Math Spring 2014 Solutions to Assignment # 8 Completion Date: Friday May 30, 2014


 Peregrine Long
 1 years ago
 Views:
Transcription
1 Math 3  Spring 4 Solutions to Assignment # 8 ompletion Date: Friday May 3, 4 Question. [p 49, #] By finding an antiderivative, evaluate each of these integrals, where the path is any contour between the indicated limits of integration: (a) i/ i e πz dz; (b) π+i ( z cos dz; ) (c) 3 (z ) 3 dz. Ans: (a) ( + i)/π; (b) e + (/e) ; (c). Solution: (a) Since d dz ( ) π eπz = e πz for all z, then F (z) = π eπz is an antiderivative of f(z) = e πz, i/ i e πz dz = i/ π eπz = π i ( e iπ/ e iπ) = ( + i). π (b) Since d ( sin (z/)) = cos (z/) for all z, then the function F (z) = sin (z/) is an antiderivative dz of f(z) = cos (z/), and π+i cos (z/) dz = sin (z/) π+i = [sin (π/ + i) sin()] = sin (π/ + i) = cos(i) = e + e. (c) Since d ( ) (z )4 = (z ) 3 for all z, then the function F (z) = dz 4 4 (z )4 is an antiderivative of f(z) = (z ) 3, and 3 (z ) 3 dz = 4 (z )4 3 = 4 [ (3 ) 4 ( ) 4] =. Question. [p 49, #5] Show that where z i denotes the principal branch z i dz = + e ( i), z i = exp(i Log z) ( z >, < Arg z < π) and where the path of integration is any contour from z = to z = that, except for its end points, lies above the real axis. (ompare with Exercise 7, Sec. 4.)
2 Suggestion: Use an antiderivative of the branch of the same power function. z i = exp(i log z) ( z >, π < arg z < 3π ) Solution: Let f(z) = z i, z >, < Arg z < π, y x then f(z) = z i = exp(i Log z) is not defined at z =, but if we consider the branch f (z) = z i = exp(i log z), where log z = ln z + iθ, π < θ < 3π, then f (z) is defined and analytic at each point of and its values coincide with the values of f(z) except at z =. y x An antiderivative of f (z) is given by F (z) = + i zi+, Therefore, and F (z) = + i exp((i + ) log z), z >, π < θ < 3π. z i dz = z i dz = + i f (z) dz = F (z) dz = F () F ( ), [ ( ) i+ ] = [ e ((+i)[ln +iπ])], + i z i dz = [ e (+i)πi] = ( e iπ e ) = i ( + e ). + i + i Question 3. [p 6, # (a)] Apply the auchygoursat theorem to show that either direction, and when f(z) = z z 3. f(z) dz = when the contour is the circle z =, in
3 Solution: Since f(z) = z is analytic inside and on the contour z =, then z 3 z z 3 dz = if the contour is traversed in either direction. z = Question 4. [p 6, # (c)] Apply the auchygoursat theorem to show that either direction, and when f(z) = Solution: Since f(z) = then z + z +. f(z) dz = when the contour is the circle z =, in z + z + = is analytic inside and on the contour z =, (z + i)(z + + i) z + z + dz = z = if the contour is traversed in either direction. Question 5. [p 6, # (f)] Apply the auchygoursat theorem to show that either direction, and when f(z) = Log(z + ). f(z) dz = when the contour is the circle z =, in Solution: Since the branch cut for f(z) = Log(z + ) extends from the point z = along the negative real axis, then f(z) is analytic inside and on the contour z =, Log(z + ) dz = if the contour is traversed in either direction. z = Question 6. [p 7, # (a)] Let denote the positively oriented boundary of the square whose sides lie along the lines x = ± and y = ±. Evaluate the integral e z z (πi/) dz. Ans: π. Solution: Let f(z) = e z, then f is analytic inside and on, and since z = πi auchy s integral formula e z πi z (πi/) dz = f (πi/) = ei/ = i, e z dz = πi( i) = π. z (πi/) is interior to, then by
4 Question 7. [p 7, # (b)] Let denote the positively oriented boundary of the square whose sides lie along the lines x = ± and y = ±. Evaluate the integral cos z z(z + 8) dz. Ans: πi/4. Solution: Let f(z) = cos z z + 8, then f is analytic inside and on, and since z = is interior to, then by auchy s integral formula πi cos z cos() z(z dz = f() = = + 8) 8 8, cos z πi z(z dz = + 8) 4. Question 8. [p 7, # (e)] Let denote the positively oriented boundary of the square whose sides lie along the lines x = ± and y = ±. Evaluate the integral tan(z/) (z x ) dz ( < x < ). Ans: iπ sec (x /). Solution: Let f(z) ( = tan(z/), then the singularities of f(z) are the zeros of cos(z/) and these occur at the points z = n + ) π, n =, ±, ±,..., all of which are outside the square. Therefore, f(z) is analytic inside and on, and since x is interior to, from auchy s integral formula we have f(x ) = πi tan(z/) (z x ) dz, and f (x ) =! πi tan(z/) (z x ) dz. Now, f (x ) = sec (x /), tan(z/) πi (z x ) dz = sec (x /), tan(z/) (z x ) dz = iπ sec (x /). Question 9. [p 7, #] Find the value of the integral of g(z) around the circle z i = in the positive sense when (a) g(z) = z + 4 ; (b) g(z) = (z + 4). Ans: (a) π/; (b) π/6.
5 Solution: (a) Let g(z) = z and f(z) = + 4 z + i, then f is analytic inside and on, and since z = i is interior to z i =, then from auchy s integral formula we have πi z i = z + 4 dz = πi z i = f(z) z i dz = f(i) = 4i, z i = z + 4 dz = π. (b) Let g(z) = (z + 4), and let f(z) = (z + i), then f is analytic inside and on, and since z = i is interior to z i =, then from auchy s integral formula we have! g(z) dz = f(z) πi z i = πi z i = (z i) dz = f (i) = (i + i) 3 = 64( i), z i = πi (z dz = + 4) 3i = π 6. Question. [p 7, #3] Let be the circle z = 3, described in the positive sense. Show that if g(w) = z z z w then g() = 8πi. What is the value of g(w) when w > 3? dz ( w 3), Solution: Let f(z) = z z, then f is analytic inside and on, and from the auchy integral formula we have z z g() = dz = πif() = πi(8 ) = 8πi. z z =3 If w > 3, then h(z) = z z z w for w > 3. is analytic inside and on, and from the auchygourat theorem, z =3 z z z w dz = Question. [p 7, #7] Let be the unit circle z = e iθ ( θ π). First show that, for any real constant a, e az dz = πi. z Then write this integral in terms of θ to derive the integration formula π e a cos θ cos(a sin θ) dθ = π.
6 Solution: Let f(z) = e az, then f is analytic inside and on, and from the auchy integral formula, we have e az πi z dz = e =, e az z Now, on, we have z = e iθ and dz = ie iθ dθ, and therefore dz = πi. e az π a(cos θ+i sin θ) z dz = e π e iθ i e iθ dθ = i e a cos θ e ai sin θ dθ, πi = π e az z Equating real and imaginary parts, we have π dz = i e a cos θ [cos(a sin θ) + i sin(a sin θ)] dθ, π e a cos θ cos(a sin θ) dθ + i e a cos θ sin(a sin θ) dθ = π. π e a cos θ cos(a sin θ) dθ = π and π e a cos θ sin(a sin θ) dθ =, and since the function h(θ) = e a cos θ cos(a sin θ) is even, then π e a cos θ cos(a sin θ) dθ = π.
(1) Let f(z) be the principal branch of z 4i. (a) Find f(i). Solution. f(i) = exp(4i Log(i)) = exp(4i(π/2)) = e 2π. (b) Show that
Let fz be the principal branch of z 4i. a Find fi. Solution. fi = exp4i Logi = exp4iπ/2 = e 2π. b Show that fz fz 2 fz z 2 fz fz 2 = λfz z 2 for all z, z 2 0, where λ =, e 8π or e 8π. Proof. We have =
More informationMath Spring 2014 Solutions to Assignment # 12 Completion Date: Thursday June 12, 2014
Math 3  Spring 4 Solutions to Assignment # Completion Date: Thursday June, 4 Question. [p 67, #] Use residues to evaluate the improper integral x + ). Ans: π/4. Solution: Let fz) = below. + z ), and for
More informationSolution for Final Review Problems 1
Solution for Final Review Problems Final time and location: Dec. Gymnasium, Rows 23, 25 5, 2, Wednesday, 92am, Main ) Let fz) be the principal branch of z i. a) Find f + i). b) Show that fz )fz 2 ) λfz
More informationResidues and Contour Integration Problems
Residues and ontour Integration Problems lassify the singularity of fz at the indicated point.. fz = cotz at z =. Ans. Simple pole. Solution. The test for a simple pole at z = is that lim z z cotz exists
More information1 Discussion on multivalued functions
Week 3 notes, Math 7651 1 Discussion on multivalued functions Log function : Note that if z is written in its polar representation: z = r e iθ, where r = z and θ = arg z, then log z log r + i θ + 2inπ
More informationComplex Variables...Review Problems (Residue Calculus Comments)...Fall Initial Draft
Complex Variables........Review Problems Residue Calculus Comments)........Fall 22 Initial Draft ) Show that the singular point of fz) is a pole; determine its order m and its residue B: a) e 2z )/z 4,
More informationMATH 106 HOMEWORK 4 SOLUTIONS. sin(2z) = 2 sin z cos z. (e zi + e zi ) 2. = 2 (ezi e zi )
MATH 16 HOMEWORK 4 SOLUTIONS 1 Show directly from the definition that sin(z) = ezi e zi i sin(z) = sin z cos z = (ezi e zi ) i (e zi + e zi ) = sin z cos z Write the following complex numbers in standard
More information18.04 Practice problems exam 1, Spring 2018 Solutions
8.4 Practice problems exam, Spring 8 Solutions Problem. omplex arithmetic (a) Find the real and imaginary part of z + z. (b) Solve z 4 i =. (c) Find all possible values of i. (d) Express cos(4x) in terms
More informationMTH 3102 Complex Variables Solutions: Practice Exam 2 Mar. 26, 2017
Name Last name, First name): MTH 31 omplex Variables Solutions: Practice Exam Mar. 6, 17 Exam Instructions: You have 1 hour & 1 minutes to complete the exam. There are a total of 7 problems. You must show
More information= 2πi Res. z=0 z (1 z) z 5. z=0. = 2πi 4 5z
MTH30 Spring 07 HW Assignment 7: From [B4]: hap. 6: Sec. 77, #3, 7; Sec. 79, #, (a); Sec. 8, #, 3, 5, Sec. 83, #5,,. The due date for this assignment is 04/5/7. Sec. 77, #3. In the example in Sec. 76,
More informationMATH 417 Homework 4 Instructor: D. Cabrera Due July 7. z c = e c log z (1 i) i = e i log(1 i) i log(1 i) = 4 + 2kπ + i ln ) cosz = eiz + e iz
MATH 47 Homework 4 Instructor: D. abrera Due July 7. Find all values of each expression below. a) i) i b) cos i) c) sin ) Solution: a) Here we use the formula z c = e c log z i) i = e i log i) The modulus
More informationSecond Midterm Exam Name: Practice Problems March 10, 2015
Math 160 1. Treibergs Second Midterm Exam Name: Practice Problems March 10, 015 1. Determine the singular points of the function and state why the function is analytic everywhere else: z 1 fz) = z + 1)z
More informationComplex 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 informationSyllabus: for Complex variables
EE2020, Spring 2009 p. 1/42 Syllabus: for omplex variables 1. Midterm, (4/27). 2. Introduction to Numerical PDE (4/30): [Ref.num]. 3. omplex variables: [Textbook]h.13h.18. omplex numbers and functions,
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 information2.5 (x + iy)(a + ib) = xa yb + i(xb + ya) = (az by) + i(bx + ay) = (a + ib)(x + iy). The middle = uses commutativity of real numbers.
Complex Analysis Sketches of Solutions to Selected Exercises Homework 2..a ( 2 i) i( 2i) = 2 i i + i 2 2 = 2 i i 2 = 2i 2..b (2, 3)( 2, ) = (2( 2) ( 3), 2() + ( 3)( 2)) = (, 8) 2.2.a Re(iz) = Re(i(x +
More informationSuggested Homework Solutions
Suggested Homework Solutions Chapter Fourteen Section #9: Real and Imaginary parts of /z: z = x + iy = x + iy x iy ( ) x iy = x #9: Real and Imaginary parts of ln z: + i ( y ) ln z = ln(re iθ ) = ln r
More informationSolutions to practice problems for the final
Solutions to practice problems for the final Holomorphicity, CauchyRiemann equations, and CauchyGoursat theorem 1. (a) Show that there is a holomorphic function on Ω = {z z > 2} whose derivative is z
More informationNPTEL 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 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 informationMath Spring 2014 Solutions to Assignment # 6 Completion Date: Friday May 23, 2014
Math 11  Spring 014 Solutions to Assignment # 6 Completion Date: Friday May, 014 Question 1. [p 109, #9] With the aid of expressions 15) 16) in Sec. 4 for sin z cos z, namely, sin z = sin x + sinh y cos
More informationNPTEL 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 informationMath 120 A Midterm 2 Solutions
Math 2 A Midterm 2 Solutions Jim Agler. Find all solutions to the equations tan z = and tan z = i. Solution. Let α be a complex number. Since the equation tan z = α becomes tan z = sin z eiz e iz cos z
More informationEE2007 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 selftest for you to determine if you if have the necessary prerequisite
More informationMTH 3102 Complex Variables Final Exam May 1, :30pm5:30pm, Skurla Hall, Room 106
Name (Last name, First name): MTH 02 omplex Variables Final Exam May, 207 :0pm5:0pm, Skurla Hall, Room 06 Exam Instructions: You have hour & 50 minutes to complete the exam. There are a total of problems.
More informationCHAPTER 3 ELEMENTARY FUNCTIONS 28. THE EXPONENTIAL FUNCTION. Definition: The exponential function: The exponential function e z by writing
CHAPTER 3 ELEMENTARY FUNCTIONS We consider here various elementary functions studied in calculus and define corresponding functions of a complex variable. To be specific, we define analytic functions of
More informationChapter 11. Cauchy s Integral Formula
hapter 11 auchy s Integral Formula If I were founding a university I would begin with a smoking room; next a dormitory; and then a decent reading room and a library. After that, if I still had more money
More informationMath 421 Midterm 2 review questions
Math 42 Midterm 2 review questions Paul Hacking November 7, 205 () Let U be an open set and f : U a continuous function. Let be a smooth curve contained in U, with endpoints α and β, oriented from α to
More informationMath 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 informationMath 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 informationMA 201 Complex Analysis Lecture 6: Elementary functions
MA 201 Complex Analysis : The Exponential Function Recall: Euler s Formula: For y R, e iy = cos y + i sin y and for any x, y R, e x+y = e x e y. Definition: If z = x + iy, then e z or exp(z) is defined
More informationMA 412 Complex Analysis Final Exam
MA 4 Complex Analysis Final Exam Summer II Session, August 9, 00.. Find all the values of ( 8i) /3. Sketch the solutions. Answer: We start by writing 8i in polar form and then we ll compute the cubic root:
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 informationMA424, S13 HW #6: Homework Problems 1. Answer the following, showing all work clearly and neatly. ONLY EXACT VALUES WILL BE ACCEPTED.
MA424, S13 HW #6: 4447 Homework Problems 1 Answer the following, showing all work clearly and neatly. ONLY EXACT VALUES WILL BE ACCEPTED. NOTATION: Recall that C r (z) is the positively oriented circle
More informationFINAL EXAM { SOLUTION
United Arab Emirates University ollege of Sciences Department of Mathematical Sciences FINAL EXAM { SOLUTION omplex Analysis I MATH 5 SETION 0 RN 56 9:0 { 0:45 on Monday & Wednesday Date: Wednesday, January
More informationONLINE EXAMINATIONS [Mid 2  M3] 1. the Machaurin's series for log (1+z)=
http://www.prsolutions.in ONLINE EXAMINATIONS [Mid 2  M3] 1. the Machaurin's series for log (1+z)= 2. 3. Expand cosz into a Taylor's series about the point z= cosz= cosz= cosz= cosz= 4. Expand the function
More information18.04 Practice problems exam 2, Spring 2018 Solutions
8.04 Practice problems exam, Spring 08 Solutions Problem. Harmonic functions (a) Show u(x, y) = x 3 3xy + 3x 3y is harmonic and find a harmonic conjugate. It s easy to compute: u x = 3x 3y + 6x, u xx =
More information6. Residue calculus. where C is any simple closed contour around z 0 and inside N ε.
6. Residue calculus Let z 0 be an isolated singularity of f(z), then there exists a certain deleted neighborhood N ε = {z : 0 < z z 0 < ε} such that f is analytic everywhere inside N ε. We define Res(f,
More informationExercises involving elementary functions
017:11:0:16:4:09 c M. K. Warby MA3614 Complex variable methods and applications 1 Exercises involving elementary functions 1. This question was in the class test in 016/7 and was worth 5 marks. a) Let
More information1. DO NOT LIFT THIS COVER PAGE UNTIL INSTRUCTED TO DO SO. Write your student number and name at the top of this page. This test has SIX pages.
Student Number Name (Printed in INK Mathematics 54 July th, 007 SIMON FRASER UNIVERSITY Department of Mathematics Faculty of Science Midterm Instructor: S. Pimentel 1. DO NOT LIFT THIS COVER PAGE UNTIL
More informationComplex varibles:contour integration examples
omple varibles:ontour integration eamples 1 Problem 1 onsider the problem d 2 + 1 If we take the substitution = tan θ then d = sec 2 θdθ, which leads to dθ = π sec 2 θ tan 2 θ + 1 dθ Net we consider the
More informationu = 0; thus v = 0 (and v = 0). Consequently,
MAT40  MANDATORY ASSIGNMENT #, FALL 00; FASIT REMINDER: The assignment must be handed in before 4:30 on Thursday October 8 at the Department of Mathematics, in the 7th floor of Niels Henrik Abels hus,
More informationAPPM 4360/5360 Homework Assignment #6 Solutions Spring 2018
APPM 436/536 Homework Assignment #6 Solutions Spring 8 Problem # ( points: onsider f (zlog z ; z r e iθ. Discuss/explain the analytic continuation of the function from R R R3 where r > and θ is in the
More informationThe Calculus of Residues
hapter 7 The alculus of Residues If fz) has a pole of order m at z = z, it can be written as Eq. 6.7), or fz) = φz) = a z z ) + a z z ) +... + a m z z ) m, 7.) where φz) is analytic in the neighborhood
More informationMan will occasionally stumble over the truth, but most of the time he will pick himself up and continue on.
hapter 3 The Residue Theorem Man will occasionally stumble over the truth, but most of the time he will pick himself up and continue on.  Winston hurchill 3. The Residue Theorem We will find that many
More information1 z n = 1. 9.(Problem) Evaluate each of the following, that is, express each in standard Cartesian form x + iy. (2 i) 3. ( 1 + i. 2 i.
. 5(b). (Problem) Show that z n = z n and z n = z n for n =,,... (b) Use polar form, i.e. let z = re iθ, then z n = r n = z n. Note e iθ = cos θ + i sin θ =. 9.(Problem) Evaluate each of the following,
More informationComplex 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 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 informationConsidering our result for the sum and product of analytic functions, this means that for (a 0, a 1,..., a N ) C N+1, the polynomial.
Lecture 3 Usual complex functions MATHGA 245.00 Complex Variables Polynomials. Construction f : z z is analytic on all of C since its real and imaginary parts satisfy the CauchyRiemann relations and
More informationEvaluation of integrals
Evaluation of certain contour integrals: Type I Type I: Integrals of the form 2π F (cos θ, sin θ) dθ If we take z = e iθ, then cos θ = 1 (z + 1 ), sin θ = 1 (z 1 dz ) and dθ = 2 z 2i z iz. Substituting
More informationMTH3101 Spring 2017 HW Assignment 4: Sec. 26: #6,7; Sec. 33: #5,7; Sec. 38: #8; Sec. 40: #2 The due date for this assignment is 2/23/17.
MTH0 Spring 07 HW Assignment : Sec. 6: #6,7; Sec. : #5,7; Sec. 8: #8; Sec. 0: # The due date for this assignment is //7. Sec. 6: #6. Use results in Sec. to verify that the function g z = ln r + iθ r >
More informationMATH243 First Semester 2013/14. Exercises 1
Complex Functions Dr Anna Pratoussevitch MATH43 First Semester 013/14 Exercises 1 Submit your solutions to questions marked with [HW] in the lecture on Monday 30/09/013 Questions or parts of questions
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 informationlim when the limit on the right exists, the improper integral is said to converge to that limit.
hapter 7 Applications of esidues  evaluation of definite and improper integrals occurring in real analysis and applied math  finding inverse Laplace transform by the methods of summing residues. 6. Evaluation
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 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 informationMATH 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 CauchyRiemann equations:
More informationLecture 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 FajardoLim, Ph.D. Department of Mathematics De La Salle University  Manila 2 ontents THE
More informationCHAPTER 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 Email : sahoopulak@gmail.com Module3:
More information1 Sum, Product, Modulus, Conjugate,
Sum, Product, Modulus, onjugate, Definition.. Given (, y) R 2, a comple number z is an epression of the form z = + iy. (.) Given a comple number of the form z = + iy we define Re z =, the real part of
More informationDefinite integrals. We shall study line integrals of f (z). In order to do this we shall need some preliminary definitions.
5. OMPLEX INTEGRATION (A) Definite integrals Integrals are extremely important in the study of functions of a complex variable. The theory is elegant, and the proofs generally simple. The theory is put
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 informationSolutions for Math 411 Assignment #10 1
Solutions for Math 4 Assignment # AA. Compute the following integrals: a) + sin θ dθ cos x b) + x dx 4 Solution of a). Let z = e iθ. By the substitution = z + z ), sin θ = i z z ) and dθ = iz dz and Residue
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 informationComplex Numbers. z = x+yi
Complex Numbers The field of complex numbers is the extension C R consisting of all expressions z = x+yi where x, y R and i = 1 We refer to x = Re(z) and y = Im(z) as the real part and the imaginary part
More informationChapter 3 Elementary Functions
Chapter 3 Elementary Functions In this chapter, we will consier elementary functions of a complex variable. We will introuce complex exponential, trigonometric, hyperbolic, an logarithmic functions. 23.
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 informationChapter II. Complex Variables
hapter II. omplex Variables Dates: October 2, 4, 7, 2002. These three lectures will cover the following sections of the text book by Keener. 6.1. omplex valued functions and branch cuts; 6.2.1. Differentiation
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 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 informationSynopsis of Complex Analysis. Ryan D. Reece
Synopsis of Complex Analysis Ryan D. Reece December 7, 2006 Chapter Complex Numbers. The Parts of a Complex Number A complex number, z, is an ordered pair of real numbers similar to the points in the real
More information3 Elementary Functions
3 Elementary Functions 3.1 The Exponential Function For z = x + iy we have where Euler s formula gives The note: e z = e x e iy iy = cos y + i sin y When y = 0 we have e x the usual exponential. When z
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 TI89 calculator says, reasonably enough, that x 1) 1/3 1 ) 3 = 8. lim
More informationMAT665:ANALYTIC FUNCTION THEORY
MAT665:ANALYTIC FUNCTION THEORY DR. RITU AGARWAL MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY JAIPUR Contents 1. About 2 2. Complex Numbers 2 3. Fundamental inequalities 2 4. Continuously differentiable functions
More informationPhysics 2400 Midterm I Sample March 2017
Physics 4 Midterm I Sample March 17 Question: 1 3 4 5 Total Points: 1 1 1 1 6 Gamma function. Leibniz s rule. 1. (1 points) Find positive x that minimizes the value of the following integral I(x) = x+1
More informationChapter 30 MSMYP1 Further Complex Variable Theory
Chapter 30 MSMYP Further Complex Variable Theory (30.) Multifunctions A multifunction is a function that may take many values at the same point. Clearly such functions are problematic for an analytic study,
More informationn } is convergent, lim n in+1
hapter 3 Series y residuos redit: This notes are 00% from chapter 6 of the book entitled A First ourse in omplex Analysis with Applications of Dennis G. Zill and Patrick D. Shanahan (2003) [2]. auchy s
More informationPSI Lectures on Complex Analysis
PSI Lectures on Complex Analysis Tibra Ali August 14, 14 Lecture 4 1 Evaluating integrals using the residue theorem ecall the residue theorem. If f (z) has singularities at z 1, z,..., z k which are enclosed
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 informationMath 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 informationComplex Variables. Cathal Ormond
Complex Variables Cathal Ormond Contents 1 Introduction 3 1.1 Definition: Polar Form.............................. 3 1.2 Definition: Length................................ 3 1.3 Definitions.....................................
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 informationEE2007: 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 informationMAT389 Fall 2016, Problem Set 11
MAT389 Fall 216, Problem Set 11 Improper integrals 11.1 In each of the following cases, establish the convergence of the given integral and calculate its value. i) x 2 x 2 + 1) 2 ii) x x 2 + 1)x 2 + 2x
More informationChapter 6: Residue Theory. Introduction. The Residue Theorem. 6.1 The Residue Theorem. 6.2 Trigonometric Integrals Over (0, 2π) Li, Yongzhao
Outline Chapter 6: Residue Theory Li, Yongzhao State Key Laboratory of Integrated Services Networks, Xidian University June 7, 2009 Introduction The Residue Theorem In the previous chapters, we have seen
More informationA REVIEW OF RESIDUES AND INTEGRATION A PROCEDURAL APPROACH
A REVIEW OF RESIDUES AND INTEGRATION A PROEDURAL APPROAH ANDREW ARHIBALD 1. Introduction When working with complex functions, it is best to understand exactly how they work. Of course, complex functions
More informationCHAPTER 4. Elementary Functions. Dr. Pulak Sahoo
CHAPTER 4 Elementary Functions BY Dr. Pulak Sahoo Assistant Professor Department of Mathematics University Of Kalyani West Bengal, India Email : sahoopulak1@gmail.com 1 Module4: Multivalued FunctionsII
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 informationComplex Series (3A) Young Won Lim 8/17/13
Complex Series (3A) 8/7/3 Copyright (c) 202, 203 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or
More informationSOLUTIONS MANUAL FOR. Advanced Engineering Mathematics with MATLAB Third Edition. Dean G. Duffy
SOLUTIONS MANUAL FOR Advanced Engineering Mathematics with MATLAB Third Edition by Dean G. Duffy SOLUTIONS MANUAL FOR Advanced Engineering Mathematics with MATLAB Third Edition by Dean G. Duffy Taylor
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 informationSOLUTION SET IV FOR FALL z 2 1
SOLUTION SET IV FOR 8.75 FALL 4.. Residues... Functions of a Complex Variable In the following, I use the notation Res zz f(z) Res(z ) Res[f(z), z ], where Res is the residue of f(z) at (the isolated singularity)
More informationSolutions to Exercises 1.1
Section 1.1 Complex Numbers 1 Solutions to Exercises 1.1 1. We have So a 0 and b 1. 5. We have So a 3 and b 4. 9. We have i 0+ 1i. i +i because i i +i 1 {}}{ 4+4i + i 3+4i. 1 + i 3 7 i 1 3 3 + i 14 1 1
More informationFunctions 45. Integrals, and Contours 55
MATH 43 COMPLEX ANALYSIS TRISTAN PHILLIPS These are notes from an introduction to complex analysis at the undergraduate level as taught by Paul Taylor at Shippensburg University during the Fall 26 semester.
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 informationComplex Analysis, Stein and Shakarchi Meromorphic Functions and the Logarithm
Complex Analysis, Stein and Shakarchi Chapter 3 Meromorphic Functions and the Logarithm YungHsiang Huang 217.11.5 Exercises 1. From the identity sin πz = eiπz e iπz 2i, it s easy to show its zeros are
More informationMa 416: Complex Variables Solutions to Homework Assignment 6
Ma 46: omplex Variables Solutions to Homework Assignment 6 Prof. Wickerhauser Due Thursday, October th, 2 Read R. P. Boas, nvitation to omplex Analysis, hapter 2, sections 9A.. Evaluate the definite integral
More informationSummary for Vector Calculus and Complex Calculus (Math 321) By Lei Li
Summary for Vector alculus and omplex alculus (Math 321) By Lei Li 1 Vector alculus 1.1 Parametrization urves, surfaces, or volumes can be parametrized. Below, I ll talk about 3D case. Suppose we use e
More informationTypes of Real Integrals
Math B: Complex Variables Types of Real Integrals p(x) I. Integrals of the form P.V. dx where p(x) and q(x) are polynomials and q(x) q(x) has no eros (for < x < ) and evaluate its integral along the fol
More informationMath 715 Homework 1 Solutions
. [arrier, Krook and Pearson Section 2 Exercise ] Show that no purely real function can be analytic, unless it is a constant. onsider a function f(z) = u(x, y) + iv(x, y) where z = x + iy and where u
More information