Instructions Answer all questions. Give your answers clearly. Do not skip intermediate steps
|
|
- Norah Thornton
- 5 years ago
- Views:
Transcription
1 Instructions Answer all questions. Give your answers clearly. Do not skip intermediate steps even they are very easy. Complete answers with no error, including expression and notation errors receive full mark. Calculators are not allowed. Cell phones have to be switched off and be kept in your pocked or in your bag. Time 90 minutes TOTAL QUESTIONS Q) 4p) Evaluate the following mathematical operations. Write your results in Cartesian form. a) 6e iπ/ i 3 b) 2 ) /3 Q2) 9p) A complex function is given as fz) = z 2 +2 z. Replace complex variable z with its Cartesian form z = x+iy in the function and obtain wx,y) = fx+iy). What are the real and imaginary parts ofwx,y). Q3) 9p) A complex function wx,y) = x 2 y 2 +2x+i2xy 2y) is given. Here x and y are real and imaginary parts of complex variablez = x+iy. Check analyticity of the function with Cauchy-Riemann equations. Is it analytic? Q4) 5p) Derive the description of the inverse cosine function w = cos z) in terms of log function. Sami Arıca Page of 9
2 Hint: Write cosw) in terms of complex exponentials.) Q5) 0p) Evaluate the following integral. 0 20t t 2 + i) 2 dt Hint: change the term in the parenthesis with a new variable.) Q6) 6p) Decide if the following domains are simply connected or not. Write your reasoning? a) b) Q7) 4p) Which of the following loops are positively oriented counter-clockwise direction)? a) y b) y x x Q8) 5p) Evaluate the following line integral. The line is : zt) = Log2)+itπ/3 π/6), 0 t. When it is necessary, take Log2) 0.7 and 3π 5.4. z+)e z dz =? Sami Arıca Page 2 of 9
3 Hint: d dz zez =? z z =? ora+ib a ib) =?) Q9) 9p) Evaluate the following integral. is a loop with any pointzon it satisfy the inequality z < 3/2. The loop traverses in the counter-clockwise direction. e z z dz =? Hint : Remember Cauchy s integral formula. Q0) 9p) Compute line integral, fz) = 0 2z z 2 2z 3 = z 3 3 z+ C fz) z z 0 dz =?) fz)dz, of the following function Here is a loop with any point z on it satisfy the inequality z < 3. The loop traverses in the counter-clockwise direction. Hint: Remember the integral; z a) n dz =?) C Sami Arıca Page 3 of 9
4 ANSWERS A) a) 6e iπ/3 = 6 cosπ/3)+i6 sinπ/3) = 3+i i 3 = 4 i ) 3 +i ) 3 i ) = 3 4 i ) 3 4 = i 3 b) 6e iπ/ i 3 = 3+i3 3+ i 3 = 4+i2 3 = e π+2πk, k Z ) /3 = e π/3+2πk/3, k = 0,,2 2 ) /3 = 2e π/3+2πk/3, k = 0,,2 ) /3 has three roots and therefore 2 ) /3 has three answers. k = 0 : 2e π/3+2πk/3 = 2e π/3 = 2 cosπ/3)+i2 sinπ/3) = +i 3 k = : 2e π/3+2πk/3 = 2e π = 2 cosπ)+i2 sinπ) = 2 k = 2 : 2e π/3+2πk/3 = 2e 5π/3 = 2 cos5π/3)+i2 sin5π/3) = i 3 A2) wx,y) = x+iy) 2 +2x+iy = x 2 +i2xy+iy) 2 +2x iy) = x 2 +i2xy y 2 +2x i2y = x 2 y 2 +2x+i2xy 2y) Re[wx,y)] = x 2 y 2 +2x Im[wx,y)] = 2xy 2y Sami Arıca Page 4 of 9
5 A3) ux,y) = Re[wx,y)] = x 2 y 2 +2x vx,y) = Im[wx,y)] = 2xy 2y ux,y) = 2x+2 x vx,y) = 2y x ux,y) = 2x y vx,y) = 2x 2 y We check if the the Cauchy-Riemann equations are satisfied. For the given function, x ux,y) y vx,y) y ux,y) = x vx,y) The equality is fulfilled in the second expression but not in the first expression for anyxandy, therefore the function is nowhere analytic. You should immediately realize that the function given in this question is the same with the one given in the second equation. This function contains z in its definition. z is nowhere analytic sofz). Recall the differentiation ofz dz dz dz dz = when dz = dx when we approach to z along real axis) = when dz = idy when we approach tozalong imaginary axis) The derivative should be independent of any direction we approach. Consequently z is nowhere analytic since there isn t any domain it is analytic in the complex plane. Sami Arıca Page 5 of 9
6 A4) We can write w = cos z) as cosw) = z to escape from the inverse function. Using Euler s formula cosw) is replaced with exponentials. e iw +e iw 2 = z Continue manipulating this equation e iw +e iw e i2w + = 2z = 2ze iw e i2w 2ze iw + = 0 We need to compute roots of the second order equation e iw ) 2 2ze iw + = 0 = z) 2 = z 2 e iw = z)+ = z+ z 2 Then we take logarithm of the both sides of the last equation to extract variablew iw = log z+ ) z 2 w = i log z+ ) z 2 This is the answer A5) Let us changet 2 + i with a new variables. t 2 + i = s 2tdt = ds t = 0 s t = s = i = 2 i Sami Arıca Page 6 of 9
7 Re-write the integral with the new variable and compute the integral 0 20t t 2 + i) 2 dt = 2 i i = 0 s 0 s 2 ds = 2 i i 2 i i 0 s 2 ds = 0 2 i + 0 i = 0+0i+20 0i 3i = 0 3i = 0+3i) 3i)+3i) = +3i A6) a) Any loop inside the domain stretches a single point in the domain. Equivalently, the interior of any loop in the domain is completely inside the domain. Therefore, this domain is simply connected. b) This loop do not meet the conditions mentioned above. It is not simply connected. A7) a) The direction of the loop is counter-clockwise opposite direction of the clockwise direction). Equivalently, while the loop direction is upward the interior of the loop is at the left side. Therefore direction of the loop is positively oriented. b) The direction of the loop is clockwise. Equivalently, while the loop direction is upward the interior of the loop is at the right side. Therefore direction of the loop is not positively oriented. A8) The functionsz+and e z are entire functions. So, z+)e z is entire. Anti-derivative of the integrandfz) = z+)e z is Fz) = ze z. Then, z+)e z dz = ze z z) z0) = z)e z) z0)e z0) Sami Arıca Page 7 of 9
8 Here, z0) = Log2)+i0 π/3 π/6) = Log2) iπ/6 z) = Log2)+i π/3 π/6) = Log2)+iπ/6 z0) = z) z0)e z0) = z)e z) = z)e z) Therefore, z)e z) z0)e z0) = i2im [ z)e z)] z)e z) = [ Log2)+iπ/6]e Log2)+iπ/6 We need to compute = Log2)e Log2)+iπ/6 +i π 6 elog2)+iπ/6 = 2Log2)e iπ/6 +i π 3 eiπ/6 Im [ z)e z)] = Im [2Log2)e iπ/6 +i π ] 3 eiπ/6 = Im [ 2Log2)e iπ/6] + Im [i π ] 3 eiπ/6 It follows = 2Log2) sinπ/6)+ π 3 cosπ/6) = Log2)+ π 3 6 z+)e z dz = i2im [ z)e z)] = i = i.4+.8) = i3.2 2Log2)+ π 3 3 ) A9) We will employ Cauchy s integral formula. C fz) z z 0 dz = 2πifz 0 ) Sami Arıca Page 8 of 9
9 The direction of loopcis counter-clockwise and encloses the pointz 0. fz) is analytic on and at the interior of the loop. e z is entire and direction of the loop is counter-clockwise and the loop encloses z =. Therefore, e z z dz = 2πie = 2πie A0) z = 3 and z = f3) = ±, f ) = ± ) are poles of fz). The loop encloses the poles and its direction is positively oriented. Then, fz)dz = fz)dz+ fz)dz C C 2 with C : r t) = 3+ǫ e i2πt and r 2 t) = +ǫ 2 e i2πt 0 t C fz)dz = C z 3 dz 3 C z+ dz = 2πi 3 0 = 2πi C 2 fz)dz = C 2 z 3 dz 3 C 2 z+ dz = 0 3 2πi = 6πi Concequently, fz)dz = 2πi 6πi = 4πi Sami Arıca Page 9 of 9
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 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 Cauchy-Riemann equations:
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 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 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 self-test for you to determine if you if have the necessary pre-requisite
More informationMATH 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 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 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 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 informationSolutions to Selected Exercises. Complex Analysis with Applications by N. Asmar and L. Grafakos
Solutions to Selected Exercises in Complex Analysis with Applications by N. Asmar and L. Grafakos Section. Complex Numbers Solutions to Exercises.. We have i + i. So a and b. 5. We have So a 3 and b 4.
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 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 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 informationIntegration 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 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 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 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 R:
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 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 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 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 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 informationP3.C8.COMPLEX NUMBERS
Recall: Within the real number system, we can solve equation of the form and b 2 4ac 0. ax 2 + bx + c =0, where a, b, c R What is R? They are real numbers on the number line e.g: 2, 4, π, 3.167, 2 3 Therefore,
More information3 Contour integrals and Cauchy s Theorem
3 ontour integrals and auchy s Theorem 3. Line integrals of complex functions Our goal here will be to discuss integration of complex functions = u + iv, with particular regard to analytic functions. Of
More informationNorth MaharashtraUniversity ; Jalgaon.
North MaharashtraUniversity ; Jalgaon. Question Bank S.Y.B.Sc. Mathematics (Sem II) MTH. Functions of a omplex Variable. Authors ; Prof. M.D.Suryawanshi (o-ordinator) Head, Department of Mathematics, S.S.V.P.S.
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 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 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 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 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 informationUNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH
UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH College of Informatics and Electronics END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MS425 SEMESTER: Autumn 25/6 MODULE TITLE: Applied Analysis DURATION OF EXAMINATION:
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 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 informationLECTURE-13 : GENERALIZED CAUCHY S THEOREM
LECTURE-3 : GENERALIZED CAUCHY S THEOREM VED V. DATAR The aim of this lecture to prove a general form of Cauchy s theorem applicable to multiply connected domains. We end with computations of some real
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 informationSolutions 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 information(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 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 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 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 informationQ You mentioned that in complex analysis we study analytic functions, or, in another name, holomorphic functions. Pray tell me, what are they?
COMPLEX ANALYSIS PART 2: ANALYTIC FUNCTIONS Q You mentioned that in complex analysis we study analytic functions, or, in another name, holomorphic functions. Pray tell me, what are they? A There are many
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 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 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 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 informationCOMPLEX DIFFERENTIAL FORMS. 1. Complex 1-forms, the -operator and the Winding Number
CHAPTER 3. COMPLEX DIFFERENTIAL FORMS 1. Complex 1-forms, the -operator and the Winding Number Now we consider differentials for complex functions. Take a complex-valued function defined on an open region
More information(a) To show f(z) is analytic, explicitly evaluate partials,, etc. and show. = 0. To find v, integrate u = v to get v = dy u =
Homework -5 Solutions Problems (a) z = + 0i, (b) z = 7 + 24i 2 f(z) = u(x, y) + iv(x, y) with u(x, y) = e 2y cos(2x) and v(x, y) = e 2y sin(2x) u (a) To show f(z) is analytic, explicitly evaluate partials,,
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 information(x 1, y 1 ) = (x 2, y 2 ) if and only if x 1 = x 2 and y 1 = y 2.
1. Complex numbers A complex number z is defined as an ordered pair z = (x, y), where x and y are a pair of real numbers. In usual notation, we write z = x + iy, where i is a symbol. The operations of
More information2.4 Lecture 7: Exponential and trigonometric
154 CHAPTER. CHAPTER II.0 1 - - 1 1 -.0 - -.0 - - - - - - - - - - 1 - - - - - -.0 - Figure.9: generalized elliptical domains; figures are shown for ǫ = 1, ǫ = 0.8, eps = 0.6, ǫ = 0.4, and ǫ = 0 the case
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 / 29 Complex Analysis Module: 4:
More informationMath 211, Fall 2014, Carleton College
A. Let v (, 2, ) (1,, ) 1, 2, and w (,, 3) (1,, ) 1,, 3. Then n v w 6, 3, 2 is perpendicular to the plane, with length 7. Thus n/ n 6/7, 3/7, 2/7 is a unit vector perpendicular to the plane. [The negation
More informationMA30056: Complex Analysis. Revision: Checklist & Previous Exam Questions I
MA30056: Complex Analysis Revision: Checklist & Previous Exam Questions I Given z C and r > 0, define B r (z) and B r (z). Define what it means for a subset A C to be open/closed. If M A C, when is M said
More information4.1 Exponential and Logarithmic Functions
. Exponential and Logarithmic Functions Joseph Heavner Honors Complex Analysis Continued) Chapter July, 05 3.) Find the derivative of f ) e i e i. d d e i e i) d d ei ) d d e i ) e i d d i) e i d d i)
More informationComplex numbers, the exponential function, and factorization over C
Complex numbers, the exponential function, and factorization over C 1 Complex Numbers Recall that for every non-zero real number x, its square x 2 = x x is always positive. Consequently, R does not contain
More informationExercises involving contour integrals and trig integrals
8::9::9:7 c M K Warby MA364 Complex variable methods applications Exercises involving contour integrals trig integrals Let = = { e it : π t π }, { e it π : t 3π } with the direction of both arcs corresponding
More informationComplex Inversion Formula for Stieltjes and Widder Transforms with Applications
Int. J. Contemp. Math. Sciences, Vol. 3, 8, no. 16, 761-77 Complex Inversion Formula for Stieltjes and Widder Transforms with Applications A. Aghili and A. Ansari Department of Mathematics, Faculty of
More informationMath Homework 1. The homework consists mostly of a selection of problems from the suggested books. 1 ± i ) 2 = 1, 2.
Math 70300 Homework 1 September 1, 006 The homework consists mostly of a selection of problems from the suggested books. 1. (a) Find the value of (1 + i) n + (1 i) n for every n N. We will use the polar
More informationR- 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 informationTopic 1 Notes Jeremy Orloff
Topic 1 Notes Jeremy Orloff 1 Complex algebra and the complex plane We will start with a review of the basic algebra and geometry of complex numbers. Most likely you have encountered this previously in
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 informationB Elements of Complex Analysis
Fourier Transform Methods in Finance By Umberto Cherubini Giovanni Della Lunga Sabrina Mulinacci Pietro Rossi Copyright 21 John Wiley & Sons Ltd B Elements of Complex Analysis B.1 COMPLEX NUMBERS The purpose
More informationPhysics 307. Mathematical Physics. Luis Anchordoqui. Wednesday, August 31, 16
Physics 307 Mathematical Physics Luis Anchordoqui 1 Bibliography L. A. Anchordoqui and T. C. Paul, ``Mathematical Models of Physics Problems (Nova Publishers, 2013) G. F. D. Duff and D. Naylor, ``Differential
More informationSolutions to Tutorial for Week 3
The University of Sydney School of Mathematics and Statistics Solutions to Tutorial for Week 3 MATH9/93: Calculus of One Variable (Advanced) Semester, 08 Web Page: sydney.edu.au/science/maths/u/ug/jm/math9/
More informationEE2 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 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 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 informationComplex varibles:contour integration examples. cos(ax) x
1 Problem 1: sinx/x omplex varibles:ontour integration examples Integration of sin x/x from to is an interesting problem 1.1 Method 1 In the first method let us consider e iax x dx = cos(ax) dx+i x sin(ax)
More informationGreen s Theorem, Cauchy s Theorem, Cauchy s Formula
Math 425 Spring 2003 Green s Theorem, Cauchy s Theorem, Cauchy s Formula These notes supplement the discussion of real line integrals and Green s Theorem presented in.6 of our text, and they discuss applications
More informationMATH 135: COMPLEX NUMBERS
MATH 135: COMPLEX NUMBERS (WINTER, 010) The complex numbers C are important in just about every branch of mathematics. These notes 1 present some basic facts about them. 1. The Complex Plane A complex
More informationCOMPLEX ANALYSIS AND RIEMANN SURFACES
COMPLEX ANALYSIS AND RIEMANN SURFACES KEATON QUINN 1 A review of complex analysis Preliminaries The complex numbers C are a 1-dimensional vector space over themselves and so a 2-dimensional vector space
More information3 + 4i 2 + 3i. 3 4i Fig 1b
The introduction of complex numbers in the 16th century was a natural step in a sequence of extensions of the positive integers, starting with the introduction of negative numbers (to solve equations of
More informationTopic 4 Notes Jeremy Orloff
Topic 4 Notes Jeremy Orloff 4 Complex numbers and exponentials 4.1 Goals 1. Do arithmetic with complex numbers.. Define and compute: magnitude, argument and complex conjugate of a complex number. 3. Euler
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 informationComplex Analysis Homework 3
Complex Analysis Homework 3 Steve Clanton David Holz March 3, 009 Problem 3 Solution. Let z = re iθ. Then, we see the mapping leaves the modulus unchanged while multiplying the argument by -3: Ω z = z
More informationHomework 3: Complex Analysis
Homework 3: Complex Analysis Course: Physics 23, Methods of Theoretical Physics (206) Instructor: Professor Flip Tanedo (flip.tanedo@ucr.edu) Due by: Friday, October 4 Corrected: 0/, problem 6 f(z) f(/z)
More informationMathematics 350: Problems to Study Solutions
Mathematics 350: Problems to Study Solutions April 25, 206. A Laurent series for cot(z centered at z 0 i converges in the annulus {z : < z i < R}. What is the largest possible value of R? Solution: The
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 informationParametric Curves. Calculus 2 Lia Vas
Calculus Lia Vas Parametric Curves In the past, we mostly worked with curves in the form y = f(x). However, this format does not encompass all the curves one encounters in applications. For example, consider
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 informationi>clicker Questions Phys 101 W2013
i>clicker Questions Phys 101 W2013 Set your frequency to AC. Hold down the power button til you see blinking. Then hit A, followed by C. Official standard is i>clicker2 (but original i>clicker will work
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 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 information1. COMPLEX NUMBERS. z 1 + z 2 := (a 1 + a 2 ) + i(b 1 + b 2 ); Multiplication by;
1. COMPLEX NUMBERS Notations: N the set of the natural numbers, Z the set of the integers, R the set of real numbers, Q := the set of the rational numbers. Given a quadratic equation ax 2 + bx + c = 0,
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 Functions of a complex variable
1 Functions of a complex variable Complex numbers have played a fundamental part in physics since the discovery of quantum mechanics, where the wave function has real arguments but complex values. The
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 informationINTRODUCTION TO COMPLEX ANALYSIS W W L CHEN
INTRODUTION TO OMPLEX NLYSIS W W L HEN c W W L hen, 1986, 28. This chapter originates from material used by the author at Imperial ollege, University of London, between 1981 and 199. It is available free
More informationHandout 1 - Contour Integration
Handout 1 - Contour Integration Will Matern September 19, 214 Abstract The purpose of this handout is to summarize what you need to know to solve the contour integration problems you will see in SBE 3.
More informationC. Complex Numbers. 1. Complex arithmetic.
C. Complex Numbers. Complex arithmetic. Most people think that complex numbers arose from attempts to solve quadratic equations, but actually it was in connection with cubic equations they first appeared.
More informationUniversity 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 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 informationMathematics of Physics and Engineering II: Homework answers You are encouraged to disagree with everything that follows. P R and i j k 2 1 1
Mathematics of Physics and Engineering II: Homework answers You are encouraged to disagree with everything that follows Homework () P Q = OQ OP =,,,, =,,, P R =,,, P S = a,, a () The vertex of the angle
More informationPHYS 3900 Homework Set #03
PHYS 3900 Homework Set #03 Part = HWP 3.0 3.04. Due: Mon. Feb. 2, 208, 4:00pm Part 2 = HWP 3.05, 3.06. Due: Mon. Feb. 9, 208, 4:00pm All textbook problems assigned, unless otherwise stated, are from the
More information1 + z 1 x (2x y)e x2 xy. xe x2 xy. x x3 e x, lim x log(x). (3 + i) 2 17i + 1. = 1 2e + e 2 = cosh(1) 1 + i, 2 + 3i, 13 exp i arctan
Complex Analysis I MT333P Problems/Homework Recommended Reading: Bak Newman: Complex Analysis Springer Conway: Functions of One Complex Variable Springer Ahlfors: Complex Analysis McGraw-Hill Jaenich:
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 information13. Complex Variables
. Complex Variables QUESTION AND ANSWERE Complex Analysis:. The residue of the function f() = ( + ) ( ) (a) (b) (c) 6 6 at = is (d) [EC: GATE-8]. (a) d Residue at = is lim ( ) f() d d = lim d + = lim +
More informationSection 5-7 : Green's Theorem
Section 5-7 : Green's Theorem In this section we are going to investigate the relationship between certain kinds of line integrals (on closed paths) and double integrals. Let s start off with a simple
More informationLecture 5. Complex Numbers and Euler s Formula
Lecture 5. Complex Numbers and Euler s Formula University of British Columbia, Vancouver Yue-Xian Li March 017 1 Main purpose: To introduce some basic knowledge of complex numbers to students so that they
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 z +
More information