Second Midterm Exam Name: Practice Problems March 10, 2015
|
|
- Harry Washington
- 5 years ago
- Views:
Transcription
1 Math Treibergs Second Midterm Exam Name: Practice Problems March 10, Determine the singular points of the function and state why the function is analytic everywhere else: z 1 fz) = z + 1)z + z + 7) Points z 0 where the function fails to be analytic, but is analytic at some point in every neighborhood of z 0 are called singular points. The function is analytic at a point z 0 if it is analytic in some neighborhood, an open set containing z 0. The given function blows up where the denominator is zero and the numerator is not zero.) For this function, this occurs at the points z 0 = 1 and z 0 = ± = ± 19 i. Away from these isolated points the denominator is nonzero so the complex derivative exists because it is a rational function of z. Thus every other point ζ is some distance r > 0 from these points, thus is analytic because the function is differentiable in the open disk of radius r about ζ. Also these points are singular because the there is no derivative at these points and every open set containing them contains other points where the function is analytic.. Let A be an open subset of and A = {z : z A} be its reflection across the real axis. Suppose that fz) is analytic on A. Show that f z) is analytic on A. First observe that A is open. That is because every point Z A has an open neighborhood U = B, namely, the reflection of the little open disk about z B A. Second, if fz) = ux, y) + ivx, y), then gz) = f z) = ux, y) ivx, y) and we show that the auchy Riemann equations hold for g. By the auchy Riemann Equations for f, Re g) = ux, y) = x x x u = x, y) y x, y) v = y [ vx, y)] = Im g) y and y Re g) = y ux, y) = y u = x, y) x v = x, y) x [ vx, y)] = Im g). x For every point of A, both real partial derivatives exist and are continuous for g because they are for f on A. Since the auchy Riemann equations hold as well, g is analytic on A.. Suppose that fz) = ux, y) + ivx, y) is an analytic function defined on a domain D. If aux, y) + bvx, y) = c, where a, b, c are real constants not all zero, then fz) is constant on A. Is the result still valid if a, b, c were allowed to be complex constants? Observe that we can t have both a = b = 0 because that implies that c = 0 too. Differentiating we find 0 = au x + bv x a b 0 = au y + bv y = bu x av x. b a = a b Since the determinant of the coefficient matrix is strictly negative if a and b are real and not both zero, we must have u x = v x = 0 everywhere in D. Hence f z) = u x + iv x = 0 and since D is connected, f is constant. 1
2 The result continues to hold if a, b.c are complex. If a + b 0 then the same argument shows f = 0 on D. But if b = a then b = ±ia, both nonzero, and the system has only one independent equation, 0 = au x ± aiv x = au x ± iv x ) = 0 = u x ± iv x. But since both u x and v x are real, the real and imaginary parts of this equation say u x = v x = 0 so f is again constant on D. 4. Let f be an entire function that equals a polynomial on [0, 1] on the real exis. Show that f is a polynomial. Let fx) = a 0 + a 1 x + + a n x n on [0, 1]. Then fz) and a 0 + a 1 z + + a n z n agree on [0, 1], and both are analytic on that is, both are entire). By the Identity Theorem, if two analytic functions agree on a line segment contained in the domain of both, then they agree over the whole domain. 5. Prove the identity: sinh e z + sinh e z = sinhcosh z) coshsinh z) Identities are proved starting from one side of the equality and deducing the other side. sinh e z + sinh e z = expez ) exp e z ) + expe z ) exp e z ) { = 1 e z + e z exp + ez e z ) exp ez + e z e z + e z + exp ez e z ) exp ez + e z ez e z ) + ez e z = 1 { exp cosh z + sinh z) exp cosh z sinh z) } + exp cosh z sinh z) exp cosh z + sinh z) = 1 { exp cosh z) exp sinh z) exp cosh z )exp sinh z) } + exp cosh z) exp sinh z) exp cosh z )exp sinh z) = 1 { }{ } exp cosh z) exp cosh z) exp sinh z )+ exp sinh z) = sinhcosh z) coshsinh z). ) } 6. Find all values z such that We write the equation as e z+i =. e x e y+1)i = e z+i = = e πi so that and so x = ln and y + 1 = π + nπ, n Z); z = 1 ln + 1 π 1 + nπ) i, n Z).
3 7. Show that logi 1/ ) = 1 log i. The logarithm is a multiple valued function. First, the three cube roots of i = e πi/ are e π/+πn)i/ n Z) so that the numbers corresponding to n = 0, 1, are the distinct roots e πi/6, e 5πi/6, e πi/, Then log e πi/6) ) 1 = ln n πi, log e 5πi/6) ) 5 = ln n πi, log e πi/) ) = ln n πi, n Z), n Z), n Z). Since 5 6 = and = 1 6 +, their union is log i 1/) = n ) πi, n Z). On the other hand, so that which is the same as log i 1/). ) 1 log i = ln n πi, n Z) 1 1 log i = 6 + n ) πi, n Z) 8. Verify the following relation for the inverse function. arccoth z are the solutions w of the equation Solving for e w yields or Finally, taking logarithm gives the solution. arccoth z = 1 log z + 1 z 1. z = coth w = cosh w sinh w = ew + e w e w e w = ew + 1 e w 1. e w + 1 = ze w 1) e w = z + 1 z 1
4 9. Show that for different choices of z 1 and z the following expression may or may not be valid: Log z 1 z = Log z 1 Log z. The issue is whether or not the differnce of arguments remains in the domain of the principal value. Thus if we have z k = r k e iθ k then Log z k = log z k in the branch π < Θ k π. Subtracting arguments in this range implies π < Θ 1 Θ < π which may put them out of range for the principal value if Θ 1 Θ π or Θ 1 Θ > π. For example if z 1 = 1 + i = e πi/4 and z = 1 + i = e πi/4 we have Log z 1 = Log i) = π [ z i = ln + π ] 4 i [ln + π4 ] i = Log z 1 Log z. On the other hand, if z 1 = 1 i = e πi/4 and z = 1 + i = e πi/4 we have Log z 1 = Log 1) = πi πi = [ln π ] z 4 i [ln + π4 ] i = Log z 1 Log z. 10. Find 1 + i) i and its principal value. 1 + i = e 1/4+n)πi for n N. Its powers are thus { 1 + i) i = e i log1+i) = exp i ln ) }) n πi, n Z) = exp π4 ) ) ln + 6nπ exp i, n Z) Its principal value corresponds to N = 0 or P. V. 1 + i) i = exp 11. Find the roots of the equation cos z =. Expressing the real and imaginary parts we find which reduces to two real equations π ) ) ln exp 4 i cos z = cos x cosh y i sin x sinh y = cos x cosh y = and sin x sinh y = 0. The second says either sinh y = 0 which implies y = 0 or sin x = 0 which implies x = nπ, n N). If y = 0 then the first equation becomes cos x cosh y = cos x cosh 0 = cos x = which has no real solution because cos x 1. Thus we have to solve the first equation in the second case x = nπ, cos x cosh y = cos nπ cosh y = 1) n cosh y =. 4
5 But cosh y 1 so that this equation only has solutions if n is even, in which case y = cosh 1, namely, real s that solves Equivalently, or = cosh s = 1 es + e s ). 0 = e s 6e s + 1 e s = 6 ± = ±. Noting that + and are reciprocals, we may write all solutions as z = nπ ± i ln + ), n Z). 1. Show that tanh z = i taniz) and d dz tanh z = sech z. We use the identities sinh z = i sin iz and cosh z = cos iz. Then tanh z = sinh z i sin iz = = i tan iz. cosh z cos iz Also, by the definitions of sinh z and cosh z and the quotient rule d dz tanh z = d sinh z dz cosh z = d e z e z dz e z + e z = ez + e z ) e z + e z ) e z + e z ) e z e z ) e z + e z ) e z + + e z) e z + e z) 4 = e z + e z ) = e z + e z ) = 1 cosh z = sech z. 1. Evaluate the following integral, assuming Re z < 0. 0 e zt dt This is an improper integral. Observing that d e zt dt z T [ e e zt dt = lim e zt zt dt = lim 0 T 0 T z For z = x + iy, the latter limit depends on the estimate ] T e zt = e xt e iyt = e xt which tends to zero as T because x = Re z < 0. = ezt we obtain 0 [ e zt = lim T z 1 ] = 1 z z. 5
6 14. let denote the right half of the circle z = in the counterclockwise direction. onsider the two given parametric representations z 1 θ) and z y) of. Find the reparameterization θ = φy) such that z y) = z 1 [φy)] and check that φ has positive derivative. z 1 θ) = e iθ, π θ π ) z y) = + iy, y ). Putting the first representation in rectangular coordinates gives Equating the ratio we find x = cos θ and y = sin θ. y = y x = sin θ cos θ = tan θ or ) y θ = φy) = Atn, π θ π ). To see that z y) = z 1 [φy)], we note that cosatn t) = 1/ 1 + t and sinatn t) = t/ 1 + t, so )) y Re z 1 φy)) = cos Atn = = = Re z y) y and )) y Im z 1 φy)) = sin Atn = y 1 + y = y 4 = Im z y). For < y < we find φ 1 y) = ) y 1 + y + = ) > 0 6
7 15. Using the contour shown in the figure, evaluate the integral Ree iz ) dz The function is the derivative in F z) = d dz e iz i = e iz Thus the integral is independent of path, and its value may be computed using the end points z 1 = 1 and z = to give ) I = Ree iz ) dz = Re fz) dz = Re F z ) F z 1 )). But so [ e iz F z ) F z 1 ) = i ] 1 = e i i ei i = i[ cos ) + i sin ) cos 1 i sin 1 ] I = sin sin Let denote the positively oriented circle z = 5 about the origin. For the branch at angle α R, with its corresponding log z = ln r + iθ where α < θ < α + π, find z / dz. We have zθ) = 5e iθ for α < θ < α + π so dz = 5ie iθ dθ. Also z / z θ) = exp ) logzθ)) 5ie iθ = 5i exp ln ) iθ As this extends continuously to the interval α θ α + π, we may integrate using real antiderivatives as usual [ ] α+π α+π z / dz = z / z θ) dθ = i e iθ/ 5 α = 5 [ e iα+π)/ e iα/] = 5 i α [ e πi/ 1 ] e iα/. 7
8 17. Let N denote the boundary of the square formed by the lines x = ± N + 1 ) π and y = ± N + 1 where N is a positive integer and N is given the counterclockwise orientation. Show that the integral tends to zero as N. dz z sin z N We bound the absolute value. The length of the square is L N = 4N + 1)π. To bound the integrand, we observe that It follows that ) π sin z = sinx + iy) = sin x cos iy + cos x sin iy = sin x cosh y + i cos x sinh y. sin z = sin x cosh y + cos x sinh y = sin x1 + sinh y) + cos x sinh y = sin x + sinh y sin x so that on the vertical lines, [ sin z sin x = ± sin N + 1 ] π) = 1. The estimate also showed that sin z sinh y so that on the horizontal lines, since sinh y is increasing in y and N 0, [ sin z sinh y = ± sinh N + 1 ] [ π) = sinh N + 1 ] ) π ) π sinh 1. So on all four sides of the square sin z 1. Thus we may estimate the integrand 1 z sin z = 1 z sin z 1 ) = M N + 1 N π because z x on the verticals and z y on the horizontals, which are both equal N + 1 ) π. Then the contour integral may be estimated N dz z sin z L 4N + 1)π NM N = ) = 16 N + 1 π N + 1) π. Thus the value of the integral tends to zero as N. 8
9 18. Find the integral, where the integrand is computed using the principal branch and is any contour from z = to z = that, except for its end points, lies above the real axis. z i dz The restriction on means that in polar coordinates, for z we have r > 0 and 0 θ π. In this range z i = exp i Log z), 0 θ π). For this range of θ s, Log z = log z where the second logarithm has the branch cut along the negative imaginary axis R π/ so π < θ < π. The function has an antiderivative on the domain D = R π/, namely F z) = z i+1 i + 1 = 1 + i exp i + 1) log z). 5 Hence the integral is found by evaluating the antiderivative function at the end points z 1 = = e 0 i and z = = e πi of the contour D z i dz = F ) F ) = 1 + i ) exp[ i + 1)ln + πi)] exp[ i + 1) ln ] 5 = 1 + i e π e i ln + e i ln ) Using the contour which is a unit square with corners 0, 1, 1 + i and i shown in the figure, evaluate the integral cos + 1 ) dz z Observe that the integrand function fz) = cos + 1 ) z is the composition of an entire function with a rational function which has only one singularity at z =. So fz) is analytic in the punctured plane, D = {}. Since is a simple, closed contour in D, by the auchy-goursat Theorem, the integral is zero. 9
10 0. Show that the integral on the real axis I = x + 4 i x + i) 4 dx = 0. Let fz) = z + 4 i z + i) 4. Since f decays quadratically in z, the function is integrable on R. For R > 0, letting R denote the line segment from R to R, we have z + 4 i I = lim R R z + i) 4 dz. Observe that the only singularity of the rational function fz) is at z = i so it is analytic in D = { i}. Let G R denote the arc of the circle z = R in the upper half plane. Thus R + G R is a simple closed contour in D so that, by the auchy Goursat Theorem, 0 = fz) dz = R +G R fz) dz + R fz) dz. G R We show that the limit of the second integral vanishes as R, thus so does the first, so I = 0. We estimate the second integral. The length of the contour is L R = πr. Also for z Γ R and R >, by the triangle inequalty Hence fz) = z + 4 i z + i 4 z + 4 i z i 4 = R + 5 R ) 4 = M R fz) dz L R M R = πrr + 5) G R R ) 4 which tends to zero as R, completing the argument. 10
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 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 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 information1 Discussion on multi-valued functions
Week 3 notes, Math 7651 1 Discussion on multi-valued 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 informationMTH 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 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 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 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 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 informationSyllabus: for Complex variables
EE-2020, 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.13-h.18. omplex numbers and functions,
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 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 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 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 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 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 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 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 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 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 Fajardo-Lim, Ph.D. Department of Mathematics De La Salle University - Manila 2 ontents THE
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 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 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 MATH-GA 245.00 Complex Variables Polynomials. Construction f : z z is analytic on all of C since its real and imaginary parts satisfy the Cauchy-Riemann relations and
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 informationMath Spring 2014 Solutions to Assignment # 8 Completion Date: Friday May 30, 2014
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
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 Analysis Homework 1: Solutions
Complex Analysis Fall 007 Homework 1: Solutions 1.1.. a) + i)4 + i) 8 ) + 1 + )i 5 + 14i b) 8 + 6i) 64 6) + 48 + 48)i 8 + 96i c) 1 + ) 1 + i 1 + 1 i) 1 + i)1 i) 1 + i ) 5 ) i 5 4 9 ) + 4 4 15 i ) 15 4
More informationFinal Exam - MATH 630: Solutions
Final Exam - MATH 630: Solutions Problem. Find all x R satisfying e xeix e ix. Solution. Comparing the moduli of both parts, we obtain e x cos x, and therefore, x cos x 0, which is possible only if x 0
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 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 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 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 informationSolution for Final Review Problems 1
Solution for Final Review Problems Final time and location: Dec. Gymnasium, Rows 23, 25 5, 2, Wednesday, 9-2am, Main ) Let fz) be the principal branch of z i. a) Find f + i). b) Show that fz )fz 2 ) λfz
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 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 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 informationFunctions of a Complex Variable and Integral Transforms
Functions of a Complex Variable and Integral Transforms Department of Mathematics Zhou Lingjun Textbook Functions of Complex Analysis with Applications to Engineering and Science, 3rd Edition. A. D. Snider
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 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 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 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 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 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 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 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 informationNATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part I
NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Comple Analsis II Lecture Notes Part I Chapter 1 Preliminar results/review of Comple Analsis I These are more detailed notes for the results
More information= 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 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 Complex Analysis Prelims Ben Strasser
Solutions to Complex Analysis Prelims Ben Strasser In preparation for the complex analysis prelim, I typed up solutions to some old exams. This document includes complete solutions to both exams in 23,
More 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 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 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 E-mail : sahoopulak1@gmail.com 1 Module-4: Multivalued Functions-II
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 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 Res z k+1 (z c), 0 =
32. COMPLEX ANALYSIS FOR APPLICATIONS Mock Final examination. (Monday June 7..am 2.pm) You may consult your handwritten notes, the book by Gamelin, and the solutions and handouts provided during the Quarter.
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 informationHomework 27. Homework 28. Homework 29. Homework 30. Prof. Girardi, Math 703, Fall 2012 Homework: Define f : C C and u, v : R 2 R by
Homework 27 Define f : C C and u, v : R 2 R by f(z) := xy where x := Re z, y := Im z u(x, y) = Re f(x + iy) v(x, y) = Im f(x + iy). Show that 1. u and v satisfies the Cauchy Riemann equations at (x, y)
More 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 Final Exam.
Math 106 - Final Exam. This is a closed book exam. No calculators are allowed. The exam consists of 8 questions worth 100 points. Good luck! Name: Acknowledgment and acceptance of honor code: Signature:
More informationChapter 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 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: 44-47 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 informationPart IB. Further Analysis. Year
Year 2004 2003 2002 2001 10 2004 2/I/4E Let τ be the topology on N consisting of the empty set and all sets X N such that N \ X is finite. Let σ be the usual topology on R, and let ρ be the topology on
More 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 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 & Integral Transforms
Complex Variables & Integral Transforms Notes taken by J.Pearson, from a S4 course at the U.Manchester. Lecture delivered by Dr.W.Parnell July 9, 007 Contents 1 Complex Variables 3 1.1 General Relations
More 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 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 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 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 informationComplex Analysis for Applications, Math 132/1, Home Work Solutions-II Masamichi Takesaki
Page 48, Problem. Complex Analysis for Applications, Math 3/, Home Work Solutions-II Masamichi Takesaki Γ Γ Γ 0 Page 9, Problem. If two contours Γ 0 and Γ are respectively shrunkable to single points in
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 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 informationComplex Integration Line Integral in the Complex Plane CHAPTER 14
HAPTER 14 omplex Integration hapter 13 laid the groundwork for the study of complex analysis, covered complex numbers in the complex plane, limits, and differentiation, and introduced the most important
More informationDepartment of Mathematics, University of California, Berkeley. GRADUATE PRELIMINARY EXAMINATION, Part A Fall Semester 2016
Department of Mathematics, University of California, Berkeley YOUR 1 OR 2 DIGIT EXAM NUMBER GRADUATE PRELIMINARY EXAMINATION, Part A Fall Semester 2016 1. Please write your 1- or 2-digit exam number on
More informationINTEGRATION WORKSHOP 2004 COMPLEX ANALYSIS EXERCISES
INTEGRATION WORKSHOP 2004 COMPLEX ANALYSIS EXERCISES PHILIP FOTH 1. Cauchy s Formula and Cauchy s Theorem 1. Suppose that γ is a piecewise smooth positively ( counterclockwise ) oriented simple closed
More 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 informationComplex Analysis. Travis Dirle. December 4, 2016
Complex Analysis 2 Complex Analysis Travis Dirle December 4, 2016 2 Contents 1 Complex Numbers and Functions 1 2 Power Series 3 3 Analytic Functions 7 4 Logarithms and Branches 13 5 Complex Integration
More 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 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 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 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 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 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 informationMTH 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 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 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 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 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 informationAssignment 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 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 informationSolutions to Exercises 6.1
34 Chapter 6 Conformal Mappings Solutions to Exercises 6.. An analytic function fz is conformal where f z. If fz = z + e z, then f z =e z z + z. We have f z = z z += z =. Thus f is conformal at all z.
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 informationComplex Analysis Math 185A, Winter 2010 Final: Solutions
Complex Analysis Math 85A, Winter 200 Final: Solutions. [25 pts] The Jacobian of two real-valued functions u(x, y), v(x, y) of (x, y) is defined by the determinant (u, v) J = (x, y) = u x u y v x v y.
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 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 informationSPRING 2006 PRELIMINARY EXAMINATION SOLUTIONS
SPRING 006 PRELIMINARY EXAMINATION SOLUTIONS 1A. Let G be the subgroup of the free abelian group Z 4 consisting of all integer vectors (x, y, z, w) such that x + 3y + 5z + 7w = 0. (a) Determine a linearly
More informationMORE CONSEQUENCES OF CAUCHY S THEOREM
MOE CONSEQUENCES OF CAUCHY S THEOEM Contents. The Mean Value Property and the Maximum-Modulus Principle 2. Morera s Theorem and some applications 3 3. The Schwarz eflection Principle 6 We have stated Cauchy
More informationComplex Functions (1A) Young Won Lim 2/22/14
Complex Functions (1A) Copyright (c) 2011-2014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or
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 information