DOCSLIB.ORG

4 Laurent's Theorem for Complex Functions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

LAURENT’S THEOREM FOR COMPLEX FUNCTIONS 487 4 LAURENT’S THEOREM FOR COMPLEX FUNCTIONS 1 Consider the function f : C n f2g ! C given by f(z) = z¡2 . By Taylor’s theorem, f has a Taylor series centered at z0 = 0 with neighborhood of con- vergence N2(0). That is, 1 X f (j)(0) f(z) = zj on N (0). j! 2 j=0 But this function f is also defined for jzj > 2, so it is natural to ask if f can be represented by some (other) series expansion centered at z0 = 0 and convergent for jzj > 2. To explore this, note that à ! 1 1 1 1 1 1 X ³2´j X 2j f(z) = = = = , z ¡ 2 z 1 ¡ 2 z z zj+1 z j=0 j=0 ¯ ¯ ¯ 2 ¯ and this series converges for z < 1, i.e., for jzj > 2. This type of power series, which involves powers of 1 , or equivalently, negative powers of z¡z0 (z ¡ z0), is known as a Laurent series. More generally, the form of a Laurent series centered at a point z0 involves both positive and negative powers of (z ¡ z0), that is, X1 X1 b f(z) = a (z ¡ z )j + j , (9.21) j 0 (z ¡ z )j j=0 j=1 0 and so it consists of two infinite series. The first series in (9.21) is often referred to as the analytic part and the second series in (9.21) is often referred to as the singular part, or the principal part of the Laurent series. For the Laurent series expansion to exist at a particular z 2 C, both the analytic part and the singular part must be convergent at z. In fact, the analytic part consisting of positive powers of (z ¡ z0) will converge for all z inside some circle of radius R, while the singular part consisting of negative powers of (z ¡ z0) will converge for all z outside some circle of radius r, as in our example. It is the overlap of the two regions of convergence associated with the analytic part and the singular part that comprises the region of convergence of the Laurent series. This region of overlap is typically the annulus centered at z0 R and denoted by Ar (z0) = fz 2 C : r < jz ¡ z0j < Rg. See Figure 9.3 for an illustration. The situation R = 1 and r = 0 corresponds to the case where the Laurent series converges everywhere in C except possibly at z0. (In our 1 particular example, we have convergence in the annulus A2 (0).) Note that since the annulus so described excludes the point z0, it is not necessary that f be differentiable at z0 for it to have a Laurent series expansion centered there. (In fact, the function f need not be differentiable anywhere within Nr(z0).) R1 R If 0 < r < r1 < R1 < R we will refer to the annulus Ar1 (z0) ½ Ar (z0) R as a proper subannulus of Ar (z0). In a manner similar to that of a convergent 488 TAYLOR SERIES, LAURENT SERIES, AND THE RESIDUE CALCULUS R Ar (z0) r R z0 Figure 9.3 The region of convergence of a Laurent series. Taylor series on subsets of its neighborhood of convergence, when a function R f has a convergent Laurent series expansion on an annulus Ar (z0), it will R R1 R1 converge absolutely on Ar (z0) and uniformly on Ar1 (z0) where Ar1 (z0) is R any proper subannulus of Ar (z0). To establish this, we must consider the convergence of a given Laurent series a bit more carefully. Since the analytic part of the Laurent series is a power series, it will converge absolutely on its neighborhood of convergence NR(z0). Recall that the convergence is uniform on NR1 (z0) for any 0 < R1 < R. To analyze the singular part, we define ½ ¾ X1 b r ´ inf jz ¡ z j : j converges . 0 (z ¡ z )j j=1 0 If r fails to exist then the singular part never converges. If r exists then we C will show that the singular part converges absolutely for (Nr(z0)) , and uni- C formly on (Nr1 (z0)) for any r1 > r. To see this, choose r1 > r. Then as indi- P1 bj cated in Figure 9.4 there exists z1 such that r · jz1¡z0j < r1 and j j=1 (z1¡z0) converges. Therefore, there exists M ¸ 0 such that ¯ ¯ ¯ ¯ ¯ bj ¯ ¯ j ¯ · M for all j ¸ 1. (z1 ¡ z0) C Now, for z 2 (Nr1 (z0)) we have ¯ ¯ ¯ ¯ µ ¶ µ ¶ ¯ ¯ ¯ ¯ j j ¯ bj ¯ ¯ bj ¯ jz1 ¡ z0j jz1 ¡ z0j ¯ j ¯ = ¯ j ¯ · M . (z ¡ z0) (z1 ¡ z0) jz ¡ z0j r1 P1 bj Since jz1¡z0j < r1 this establishes via the Weierstrass M-test that j j=1 (z¡z0) C C converges absolutely on (Nr(z0)) and uniformly on (Nr1 (z0)) . Since r1 > r was arbitrary, this result holds for any r1 > r. We leave it to the reader to P1 bj show that j diverges on Nr(z0). As with a Taylor series, the points j=1 (z¡z0) on the boundary Cr(z0) must be studied individually. LAURENT’S THEOREM FOR COMPLEX FUNCTIONS 489 r r z0 1 Figure 9.4 The regions of convergence and divergence of the singular part of a Laurent series. P1 bj I 9.42 As claimed above, show that j diverges on Nr(z0). j=1 (z¡z0) The above discussion and exercise establish the following result. Proposition 4.1 Suppose f : D ! C has a Laurent series expansion f(z) = P1 j P1 bj R aj(z ¡ z0) + j on the annulus A (z0) ½ D (where r ¸ 0 and R j=0 j=1 (z¡z0) r R1 R may be 1). Then for any proper subannulus Ar1 (z0) ½ Ar (z0) the given Laurent R1 R1 series expansion for f converges absolutely on Ar1 (z0) and uniformly on Ar1 (z0). This result, in turn, implies the following. It will be instrumental in proving part of the key result of this section. P1 bj Proposition 4.2 Let j be the singular part of a Laurent series expan- j=1 (z¡z0) R P1 bj sion for f : D ! C on A (z0) ½ D. Then j represents a continuous r j=1 (z¡z0) C C function on (Nr(z0)) , and for any contour C ½ (Nr(z0)) we have Z µ 1 ¶ 1 Z X bj X bj j dz = j dz. (z ¡ z0) (z ¡ z0) C j=1 j=1 C I 9.43 Prove the above proposition. It is also true, and of great practical value as we will see, that when a Laurent series expansion exists for a function it is unique. To see this, suppose f is R differentiable on the annulus Ar (z0) and suppose too that it has a convergent Laurent series there given by 1 1 X X b f(z) = a (z ¡ z )k + k . (9.22) k 0 (z ¡ z )k k=0 k=1 0 490 TAYLOR SERIES, LAURENT SERIES, AND THE RESIDUE CALCULUS Recall that we established the uniqueness of the Taylor series representation of a function centered at a point by differentiating the series term-by-term; here, we will establish the uniqueness of the Laurent series representation by R integrating term-by-term. Let C be any simple closed contour in Ar (z0) with R1 R nC (z0) = 1, and note that C ½ Ar1 (z0) ½ Ar (z0) for some proper subannulus R1 R Ar1 (z0) ½ Ar (z0). Also note that for any k ¸ 0, I I 1 I 1 f(³) X X b d³ = a (³ ¡ z )j¡k¡1 d³ + j d³ k+1 j 0 j+k+1 (³ ¡ z0) (³ ¡ z0) C C j=0 C j=1 I I X1 X1 b = a (³ ¡ z )j¡k¡1 d³ + j d³ (9.23) j 0 j+k+1 (³ ¡ z0) j=0 C j=1 C = 2¼i ak nC (z0) = 2¼i ak. Note in (9.23) above that the integral of each summand corresponding to aj for j 6= k, and for bj for j ¸ 1, vanishes due to the integrand having an antiderivative within NR1 (z0). Also, in integrating the singular part term- by-term, we have applied Proposition 4.2. This shows that each ak in (9.22) is uniquely determined by f and z0. A similar argument can be made for the uniqueness of each b for k ¸ 1 in (9.22) by considering the integral H k f(³) d³ for any fixed k ¸ 1. We leave this to the reader. (³¡z )¡k+1 C 0 I 9.44 Establish the uniqueness of the bk terms in the Laurent series representation H f(³) (9.22) by carrying out the integral ¡k+1 d³ for any fixed k ¸ 1. (³¡z0) C With these important facts about an existing Laurent series established, we now state and prove Laurent’s theorem, the key result of this section. It is a kind of sibling to Taylor’s theorem in complex function theory. It gives conditions under which a function f : D ! C is guaranteed a Laurent series R representation convergent on an annulus Ar (z0) ½ D. Theorem 4.3 (Laurent’s Theorem) R Let f : D ! C be differentiable on the annulus Ar (z0) = fz : r < jz ¡ z0j < Rg ½ D (where r ¸ 0 and R may be 1). Then f(z) can be expressed uniquely by X1 X1 b f(z) = a (z ¡ z )j + j , j 0 (z ¡ z )j j=0 j=1 0 R R on Ar (z0).
Recommended publications
  • 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

    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.
  • Topic 7 Notes 7 Taylor and Laurent Series

    Topic 7 Notes 7 Taylor and Laurent Series

    Topic 7 Notes Jeremy Orloff 7 Taylor and Laurent series 7.1 Introduction We originally defined an analytic function as one where the derivative, defined as a limit of ratios, existed. We went on to prove Cauchy's theorem and Cauchy's integral formula. These revealed some deep properties of analytic functions, e.g. the existence of derivatives of all orders. Our goal in this topic is to express analytic functions as infinite power series. This will lead us to Taylor series. When a complex function has an isolated singularity at a point we will replace Taylor series by Laurent series. Not surprisingly we will derive these series from Cauchy's integral formula. Although we come to power series representations after exploring other properties of analytic functions, they will be one of our main tools in understanding and computing with analytic functions. 7.2 Geometric series Having a detailed understanding of geometric series will enable us to use Cauchy's integral formula to understand power series representations of analytic functions. We start with the definition: Definition. A finite geometric series has one of the following (all equivalent) forms. 2 3 n Sn = a(1 + r + r + r + ::: + r ) = a + ar + ar2 + ar3 + ::: + arn n X = arj j=0 n X = a rj j=0 The number r is called the ratio of the geometric series because it is the ratio of consecutive terms of the series. Theorem. The sum of a finite geometric series is given by a(1 − rn+1) S = a(1 + r + r2 + r3 + ::: + rn) = : (1) n 1 − r Proof.
  • Math 336 Sample Problems

    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.
  • Complex Analysis Class 24: Wednesday April 2

    Complex Analysis Class 24: Wednesday April 2

    Complex Analysis Math 214 Spring 2014 Fowler 307 MWF 3:00pm - 3:55pm c 2014 Ron Buckmire http://faculty.oxy.edu/ron/math/312/14/ Class 24: Wednesday April 2 TITLE Classifying Singularities using Laurent Series CURRENT READING Zill & Shanahan, §6.2-6.3 HOMEWORK Zill & Shanahan, §6.2 3, 15, 20, 24 33*. §6.3 7, 8, 9, 10. SUMMARY We shall be introduced to Laurent Series and learn how to use them to classify different various kinds of singularities (locations where complex functions are no longer analytic). Classifying Singularities There are basically three types of singularities (points where f(z) is not analytic) in the complex plane. Isolated Singularity An isolated singularity of a function f(z) is a point z0 such that f(z) is analytic on the punctured disc 0 < |z − z0| <rbut is undefined at z = z0. We usually call isolated singularities poles. An example is z = i for the function z/(z − i). Removable Singularity A removable singularity is a point z0 where the function f(z0) appears to be undefined but if we assign f(z0) the value w0 with the knowledge that lim f(z)=w0 then we can say that we z→z0 have “removed” the singularity. An example would be the point z = 0 for f(z) = sin(z)/z. Branch Singularity A branch singularity is a point z0 through which all possible branch cuts of a multi-valued function can be drawn to produce a single-valued function. An example of such a point would be the point z = 0 for Log (z).
  • Residue Theorem

    Residue Theorem

    Topic 8 Notes Jeremy Orloff 8 Residue Theorem 8.1 Poles and zeros f z z We remind you of the following terminology: Suppose . / is analytic at 0 and f z a z z n a z z n+1 ; . / = n. * 0/ + n+1. * 0/ + § a ≠ f n z n z with n 0. Then we say has a zero of order at 0. If = 1 we say 0 is a simple zero. f z Suppose has an isolated singularity at 0 and Laurent series b b b n n*1 1 f .z/ = + + § + + a + a .z * z / + § z z n z z n*1 z z 0 1 0 . * 0/ . * 0/ * 0 < z z < R b ≠ f n z which converges on 0 * 0 and with n 0. Then we say has a pole of order at 0. n z If = 1 we say 0 is a simple pole. There are several examples in the Topic 7 notes. Here is one more Example 8.1. z + 1 f .z/ = z3.z2 + 1/ has isolated singularities at z = 0; ,i and a zero at z = *1. We will show that z = 0 is a pole of order 3, z = ,i are poles of order 1 and z = *1 is a zero of order 1. The style of argument is the same in each case. At z = 0: 1 z + 1 f .z/ = ⋅ : z3 z2 + 1 Call the second factor g.z/. Since g.z/ is analytic at z = 0 and g.0/ = 1, it has a Taylor series z + 1 g.z/ = = 1 + a z + a z2 + § z2 + 1 1 2 Therefore 1 a a f .z/ = + 1 +2 + § : z3 z2 z This shows z = 0 is a pole of order 3.
  • A Formal Proof of Cauchy's Residue Theorem

    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).
  • Chapter 2 Complex Analysis

    Chapter 2 Complex Analysis

    Chapter 2 Complex Analysis In this part of the course we will study some basic complex analysis. This is an extremely useful and beautiful part of mathematics and forms the basis of many techniques employed in many branches of mathematics and physics. We will extend the notions of derivatives and integrals, familiar from calculus, to the case of complex functions of a complex variable. In so doing we will come across analytic functions, which form the centerpiece of this part of the course. In fact, to a large extent complex analysis is the study of analytic functions. After a brief review of complex numbers as points in the complex plane, we will ¯rst discuss analyticity and give plenty of examples of analytic functions. We will then discuss complex integration, culminating with the generalised Cauchy Integral Formula, and some of its applications. We then go on to discuss the power series representations of analytic functions and the residue calculus, which will allow us to compute many real integrals and in¯nite sums very easily via complex integration. 2.1 Analytic functions In this section we will study complex functions of a complex variable. We will see that di®erentiability of such a function is a non-trivial property, giving rise to the concept of an analytic function. We will then study many examples of analytic functions. In fact, the construction of analytic functions will form a basic leitmotif for this part of the course. 2.1.1 The complex plane We already discussed complex numbers briefly in Section 1.3.5.
  • Lecture Note for Math 220A Complex Analysis of One Variable

    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 .
  • Compex Analysis

    Compex Analysis

    ( ! ) 1. The amplitu.e of a comple2 number isisis a) 0 b) 89: c) 8 .) :8 2.The amplitu.e of the quotient of two comple2 numbers is a) the sum of their amplitu.es b) the .ifference of their amplitu.es c) the pro.uct of their amplitu.es .) the quotient of their amplitu.es 3. The ? = A B then ?C = : : : : : : a) AB b) A B c) DB .) imaginary J ? K ?: J 4. If ? I ?: are any two comple2 numbers, then J ?.?: J isisis J ? K ?: J J ? K ?: J J ? K ?: J J J J J a)a)a) J ?: J b) J ?. J c) J ?: JJ? J .) J ? J J ?: J : : 555.5...MNO?P:Q: A B R = a) S b) DS c) T .) U 666.The6.The equation J? A UJ A J? D UJ = W represents a) Circle b) ellipse c) parparabolaabola .) hyperbola 777.The7.The equation J? A ZJ = J? D ZJ represents a) Circle b) ellipse c) parparabolaabola .) Real a2is 888.The8.The equation J? A J = ]:J? D J represents a) Circle b) ellipse c) parparabolaabola .) hyperbola 999.The9.The equation J?A:J A J?D:J = U represents a) Circle b) a st. line c) parabola .) hyperbohyperbolalalala 101010.The10.The equation J:? D J = J?D:J represents a) Circle b) a st. line c) parabola .) hyperbola : : 111111.The11.The equation J? D J A J?D:J = U represents a) Circle b) a st. line c) parabola .) hyperbola ?a 121212.The12.The equation !_ `?a b = c represents a) Circle b) a st.
  • Lecture 18: Analytic Functions and Integrals

    Lecture 18: Analytic Functions and Integrals

    Lecture 17: Analytic Functions and Integrals (See Chapter 14 in Boas) This is a good point to take a brief detour and expand on our previous discussions of complex variables and complex functions of complex variables. In particular, we want to consider the properties of so-called analytic functions, especially those properties that allow us to perform integrals relevant to evaluating an inverse Fourier transform. More generally we want to motivate the process of thinking about the functions that appear in physics in terms of their properties in the complex plane (where “real life” is the real axis). In the complex plane we can more usefully define a well “behaved function”; we want both the function and its derivative to be well defined. Consider a complex valued function of a complex variable defined by F z U x,,, y iV x y (17.1) where UV, are real functions of the real variables xy, . The function Fz is said to be analytic (or regular or holomorphic) in a region of the complex plane if, at every (complex) point z in the region, it possesses a unique (finite) derivative that is independent of the direction in the complex plane along which we take the derivative. (This implies that there is always a, perhaps small, region around every point of analyticity within which the derivatives exist.) This property of the function is summarized by the Cauchy-Riemann conditions, i.e., a function is analytic/regular at the point z x iy if and only if (iff) UVVU , (17.2) x y x y at that point.
  • Residue Thry

    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,..
  • COMPLEX ANALYSIS Notes Lent 2006 T. K. Carne

    COMPLEX ANALYSIS Notes Lent 2006 T. K. Carne

    Department of Pure Mathematics and Mathematical Statistics University of Cambridge COMPLEX ANALYSIS Notes Lent 2006 T. K. Carne. [email protected] c Copyright. Not for distribution outside Cambridge University. CONTENTS 1. ANALYTIC FUNCTIONS 1 Domains 1 Analytic Functions 1 Cauchy – Riemann Equations 1 2. POWER SERIES 3 Proposition 2.1 Radius of convergence 3 Proposition 2.2 Power series are differentiable. 3 Corollary 2.3 Power series are infinitely differentiable 4 The Exponential Function 4 Proposition 2.4 Products of exponentials 5 Corollary 2.5 Properties of the exponential 5 Logarithms 6 Branches of the logarithm 6 Logarithmic singularity 6 Powers 7 Branches of powers 7 Branch point 7 Conformal Maps 8 3. INTEGRATION ALONG CURVES 9 Definition of curves 9 Integral along a curve 9 Integral with respect to arc length 9 Proposition 3.1 10 Proposition 3.2 Fundamental Theorem of Calculus 11 Closed curves 11 Winding Numbers 11 Definition of the winding number 11 Lemma 3.3 12 Proposition 3.4 Winding numbers under perturbation 13 Proposition 3.5 Winding number constant on each component 13 Homotopy 13 Definition of homotopy 13 Definition of simply-connected 14 Definition of star domains 14 Proposition 3.6 Winding number and homotopy 14 Chains and Cycles 14 4 CAUCHY’S THEOREM 15 Proposition 4.1 Cauchy’s theorem for triangles 15 Theorem 4.2 Cauchy’s theorem for a star domain 16 Proposition 4.10 Cauchy’s theorem for triangles 17 Theorem 4.20 Cauchy’s theorem for a star domain 18 Theorem 4.3 Cauchy’s Representation Formula 18 Theorem 4.4 Liouville’s theorem 19 Corollary 4.5 The Fundamental Theorem of Algebra 19 Homotopy form of Cauchy’s Theorem.