DOCSLIB.ORG

Advanced Mathematical Methods in Theoretical Physics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Advanced Mathematical Methods in Theoretical Physics Advanced Mathematical Methods in Theoretical Physics Wintersemester 2017/2018 Technische Universit¨atBerlin PD Dr. Gernot Schaller March 15, 2021 i This lecture aims at providing physics students and neighboring disciplines with a heuristic toolbox that can be used to tackle all kinds of problems. As time and resources are limited, we will explore only small fractions of different much wider fields, and students are invited to take the lecture only as a starting point. The lecture will try to be as self-contained as possible and aims at providing rather recipes than strict mathematical proofs. It is not an original contribution but an excerpt of many papers, books, other lectures, and own experiences. Only a few of these have been included as references (footnotes describing famous scientists originate from Wikipedia and solely serve for better identification). As successful learning requires practice, the lecture (Mondays) will be accompanied by exer- cises. Students should turn in individual solutions to the exercises at the beginning of Wednesday seminars (computer algebra may be used if applicable). Students may earn points for the exercise sheets, to be admitted to the final exam, they should have earned 50% of the points. After passing the final exam, the students earn in total six ECTS credit points. To get these credit points, students therefore need to get 50 % of the exercise points and 50 % to pass the final exam. The lecture script will be made available online at http://www1.itp.tu-berlin.de/schaller/. Corrections and suggestions for improvements should be addressed to [email protected]. The tentative content of the lecture is as follows: Integration Integral Transforms Ordinary Differential Equations Partial Differential Equations Statistics Operators ii Contents 1 Integration of Functions1 1.1 Heuristic Introduction into Complex Analysis . .1 1.1.1 Differentiation over C ...............................1 1.1.2 Integration in the complex plane C .......................4 1.1.3 Cauchy integral theorem . .5 1.1.4 Cauchy's integral formula . .6 1.1.5 Examples . .7 1.1.6 The Laurent Expansion . .9 1.1.7 The Residue Theorem . 11 1.1.8 Principal Value integrals . 14 1.2 Useful Integration Tricks . 16 1.2.1 Standard Integrals . 16 1.2.2 Integration by Differentiation . 17 1.2.3 Saddle point approximation . 18 1.3 The Euler MacLaurin summation formula . 19 1.3.1 Bernoulli functions . 19 1.3.2 The Euler-MacLaurin formula . 21 1.3.3 Application Examples . 22 1.4 Numerical Integration . 24 1.4.1 The Trapezoidal Rule . 25 1.4.2 The Simpson Rule . 26 1.4.3 Monte-Carlo integration . 29 2 Integral Transforms 31 2.1 Fourier Transform . 31 2.1.1 Continuous Fourier Transform . 31 2.1.2 Important Fourier Transforms . 33 2.1.3 Applications of the convolution theorem . 34 2.1.4 Discrete Fourier Transform . 35 2.2 Laplace Transform . 36 2.2.1 Definition . 36 2.2.2 Properties . 40 2.2.3 Applications . 43 3 Ordinary Differential Equations 51 3.1 Linear ODEs with constant coefficients . 52 3.1.1 Properties of the Matrix Exponential . 52 iii iv CONTENTS 3.1.2 Numerical Computation of the matrix exponential . 55 3.2 The adiabatically driven Schr¨odingerequation . 57 3.3 Periodic Linear ODEs . 60 3.4 Nonlinear ODEs . 62 3.4.1 Separable nonlinear ODEs . 62 3.4.2 Fixed-Point Analysis . 63 3.5 Numerical Solution . 69 3.5.1 Runge-Kutta algorithm . 71 3.5.2 Leapfrog integration . 71 3.5.3 Adaptive stepsize control . 72 3.6 A note on Large Systems . 73 4 Special Partial Differential Equations 75 4.1 Separation Ansatz . 76 4.1.1 Diffusion Equation . 76 4.1.2 Damped Wave Equation . 78 4.2 Fourier Transform . 81 4.2.1 Example: Reaction-Diffusion Equation . 81 4.2.2 Example: Unbounded Wave Equation . 82 4.2.3 Example: Fokker-Planck equation . 82 4.3 Green's functions . 84 4.3.1 Example: Poisson Equation . 84 4.3.2 Example: Wave Equation . 85 4.4 Nonlinear Equations . 88 4.4.1 Example: Fisher-Kolmogorov-Equation . 88 4.4.2 Example: Korteweg-de-Vries equation . 89 4.5 Numerical Solution: Finite Differences . 90 4.5.1 Explicit Forward-Time Discretization . 91 4.5.2 Implicit Centered-Time discretization . 92 4.5.3 Indexing in higher dimensions . 94 4.5.4 Nonlinear PDEs . 94 5 Master Equations 99 5.1 Rate Equations . 99 5.1.1 Example 1: Fluctuating two-level system . 100 5.1.2 Example 2: Interacting quantum dots . 100 5.2 Density Matrix Formalism . 101 5.2.1 Density Matrix . 101 5.2.2 Dynamical Evolution in a closed system . 102 5.2.3 Most general evolution in an open system . 104 5.3 Lindblad Master Equation . 104 5.3.1 Example: Master Equation for a driven cavity . 106 5.3.2 Superoperator notation . 107 5.4 Full Counting Statistics in master equations . 108 5.4.1 Phenomenologic Identification of Jump Terms . 108 5.4.2 Example: Single-Electron-Transistor . 111 5.5 Entropy and Thermodynamics . 112 CONTENTS v 5.5.1 Spohn's inequality . 112 5.5.2 Phenomenologic definition of currents . 113 5.5.3 Thermodynamics of Lindblad equations . 114 5.5.4 Nonequilibrium thermodynamics . 116 6 Canonical Operator Transformations 119 6.1 Bogoliubov transformations . 119 6.1.1 Example: Diagonalization of a homogeneous chain . 121 6.2 Jordan-Wigner Transform . 121 6.3 Collective Spin Models . 128 6.3.1 Example: The Lipkin-Meshkov-Glick model . 131 6.3.2 Holstein-Primakoff transform . 133 vi CONTENTS Chapter 1 Integration of Functions In this chapter, we first briefly review the properties of complex numbers and functions of complex numbers. We will find that these properties can be used to solve e.g. challenging integrals in an elegant fashion. 1.1 Heuristic Introduction into Complex Analysis Complex analysis treats complex-valued functions of a complex variable, i.e., maps from C to C. As will become obvious in the following, there are fundamental differences compared to real functions, which however can be used to simplify many relationships extremely. In the following, we will therefore for a complex-valued function f(z) always use the partition z = x + iy : fx; y 2 Rg 2 f(z) = u(x; y) + iv(x; y): fu; v : R ! Rg ; (1.1) where x and y denote real and imaginary part of the complex variable z, and u and v the real and imaginary part of the function f(z), respectively. 1.1.1 Differentiation over C The differentiability of complex-valued functions is treated in complete analogy to the real case. One fundamental difference however is that the limit f(z0) − f(z) f 0(z) := lim (1.2) z0!z z0 − z has in the complex plane an infinite number of ways to approach z0 ! z, whereas in the real case there are just two such paths (left and right limit). In analogy to the real case we define the derivative of a complex function f(z) via the difference quotient when the limit in Eq. (1.2) a.) exists and b.) is independent of the chosen path. In this case, the function f(z) is then called in z complex differentiable. If f(z) is complex- differentiable in a U"-environment of z0, it is also called holomorphic in z0. Demanding that a function is holomorphic has strong consequences, see Fig. 1.1. It implies 1 2 CHAPTER 1. INTEGRATION OF FUNCTIONS iy z’=z+i ∆ y Figure 1.1: For the complex-differentiable func- tion f(x + iy) = u(x; y) + iv(x; y) we consider the 0 path z ! z once parallel to the real and once z z’=z+ ∆ x parallel to the imaginary axis. For a complex- differentiable function, the result must be inde- pendent of the chosen path. x that the derivative can be expressed by u(x + ∆x; y) + iv(x + ∆x; y) − u(x; y) − iv(x; y) f 0(z) = lim ∆x!0 ∆x u(x + ∆x; y) − u(x; y) v(x + ∆x; y) − v(x; y) = lim + i lim ∆x!0 ∆x ∆x!0 ∆x @u @v = + i : (1.3) @x @x On the other hand one must also have u(x; y + ∆y) + iv(x; y + ∆y) − u(x; y) − iv(x; y) f 0(z) = lim ∆y!0 i∆y u(x; y + ∆y) − u(x; y) v(x; y + ∆y) − v(x; y) = lim + i lim ∆y!0 i∆y ∆y!0 i∆y @u @v = −i + ; (1.4) @y @y since f(z) has been assumed complex differentiable. Now, comparing the real and imaginary parts of Eqns. (1.3) and (1.4) one obtains the Cauchy 1-Riemann 2 differential equations @u @v @u @v = ; = − ; (1.5) @x @y @y @x.
Load more

Recommended publications
  • The Hilbert Transform
    (February 14, 2017) The Hilbert transform Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ [This document is http:=/www.math.umn.edu/egarrett/m/fun/notes 2016-17/hilbert transform.pdf] 1. The principal-value functional 2. Other descriptions of the principal-value functional 3. Uniqueness of odd/even homogeneous distributions 4. Hilbert transforms 5. Examples 6. ... 7. Appendix: uniqueness of equivariant functionals 8. Appendix: distributions supported at f0g 9. Appendix: the snake lemma Formulaically, the Cauchy principal-value functional η is Z 1 f(x) Z f(x) ηf = principal-value functional of f = P:V: dx = lim dx −∞ x "!0+ jxj>" x This is a somewhat fragile presentation. In particular, the apparent integral is not a literal integral! The uniqueness proven below helps prove plausible properties like the Sokhotski-Plemelj theorem from [Sokhotski 1871], [Plemelji 1908], with a possibly unexpected leading term: Z 1 f(x) Z 1 f(x) lim dx = −iπ f(0) + P:V: dx "!0+ −∞ x + i" −∞ x See also [Gelfand-Silov 1964]. Other plausible identities are best certified using uniqueness, such as Z f(x) − f(−x) ηf = 1 dx (for f 2 C1( ) \ L1( )) 2 x R R R f(x)−f(−x) using the canonical continuous extension of x at 0. Also, 2 Z f(x) − f(0) · e−x ηf = dx (for f 2 Co( ) \ L1( )) x R R R The Hilbert transform of a function f on R is awkwardly described as a principal-value integral 1 Z 1 f(t) 1 Z f(t) (Hf)(x) = P:V: dt = lim dt π −∞ x − t π "!0+ jt−xj>" x − t with the leading constant 1/π understandable with sufficient hindsight: we will see that this adjustment makes H extend to a unitary operator on L2(R).
    [Show full text]
  • MATH 305 Complex Analysis, Spring 2016 Using Residues to Evaluate Improper Integrals Worksheet for Sections 78 and 79
    MATH 305 Complex Analysis, Spring 2016 Using Residues to Evaluate Improper Integrals Worksheet for Sections 78 and 79 One of the interesting applications of Cauchy's Residue Theorem is to find exact values of real improper integrals. The idea is to integrate a complex rational function around a closed contour C that can be arbitrarily large. As the size of the contour becomes infinite, the piece in the complex plane (typically an arc of a circle) contributes 0 to the integral, while the part remaining covers the entire real axis (e.g., an improper integral from −∞ to 1). An Example Let us use residues to derive the formula p Z 1 x2 2 π 4 dx = : (1) 0 x + 1 4 Note the somewhat surprising appearance of π for the value of this integral. z2 First, let f(z) = and let C = L + C be the contour that consists of the line segment L z4 + 1 R R R on the real axis from −R to R, followed by the semi-circle CR of radius R traversed CCW (see figure below). Note that C is a positively oriented, simple, closed contour. We will assume that R > 1. Next, notice that f(z) has two singular points (simple poles) inside C. Call them z0 and z1, as shown in the figure. By Cauchy's Residue Theorem. we have I f(z) dz = 2πi Res f(z) + Res f(z) C z=z0 z=z1 On the other hand, we can parametrize the line segment LR by z = x; −R ≤ x ≤ R, so that I Z R x2 Z z2 f(z) dz = 4 dx + 4 dz; C −R x + 1 CR z + 1 since C = LR + CR.
    [Show full text]
  • Math 336 Sample Problems
    Math 336 Sample Problems One notebook sized page of notes will be allowed on the test. The test will cover up to section 2.6 in the text. 1. Suppose that v is the harmonic conjugate of u and u is the harmonic conjugate of v. Show that u and v must be constant. ∞ X 2 P∞ n 2. Suppose |an| converges. Prove that f(z) = 0 anz is analytic 0 Z 2π for |z| < 1. Compute lim |f(reit)|2dt. r→1 0 z − a 3. Let a be a complex number and suppose |a| < 1. Let f(z) = . 1 − az Prove the following statments. (a) |f(z)| < 1, if |z| < 1. (b) |f(z)| = 1, if |z| = 1. 2πij 4. Let zj = e n denote the n roots of unity. Let cj = |1 − zj| be the n − 1 chord lengths from 1 to the points zj, j = 1, . .n − 1. Prove n that the product c1 · c2 ··· cn−1 = n. Hint: Consider z − 1. 5. Let f(z) = x + i(x2 − y2). Find the points at which f is complex differentiable. Find the points at which f is complex analytic. 1 6. Find the Laurent series of the function z in the annulus D = {z : 2 < |z − 1| < ∞}. 1 Sample Problems 2 7. Using the calculus of residues, compute Z +∞ dx 4 −∞ 1 + x n p0(z) Y 8. Let f(z) = , where p(z) = (z −z ) and the z are distinct and zp(z) j j j=1 different from 0. Find all the poles of f and compute the residues of f at these poles.
    [Show full text]
  • A Formal Proof of Cauchy's Residue Theorem
    A Formal Proof of Cauchy's Residue Theorem Wenda Li and Lawrence C. Paulson Computer Laboratory, University of Cambridge fwl302,[email protected] Abstract. We present a formalization of Cauchy's residue theorem and two of its corollaries: the argument principle and Rouch´e'stheorem. These results have applications to verify algorithms in computer alge- bra and demonstrate Isabelle/HOL's complex analysis library. 1 Introduction Cauchy's residue theorem | along with its immediate consequences, the ar- gument principle and Rouch´e'stheorem | are important results for reasoning about isolated singularities and zeros of holomorphic functions in complex anal- ysis. They are described in almost every textbook in complex analysis [3, 15, 16]. Our main motivation of this formalization is to certify the standard quantifier elimination procedure for real arithmetic: cylindrical algebraic decomposition [4]. Rouch´e'stheorem can be used to verify a key step of this procedure: Collins' projection operation [8]. Moreover, Cauchy's residue theorem can be used to evaluate improper integrals like Z 1 itz e −|tj 2 dz = πe −∞ z + 1 Our main contribution1 is two-fold: { Our machine-assisted formalization of Cauchy's residue theorem and two of its corollaries is new, as far as we know. { This paper also illustrates the second author's achievement of porting major analytic results, such as Cauchy's integral theorem and Cauchy's integral formula, from HOL Light [12]. The paper begins with some background on complex analysis (Sect. 2), fol- lowed by a proof of the residue theorem, then the argument principle and Rouch´e'stheorem (3{5).
    [Show full text]
  • Lecture Note for Math 220A Complex Analysis of One Variable
    Lecture Note for Math 220A Complex Analysis of One Variable Song-Ying Li University of California, Irvine Contents 1 Complex numbers and geometry 2 1.1 Complex number field . 2 1.2 Geometry of the complex numbers . 3 1.2.1 Euler's Formula . 3 1.3 Holomorphic linear factional maps . 6 1.3.1 Self-maps of unit circle and the unit disc. 6 1.3.2 Maps from line to circle and upper half plane to disc. 7 2 Smooth functions on domains in C 8 2.1 Notation and definitions . 8 2.2 Polynomial of degree n ...................... 9 2.3 Rules of differentiations . 11 3 Holomorphic, harmonic functions 14 3.1 Holomorphic functions and C-R equations . 14 3.2 Harmonic functions . 15 3.3 Translation formula for Laplacian . 17 4 Line integral and cohomology group 18 4.1 Line integrals . 18 4.2 Cohomology group . 19 4.3 Harmonic conjugate . 21 1 5 Complex line integrals 23 5.1 Definition and examples . 23 5.2 Green's theorem for complex line integral . 25 6 Complex differentiation 26 6.1 Definition of complex differentiation . 26 6.2 Properties of complex derivatives . 26 6.3 Complex anti-derivative . 27 7 Cauchy's theorem and Morera's theorem 31 7.1 Cauchy's theorems . 31 7.2 Morera's theorem . 33 8 Cauchy integral formula 34 8.1 Integral formula for C1 and holomorphic functions . 34 8.2 Examples of evaluating line integrals . 35 8.3 Cauchy integral for kth derivative f (k)(z) . 36 9 Application of the Cauchy integral formula 36 9.1 Mean value properties .
    [Show full text]
  • Residue Thry
    17 Residue Theory “Residue theory” is basically a theory for computing integrals by looking at certain terms in the Laurent series of the integrated functions about appropriate points on the complex plane. We will develop the basic theorem by applying the Cauchy integral theorem and the Cauchy integral formulas along with Laurent series expansions of functions about the singular points. We will then apply it to compute many, many integrals that cannot be easily evaluated otherwise. Most of these integrals will be over subintervals of the real line. 17.1 Basic Residue Theory The Residue Theorem Suppose f is a function that, except for isolated singularities, is single-valued and analytic on some simply-connected region R . Our initial interest is in evaluating the integral f (z) dz . C I 0 where C0 is a circle centered at a point z0 at which f may have a pole or essential singularity. We will assume the radius of C0 is small enough that no other singularity of f is on or enclosed by this circle. As usual, we also assume C0 is oriented counterclockwise. In the region right around z0 , we can express f (z) as a Laurent series ∞ k f (z) ak (z z0) , = − k =−∞X and, as noted earlier somewhere, this series converges uniformly in a region containing C0 . So ∞ k ∞ k f (z) dz ak (z z0) dz ak (z z0) dz . C = C − = C − 0 0 k k 0 I I =−∞X =−∞X I But we’ve seen k (z z0) dz for k 0, 1, 2, 3,..
    [Show full text]
  • Laurent Series and Residue Calculus
    Laurent Series and Residue Calculus Nikhil Srivastava March 19, 2015 If f is analytic at z0, then it may be written as a power series: 2 f(z) = a0 + a1(z − z0) + a2(z − z0) + ::: which converges in an open disk around z0. In fact, this power series is simply the Taylor series of f at z0, and its coefficients are given by I 1 (n) 1 f(z) an = f (z0) = n+1 ; n! 2πi (z − z0) where the latter equality comes from Cauchy's integral formula, and the integral is over a positively oriented contour containing z0 contained in the disk where it f(z) is analytic. The existence of this power series is an extremely useful characterization of f near z0, and from it many other useful properties may be deduced (such as the existence of infinitely many derivatives, vanishing of simple closed contour integrals around z0 contained in the disk of convergence, and many more). The situation is not much worse when z0 is an isolated singularity of f, i.e., f(z) is analytic in a puncured disk 0 < jz − z0j < r for some r. In this case, we have: Laurent's Theorem. If z0 is an isolated singularity of f and f(z) is analytic in the annulus 0 < jz − z0j < r, then 2 b1 b2 bn f(z) = a0 + a1(z − z0) + a2(z − z0) + ::: + + 2 + ::: + n + :::; (∗) z − z0 (z − z0) (z − z0) where the series converges absolutely in the annulus. In class I described how this can be done for any annulus, but the most useful case is a punctured disk around an isolated singularity.
    [Show full text]
  • An Extension of Hewitt's Inversion Formula and Its Application to Fluctuation Theory
    AN EXTENSION OF HEWITT’S INVERSION FORMULA AND ITS APPLICATION TO FLUCTUATION THEORY S¸erban E. B˘adil˘a1 Abstract. We analyze fluctuations of random walks with generally distributed increments. Integral representations for key performance measures are obtained by extending an inversion theorem of Hewitt [11] for Laplace-Stieltjes transforms. Another important step in the analysis involves introducing the so-called harmonic measures associated to the walk. It is also pointed out that such representations can be explicitly calculated, if one assumes a form of rational structure for the increment transform. Applications include, but are not restricted to, queueing and insurance risk problems. Keywords: fluctuations of random walks, harmonic analysis, singular integrals 2010 Mathematics Subject Classification: Primary 60G50 Secondary 30E20 1. Introduction A standard assumption in the theory of fluctuations of random walks as they appear in, e.g., queueing or insurance applications, is that the increment of the random walk can be represented as the difference of two independent random variables. In the context of queueing theory, random walks with increments which do not have this property appear as embedded at arrival epochs of customers in a single server queue in which the service requirement of the current customer is correlated with the time until the next arrival. In risk reserve processes that appear in insurance, the independence assumption is violated when the current claim size depends on the time elapsed since the previous arrival, and hence on the premium gained meanwhile. The purpose of the present paper is to show how much can be done for random walks which do not satisfy this independence assumption, regarding their maxima (as in the waiting time/maximum aggregate loss), their minima (idle periods/deficit at ruin) and their excur- arXiv:1508.00751v1 [math.PR] 4 Aug 2015 sions which are related to busy periods or the time to ruin.
    [Show full text]
  • 21. the Residue Theorem Definition 21.1. Let F Be a Holomorphic Function with an Isolated Singularity at A. the Residue of F At
    21. The residue theorem Definition 21.1. Let f be a holomorphic function with an isolated singularity at a. The residue of f at a, denoted Resa f(z), is equal to a−1, the coef- ficient of 1=(z − a) in the Laurent series expansion of f in a punctured neighbourhood of a. Theorem 21.2 (Residue Theorem). Let U be a bounded domain with piecewise differentiable boundary and let f be a holomorphic function on U [@U except at finitely many isolated singular points a1; a2; : : : ; an belonging to U. We have n Z X f(z) dz = 2πi Resai f(z): @U j=1 Before we prove this theorem, we give some applications. Example 21.3. Compute: I 1 dz jzj=3 (z − 1)(z + 1) The function 1 (z − 1)(z + 1) has isolated singularities at z = 1 and z = −1 and is otherwise holo- morphic. Both of these points belong to the open unit disk of radius 3 centred at 0. The boundary of this disk is the circle of radius 3, centred at 0. So we have to compute the residue at these two points and then apply the residue theorem. We use the method of partial fractions 1 1 1 1 1 = − : (z − 1)(z + 1) 2 z − 1 2 z + 1 We have 1 1 Res = : 1 (z − 1)(z + 1) 2 and 1 1 Res = − : −1 (z − 1)(z + 1) 2 Thus I 1 dz = 0; jzj=3 (z − 1)(z + 1) by the residue theorem. 1 To apply the residue theorem, we need to be able to compute residues efficiently.
    [Show full text]
  • 12 Lecture 12: Holomorphic Functions
    12 Lecture 12: Holomorphic functions For the remainder of this course we will be thinking hard about how the following theorem allows one to explicitly evaluate a large class of Fourier transforms. This will enable us to write down explicit solutions to a large class of ODEs and PDEs. The Cauchy Residue Theorem: Let C ⊂ C be a simple closed contour. Let f : C ! C be a complex function which is holomorphic along C and inside C except possibly at a finite num- ber of points a1; : : : ; am at which f has a pole. Then, with C oriented in an anti-clockwise sense, m Z X f(z) dz = 2πi res(f; ak); (12.1) C k=1 where res(f; ak) is the residue of the function f at the pole ak 2 C. You are probably not yet familiar with the meaning of the various components in the statement of this theorem, in particular the underlined terms and what R is meant by the contour integral C f(z) dz, and so our first task will be to explain the terminology. The Cauchy Residue theorem has wide application in many areas of pure and applied mathematics, it is a basic tool both in engineering mathematics and also in the purest parts of geometric analysis. The idea is that the right-side of (12.1), which is just a finite sum of complex numbers, gives a simple method for evaluating the contour integral; on the other hand, sometimes one can play the reverse game and use an `easy' contour integral and (12.1) to evaluate a difficult infinite sum (allowing m ! 1).
    [Show full text]
  • Lecture #31: the Cauchy Residue Theorem
    Mathematics312(Fall2013) November25,2013 Prof. Michael Kozdron Lecture #31: The Cauchy Residue Theorem Recall that last class we showed that a function f(z)hasapoleoforderm at z0 if and only if g(z) f(z)= (z z )m − 0 for some function g(z)thatisanalyticinaneighbourhoodofz and has g(z ) =0.Wealso 0 0 derived a formula for Res(f; z0). Theorem 31.1. If f(z) is analytic for 0 < z z0 <Rand has a pole of order m at z0, then | − | m 1 m 1 1 d − m 1 d − m Res(f; z0)= m 1 (z z0) f(z) = lim m 1 (z z0) f(z). (m 1)! dz − − (m 1)! z z0 dz − − z=z0 → − − In particular, if z0 is a simple pole, then Res(f; z0)=(z z0)f(z) =lim(z z0)f(z). − z z0 − z=z0 → Example 31.2. Suppose that sin z f(z)= . (z2 1)2 − Determine the order of the pole at z0 =1. Solution. Observe that z2 1=(z 1)(z +1)andso − − sin z sin z sin z/(z +1)2 f(z)= = = . (z2 1)2 (z 1)2(z +1)2 (z 1)2 − − − Since sin z g(z)= (z +1)2 2 is analytic at 1 and g(1) = 2− sin(1) =0,weconcludethatz =1isapoleoforder2. 0 Example 31.3. Determine the residue at z0 =1of sin z f(z)= (z2 1)2 − and compute f(z)dz C where C = z 1 =1/2 is the circle of radius 1/2centredat1orientedcounterclockwise. {| − | } 31–1 Solution. Since we can write (z 1)2f(z)=g(z)where − sin z g(z)= (z +1)2 is analytic at z =1withg(1) =0,theresidueoff(z)atz =1is 0 0 1 d2 1 d Res(f;1)= − (z 1)2f(z) = (z 1)2f(z) (2 1)! dz2 1 − dz − − − z=1 z=1 d sin z = dz (z +1)2 z=1 (z +1)2 cosz 2(z +1)sinz = − (z +1)4 z=1 4cos1 4sin1 = − 16 cos 1 sin 1 = − .
    [Show full text]
  • Calculus of Complex Functions. Laurent Series and Residue Theorem
    Math Methods I Lia Vas Calculus of Complex functions. Laurent Series and Residue Theorem Review of complex numbers.A complex number is any expression of the form x+iy wherep x and y are real numbers, called the real part and the imaginary part of x + iy; and i is −1: Thus, i2 = −1. Other powers of i can be determined using the relation i2 = −1: For example, i3 = i2i = −i and i10 = (i2)5 = (−1)5 = −1: Two complex numbers can be added by adding similar terms and they can be multiplied by foiling. The complex number x − iy is said to be complex conjugate of the number x + iy: To find the quotient of two complex numbers, one multiplies both the numerator and the denominator by the complex conjugate of the denominator. In this way, the answer is again in x + iy form. Example 1. Find the sum, product and quotient of 2 + i and 3 − 4i: Solutions. The sum is 2 + i + 3 − 4i = 2 + 3 + i − 4i = 5 − 3i: The product is (2 + i)(3 − 4i) = 6 − 8i + 3i − 4i2 = 6 − 8i + 3i + 4 = 10 − 5i: The complex number 3 + 4i is the conjugate of 3 − 4i; so the quotient is 2 + i 2 + i 3 + 4i (2 + i)(3 + 4i) 6 + 8i + 3i − 4 2 + 11i 2 11 = = = = = + i 3 − 4i 3 − 4i 3 + 4i (3 − 4i)(3 + 4i) 9 + 12i − 12i + 16 25 25 25 Trigonometric Representations. Let us recall the polar coordinates x = r cos θ and y = r sin θ: Using this representation, we have that z = x + iy = r cos θ + ir sin θ: The value r is the distance from the point (x; y) in the plane to the origin.
    [Show full text]