Gibbon (Professor J. D Gibbon 1, Dept of Mathematics) j.d.gibbon@ic.ac.uk http://www.imperial.ac.uk/ jdg These notes are not identical word-for-word with my lectures

More information

Faculteit Wiskunde en Informatica

Faculteit Wiskunde en Informatica Faculteit Wiskunde en Informatica Lecture notes for courses on Complex Analysis, Fourier Analysis and Asymptotic Analysis of Integrals S.W. Rienstra Originally based on the lecture notes for complex function

More information

1 Fundamental Concepts From Algebra & Precalculus

1 Fundamental Concepts From Algebra & Precalculus Fundamental Concepts From Algebra & Precalculus. Review Exercises.. Simplify eac expression.. 5 7) [ 5)) ]. ) 5) 7) 9 + 8 5. 8 [ 5) 8 6)] [9 + 8 5 ]. 9 + 8 5 ) 8) + 5. 5 + [ )6)] 7) 7 + 6 5 6. 8 5 ) 6

More information

Math 423/823 Exam 1 Topics Covered

Math 423/823 Exam 1 Topics Covered Math 423/823 Exam 1 Topics Covered Complex numbers: C: z = x+yi, where i 2 = 1; addition and multiplication behaves like reals. Formally, x + yi (x,y), with (x,y) + (a,b) = (x + a,y + b) and (x,y)(a,b)

More information

ENGI Gradient, Divergence, Curl Page 5.01

ENGI Gradient, Divergence, Curl Page 5.01 ENGI 94 5. - Gradient, Divergence, Curl Page 5. 5. The Gradient Operator A brief review is provided here for the gradient operator in both Cartesian and orthogonal non-cartesian coordinate systems. Sections

More information

February 11, 2019 LECTURE 5: THE RULES OF DIFFERENTIATION.

February 11, 2019 LECTURE 5: THE RULES OF DIFFERENTIATION. February 11, 2019 LECTURE 5: THE RULES OF DIFFERENTIATION 110211 HONORS MULTIVARIABLE CALCULUS PROFESSOR RICHARD BROWN Synopsis Here, we bring back the rules for differentiation used to derive new functions

More information

1. A polynomial p(x) in one variable x is an algebraic expression in x of the form

1. A polynomial p(x) in one variable x is an algebraic expression in x of the form POLYNOMIALS Important Points 1. A polynomial p(x) in one variable x is an algebraic expression in x of the form p(x) = a nx n +a n-1x n-1 + a 2x 2 +a 1x 1 +a 0x 0 where a 0, a 1, a 2 a n are constants

More information

Conformal maps. Lent 2019 COMPLEX METHODS G. Taylor. A star means optional and not necessarily harder.

Conformal 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 information

( ) y 2! 4. ( )( y! 2)

( ) y 2! 4. ( )( y! 2) 1. Dividing: 4x3! 8x 2 + 6x 2x 5.7 Division of Polynomials = 4x3 2x! 8x2 2x + 6x 2x = 2x2! 4 3. Dividing: 1x4 + 15x 3! 2x 2!5x 2 = 1x4!5x 2 + 15x3!5x 2! 2x2!5x 2 =!2x2! 3x + 4 5. Dividing: 8y5 + 1y 3!

More information

Section 21. The Metric Topology (Continued)

Section 21. The Metric Topology (Continued) 21. The Metric Topology (cont.) 1 Section 21. The Metric Topology (Continued) Note. In this section we give a number of results for metric spaces which are familar from calculus and real analysis. We also

More information

Chapter 9. Analytic Continuation. 9.1 Analytic Continuation. For every complex problem, there is a solution that is simple, neat, and wrong.

Chapter 9. Analytic Continuation. 9.1 Analytic Continuation. For every complex problem, there is a solution that is simple, neat, and wrong. Chapter 9 Analytic Continuation For every complex problem, there is a solution that is simple, neat, and wrong. - H. L. Mencken 9.1 Analytic Continuation Suppose there is a function, f 1 (z) that is analytic

More information

Integration in the Complex Plane (Zill & Wright Chapter 18)

Integration in the Complex Plane (Zill & Wright Chapter 18) Integration in the omplex Plane Zill & Wright hapter 18) 116-4-: omplex Variables Fall 11 ontents 1 ontour Integrals 1.1 Definition and Properties............................. 1. Evaluation.....................................

More information

Complex Analysis Slide 9: Power Series

Complex Analysis Slide 9: Power Series Complex Analysis Slide 9: Power Series MA201 Mathematics III Department of Mathematics IIT Guwahati August 2015 Complex Analysis Slide 9: Power Series 1 / 37 Learning Outcome of this Lecture We learn Sequence

More information

MATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5

MATH 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 information

Math 411, Complex Analysis Definitions, Formulas and Theorems Winter y = sinα

Math 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 information

MATH 19520/51 Class 5

MATH 19520/51 Class 5 MATH 19520/51 Class 5 Minh-Tam Trinh University of Chicago 2017-10-04 1 Definition of partial derivatives. 2 Geometry of partial derivatives. 3 Higher derivatives. 4 Definition of a partial differential

More information

1 Holomorphic functions

1 Holomorphic functions Robert Oeckl CA NOTES 1 15/09/2009 1 1 Holomorphic functions 11 The complex derivative The basic objects of complex analysis are the holomorphic functions These are functions that posses a complex derivative

More information

Complex Analysis MATH 6300 Fall 2013 Homework 4

Complex Analysis MATH 6300 Fall 2013 Homework 4 Complex Analysis MATH 6300 Fall 2013 Homework 4 Due Wednesday, December 11 at 5 PM Note that to get full credit on any problem in this class, you must solve the problems in an efficient and elegant manner,

More information

Homework 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. 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 information

3 Applications of partial differentiation

3 Applications of partial differentiation Advanced Calculus Chapter 3 Applications of partial differentiation 37 3 Applications of partial differentiation 3.1 Stationary points Higher derivatives Let U R 2 and f : U R. The partial derivatives

More information

= 2 x y 2. (1)

= 2 x y 2. (1) COMPLEX ANALYSIS PART 5: HARMONIC FUNCTIONS A Let me start by asking you a question. Suppose that f is an analytic function so that the CR-equation f/ z = 0 is satisfied. Let us write u and v for the real

More information

A DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any

A DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any Y Y Y X X «/ YY Y Y ««Y x ) & \ & & } # Y \#$& / Y Y X» \\ / X X X x & Y Y X «q «z \x» = q Y # % \ & [ & Z \ & { + % ) / / «q zy» / & / / / & x x X / % % ) Y x X Y $ Z % Y Y x x } / % «] «] # z» & Y X»

More information

Part IB Complex Analysis

Part 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 information

Selected Solutions To Problems in Complex Analysis

Selected 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 information

Complex Homework Summer 2014

Complex Homework Summer 2014 omplex Homework Summer 24 Based on Brown hurchill 7th Edition June 2, 24 ontents hw, omplex Arithmetic, onjugates, Polar Form 2 2 hw2 nth roots, Domains, Functions 2 3 hw3 Images, Transformations 3 4 hw4

More information

Two Posts to Fill On School Board

Two Posts to Fill On School Board Y Y 9 86 4 4 qz 86 x : ( ) z 7 854 Y x 4 z z x x 4 87 88 Y 5 x q x 8 Y 8 x x : 6 ; : 5 x ; 4 ( z ; ( ) ) x ; z 94 ; x 3 3 3 5 94 ; ; ; ; 3 x : 5 89 q ; ; x ; x ; ; x : ; ; ; ; ; ; 87 47% : () : / : 83

More information

Polynomials. In many problems, it is useful to write polynomials as products. For example, when solving equations: Example:

Polynomials. In many problems, it is useful to write polynomials as products. For example, when solving equations: Example: Polynomials Monomials: 10, 5x, 3x 2, x 3, 4x 2 y 6, or 5xyz 2. A monomial is a product of quantities some of which are unknown. Polynomials: 10 + 5x 3x 2 + x 3, or 4x 2 y 6 + 5xyz 2. A polynomial is a

More information

Lecture notes: Introduction to Partial Differential Equations

Lecture notes: Introduction to Partial Differential Equations Lecture notes: Introduction to Partial Differential Equations Sergei V. Shabanov Department of Mathematics, University of Florida, Gainesville, FL 32611 USA CHAPTER 1 Classification of Partial Differential

More information

D(f/g)(P ) = D(f)(P )g(p ) f(p )D(g)(P ). g 2 (P )

D(f/g)(P ) = D(f)(P )g(p ) f(p )D(g)(P ). g 2 (P ) We first record a very useful: 11. Higher derivatives Theorem 11.1. Let A R n be an open subset. Let f : A R m and g : A R m be two functions and suppose that P A. Let λ A be a scalar. If f and g are differentiable

More information

Partial Derivatives October 2013

Partial Derivatives October 2013 Partial Derivatives 14.3 02 October 2013 Derivative in one variable. Recall for a function of one variable, f (a) = lim h 0 f (a + h) f (a) h slope f (a + h) f (a) h a a + h Partial derivatives. For a

More information

OR MSc Maths Revision Course

OR MSc Maths Revision Course OR MSc Maths Revision Course Tom Byrne School of Mathematics University of Edinburgh t.m.byrne@sms.ed.ac.uk 15 September 2017 General Information Today JCMB Lecture Theatre A, 09:30-12:30 Mathematics revision

More information

Another consequence of the Cauchy Schwarz inequality is the continuity of the inner product.

Another consequence of the Cauchy Schwarz inequality is the continuity of the inner product. . Inner product spaces 1 Theorem.1 (Cauchy Schwarz inequality). If X is an inner product space then x,y x y. (.) Proof. First note that 0 u v v u = u v u v Re u,v. (.3) Therefore, Re u,v u v (.) for all

More information

Functions of a Complex Variable (Zill & Wright Chapter 17)

Functions of a Complex Variable (Zill & Wright Chapter 17) Functions of a Complex Variable (Zill & Wright Chapter 17) 1016-40-0: Complex Variables Winter 01-013 Contents 0 Administrata 0.1 Outline........................................... 3 1 Complex Numbers

More information

MATH MIDTERM 1 SOLUTION. 1. (5 points) Determine whether the following statements are true of false, no justification is required.

MATH MIDTERM 1 SOLUTION. 1. (5 points) Determine whether the following statements are true of false, no justification is required. MATH 185-4 MIDTERM 1 SOLUTION 1. (5 points Determine whether the following statements are true of false, no justification is required. (1 (1pointTheprincipalbranchoflogarithmfunctionf(z = Logz iscontinuous

More information

Economics 204 Fall 2013 Problem Set 5 Suggested Solutions

Economics 204 Fall 2013 Problem Set 5 Suggested Solutions Economics 204 Fall 2013 Problem Set 5 Suggested Solutions 1. Let A and B be n n matrices such that A 2 = A and B 2 = B. Suppose that A and B have the same rank. Prove that A and B are similar. Solution.

More information

Part IB. Complex Analysis. Year

Part 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 information

Example: Limit definition. Geometric meaning. Geometric meaning, y. Notes. Notes. Notes. f (x, y) = x 2 y 3 :

Example: Limit definition. Geometric meaning. Geometric meaning, y. Notes. Notes. Notes. f (x, y) = x 2 y 3 : Partial Derivatives 14.3 02 October 2013 Derivative in one variable. Recall for a function of one variable, f (a) = lim h 0 f (a + h) f (a) h slope f (a + h) f (a) h a a + h Partial derivatives. For a

More information

Quasi-conformal maps and Beltrami equation

Quasi-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 information

Derivatives in 2D. Outline. James K. Peterson. November 9, Derivatives in 2D! Chain Rule

Derivatives in 2D. Outline. James K. Peterson. November 9, Derivatives in 2D! Chain Rule Derivatives in 2D James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 9, 2016 Outline Derivatives in 2D! Chain Rule Let s go back to

More information

RESEARCH STATEMENT. Introduction

RESEARCH STATEMENT. Introduction RESEARCH STATEMENT PRITHA CHAKRABORTY Introduction My primary research interests lie in complex analysis (in one variable), especially in complex-valued analytic function spaces and their applications

More information

Solution/Correction standard, second Test Mathematics A + B1; November 7, 2014.

Solution/Correction standard, second Test Mathematics A + B1; November 7, 2014. Solution/Correction standard, second Test Mathematics A + B1; November 7, 014. Kenmerk : Leibniz/toetsen/Re-Exam-Math-A-B1-141-Solutions Course : Mathematics A + B1 (Leibniz) Vakcode : 1911010 Date : November

More information

MATH SPRING UC BERKELEY

MATH 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 information

Complex Analysis Qualifying Exam Solutions

Complex 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 information

Math 312 Fall 2013 Final Exam Solutions (2 + i)(i + 1) = (i 1)(i + 1) = 2i i2 + i. i 2 1

Math 312 Fall 2013 Final Exam Solutions (2 + i)(i + 1) = (i 1)(i + 1) = 2i i2 + i. i 2 1 . (a) We have 2 + i i Math 32 Fall 203 Final Exam Solutions (2 + i)(i + ) (i )(i + ) 2i + 2 + i2 + i i 2 3i + 2 2 3 2 i.. (b) Note that + i 2e iπ/4 so that Arg( + i) π/4. This implies 2 log 2 + π 4 i..

More information

Math 594. Solutions 5

Math 594. Solutions 5 Math 594. Solutions 5 Book problems 6.1: 7. Prove that subgroups and quotient groups of nilpotent groups are nilpotent (your proof should work for infinite groups). Give an example of a group G which possesses

More information

1 Complex Numbers. 1.1 Sums and Products

1 Complex Numbers. 1.1 Sums and Products 1 Complex Numbers 1.1 Sums Products Definition: The complex plane, denoted C is the set of all ordered pairs (x, y) with x, y R, where Re z = x is called the real part Imz = y is called the imaginary part.

More information