Chapter 10: Power Series
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Integration in the Complex Plane (Zill & Wright Chapter
Integration in the Complex Plane (Zill & Wright Chapter 18) 1016-420-02: Complex Variables∗ Winter 2012-2013 Contents 1 Contour Integrals 2 1.1 Definition and Properties . 2 1.2 Evaluation . 3 1.2.1 Example: R z¯ dz ............................. 3 C1 1.2.2 Example: R z¯ dz ............................. 4 C2 R 2 1.2.3 Example: C z dz ............................. 4 1.3 The ML Limit . 5 1.4 Circulation and Flux . 5 2 The Cauchy-Goursat Theorem 7 2.1 Integral Around a Closed Loop . 7 2.2 Independence of Path for Analytic Functions . 8 2.3 Deformation of Closed Contours . 9 2.4 The Antiderivative . 10 3 Cauchy's Integral Formulas 12 3.1 Cauchy's Integral Formula . 12 3.1.1 Example #1 . 13 3.1.2 Example #2 . 13 3.2 Cauchy's Integral Formula for Derivatives . 14 3.3 Consequences of Cauchy's Integral Formulas . 16 3.3.1 Cauchy's Inequality . 16 3.3.2 Liouville's Theorem . 16 ∗Copyright 2013, John T. Whelan, and all that 1 Tuesday 18 December 2012 1 Contour Integrals 1.1 Definition and Properties Recall the definition of the definite integral Z xF X f(x) dx = lim f(xk) ∆xk (1.1) ∆xk!0 xI k We'd like to define a similar concept, integrating a function f(z) from some point zI to another point zF . The problem is that, since zI and zF are points in the complex plane, there are different ways to get between them, and adding up the value of the function along one path will not give the same result as doing it along another path, even if they have the same endpoints. -
Class 1/28 1 Zeros of an Analytic Function
Math 752 Spring 2011 Class 1/28 1 Zeros of an analytic function Towards the fundamental theorem of algebra and its statement for analytic functions. Definition 1. Let f : G → C be analytic and f(a) = 0. a is said to have multiplicity m ≥ 1 if there exists an analytic function g : G → C with g(a) 6= 0 so that f(z) = (z − a)mg(z). Definition 2. If f is analytic in C it is called entire. An entire function has a power series expansion with infinite radius of convergence. Theorem 1 (Liouville’s Theorem). If f is a bounded entire function then f is constant. 0 Proof. Assume |f(z)| ≤ M for all z ∈ C. Use Cauchy’s estimate for f to obtain that |f 0(z)| ≤ M/R for every R > 0 and hence equal to 0. Theorem 2 (Fundamental theorem of algebra). For every non-constant polynomial there exists a ∈ C with p(a) = 0. Proof. Two facts: If p has degree ≥ 1 then lim p(z) = ∞ z→∞ where the limit is taken along any path to ∞ in C∞. (Sometimes also written as |z| → ∞.) If p has no zero, its reciprocal is therefore entire and bounded. Invoke Liouville’s theorem. Corollary 1. If p is a polynomial with zeros aj (multiplicity kj) then p(z) = k k km c(z − a1) 1 (z − a2) 2 ...(z − am) . Proof. Induction, and the fact that p(z)/(z − a) is a polynomial of degree n − 1 if p(a) = 0. 1 The zero function is the only analytic function that has a zero of infinite order. -
Informal Lecture Notes for Complex Analysis
Informal lecture notes for complex analysis Robert Neel I'll assume you're familiar with the review of complex numbers and their algebra as contained in Appendix G of Stewart's book, so we'll pick up where that leaves off. 1 Elementary complex functions In one-variable real calculus, we have a collection of basic functions, like poly- nomials, rational functions, the exponential and log functions, and the trig functions, which we understand well and which serve as the building blocks for more general functions. The same is true in one complex variable; in fact, the real functions we just listed can be extended to complex functions. 1.1 Polynomials and rational functions We start with polynomials and rational functions. We know how to multiply and add complex numbers, and thus we understand polynomial functions. To be specific, a degree n polynomial, for some non-negative integer n, is a function of the form n n−1 f(z) = cnz + cn−1z + ··· + c1z + c0; 3 where the ci are complex numbers with cn 6= 0. For example, f(z) = 2z + (1 − i)z + 2i is a degree three (complex) polynomial. Polynomials are clearly defined on all of C. A rational function is the quotient of two polynomials, and it is defined everywhere where the denominator is non-zero. z2+1 Example: The function f(z) = z2−1 is a rational function. The denomina- tor will be zero precisely when z2 = 1. We know that every non-zero complex number has n distinct nth roots, and thus there will be two points at which the denominator is zero. -
Calculus and Differential Equations II
Calculus and Differential Equations II MATH 250 B Sequences and series Sequences and series Calculus and Differential Equations II Sequences A sequence is an infinite list of numbers, s1; s2;:::; sn;::: , indexed by integers. 1n Example 1: Find the first five terms of s = (−1)n , n 3 n ≥ 1. Example 2: Find a formula for sn, n ≥ 1, given that its first five terms are 0; 2; 6; 14; 30. Some sequences are defined recursively. For instance, sn = 2 sn−1 + 3, n > 1, with s1 = 1. If lim sn = L, where L is a number, we say that the sequence n!1 (sn) converges to L. If such a limit does not exist or if L = ±∞, one says that the sequence diverges. Sequences and series Calculus and Differential Equations II Sequences (continued) 2n Example 3: Does the sequence converge? 5n 1 Yes 2 No n 5 Example 4: Does the sequence + converge? 2 n 1 Yes 2 No sin(2n) Example 5: Does the sequence converge? n Remarks: 1 A convergent sequence is bounded, i.e. one can find two numbers M and N such that M < sn < N, for all n's. 2 If a sequence is bounded and monotone, then it converges. Sequences and series Calculus and Differential Equations II Series A series is a pair of sequences, (Sn) and (un) such that n X Sn = uk : k=1 A geometric series is of the form 2 3 n−1 k−1 Sn = a + ax + ax + ax + ··· + ax ; uk = ax 1 − xn One can show that if x 6= 1, S = a . -
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 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. -
1 the Complex Plane
Math 135A, Winter 2012 Complex numbers 1 The complex numbers C are important in just about every branch of mathematics. These notes present some basic facts about them. 1 The Complex Plane A complex number z is given by a pair of real numbers x and y and is written in the form z = x+iy, where i satisfies i2 = −1. The complex numbers may be represented as points in the plane, with the real number 1 represented by the point (1; 0), and the complex number i represented by the point (0; 1). The x-axis is called the \real axis," and the y-axis is called the \imaginary axis." For example, the complex numbers 1, i, 3 + 4i and 3 − 4i are illustrated in Fig 1a. 3 + 4i 6 + 4i 2 + 3i imag i 4 + i 1 real 3 − 4i Fig 1a Fig 1b Complex numbers are added in a natural way: If z1 = x1 + iy1 and z2 = x2 + iy2, then z1 + z2 = (x1 + x2) + i(y1 + y2) (1) It's just vector addition. Fig 1b illustrates the addition (4 + i) + (2 + 3i) = (6 + 4i). Multiplication is given by z1z2 = (x1x2 − y1y2) + i(x1y2 + x2y1) Note that the product behaves exactly like the product of any two algebraic expressions, keeping in mind that i2 = −1. Thus, (2 + i)(−2 + 4i) = 2(−2) + 8i − 2i + 4i2 = −8 + 6i We call x the real part of z and y the imaginary part, and we write x = Re z, y = Im z.(Remember: Im z is a real number.) The term \imaginary" is a historical holdover; it took mathematicians some time to accept the fact that i (for \imaginary," naturally) was a perfectly good mathematical object. -
[Math.CA] 1 Jul 1992 Napiain.Freape Ecnlet Can We Example, for Applications
Convolution Polynomials Donald E. Knuth Computer Science Department Stanford, California 94305–2140 Abstract. The polynomials that arise as coefficients when a power series is raised to the power x include many important special cases, which have surprising properties that are not widely known. This paper explains how to recognize and use such properties, and it closes with a general result about approximating such polynomials asymptotically. A family of polynomials F (x), F (x), F (x),... forms a convolution family if F (x) has degree n 0 1 2 n ≤ and if the convolution condition F (x + y)= F (x)F (y)+ F − (x)F (y)+ + F (x)F − (y)+ F (x)F (y) n n 0 n 1 1 · · · 1 n 1 0 n holds for all x and y and for all n 0. Many such families are known, and they appear frequently ≥ n in applications. For example, we can let Fn(x)= x /n!; the condition (x + y)n n xk yn−k = n! k! (n k)! kX=0 − is equivalent to the binomial theorem for integer exponents. Or we can let Fn(x) be the binomial x coefficient n ; the corresponding identity n x + y x y = n k n k Xk=0 − is commonly called Vandermonde’s convolution. How special is the convolution condition? Mathematica will readily find all sequences of polynomials that work for, say, 0 n 4: ≤ ≤ F[n_,x_]:=Sum[f[n,j]x^j,{j,0,n}]/n! arXiv:math/9207221v1 [math.CA] 1 Jul 1992 conv[n_]:=LogicalExpand[Series[F[n,x+y],{x,0,n},{y,0,n}] ==Series[Sum[F[k,x]F[n-k,y],{k,0,n}],{x,0,n},{y,0,n}]] Solve[Table[conv[n],{n,0,4}], [Flatten[Table[f[i,j],{i,0,4},{j,0,4}]]]] Mathematica replies that the F ’s are either identically zero or the coefficients of Fn(x) = fn0 + f x + f x2 + + f xn /n! satisfy n1 n2 · · · nn f00 = 1 , f10 = f20 = f30 = f40 = 0 , 2 3 f22 = f11 , f32 = 3f11f21 , f33 = f11 , 2 2 4 f42 = 4f11f31 + 3f21 , f43 = 6f11f21 , f44 = f11 . -
Arxiv:1207.1472V2 [Math.CV]
SOME SIMPLIFICATIONS IN THE PRESENTATIONS OF COMPLEX POWER SERIES AND UNORDERED SUMS OSWALDO RIO BRANCO DE OLIVEIRA Abstract. This text provides very easy and short proofs of some basic prop- erties of complex power series (addition, subtraction, multiplication, division, rearrangement, composition, differentiation, uniqueness, Taylor’s series, Prin- ciple of Identity, Principle of Isolated Zeros, and Binomial Series). This is done by simplifying the usual presentation of unordered sums of a (countable) family of complex numbers. All the proofs avoid formal power series, double series, iterated series, partial series, asymptotic arguments, complex integra- tion theory, and uniform continuity. The use of function continuity as well as epsilons and deltas is kept to a mininum. Mathematics Subject Classification: 30B10, 40B05, 40C15, 40-01, 97I30, 97I80 Key words and phrases: Power Series, Multiple Sequences, Series, Summability, Complex Analysis, Functions of a Complex Variable. Contents 1. Introduction 1 2. Preliminaries 2 3. Absolutely Convergent Series and Commutativity 3 4. Unordered Countable Sums and Commutativity 5 5. Unordered Countable Sums and Associativity. 9 6. Sum of a Double Sequence and The Cauchy Product 10 7. Power Series - Algebraic Properties 11 8. Power Series - Analytic Properties 14 References 17 arXiv:1207.1472v2 [math.CV] 27 Jul 2012 1. Introduction The objective of this work is to provide a simplification of the theory of un- ordered sums of a family of complex numbers (in particular, for a countable family of complex numbers) as well as very easy proofs of basic operations and properties concerning complex power series, such as addition, scalar multiplication, multipli- cation, division, rearrangement, composition, differentiation (see Apostol [2] and Vyborny [21]), Taylor’s formula, principle of isolated zeros, uniqueness, principle of identity, and binomial series. -
Formal Power Series - Wikipedia, the Free Encyclopedia
Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series -
The Integral Resulting in a Logarithm of a Complex Number Dr. E. Jacobs in first Year Calculus You Learned About the Formula: Z B 1 Dx = Ln |B| − Ln |A| a X
The Integral Resulting in a Logarithm of a Complex Number Dr. E. Jacobs In first year calculus you learned about the formula: Z b 1 dx = ln jbj ¡ ln jaj a x It is natural to want to apply this formula to contour integrals. If z1 and z2 are complex numbers and C is a path connecting z1 to z2, we would expect that: Z 1 dz = log z2 ¡ log z1 C z However, you are about to see that there is an interesting complication when we take the formula for the integral resulting in logarithm and try to extend it to the complex case. First of all, recall that if f(z) is an analytic function on a domain D and C is a closed loop in D then : I f(z) dz = 0 This is referred to as the Cauchy-Integral Theorem. So, for example, if C His a circle of radius r around the origin we may conclude immediately that 2 2 C z dz = 0 because z is an analytic function for all values of z. H If f(z) is not analytic then C f(z) dz is not necessarily zero. Let’s do an 1 explicit calculation for f(z) = z . If we are integrating around a circle C centered at the origin, then any point z on this path may be written as: z = reiθ H 1 iθ Let’s use this to integrate C z dz. If z = re then dz = ireiθ dθ H 1 Now, substitute this into the integral C z dz. -
Complex Sinusoids Copyright C Barry Van Veen 2014
Complex Sinusoids Copyright c Barry Van Veen 2014 Feel free to pass this ebook around the web... but please do not modify any of its contents. Thanks! AllSignalProcessing.com Key Concepts 1) The real part of a complex sinusoid is a cosine wave and the imag- inary part is a sine wave. 2) A complex sinusoid x(t) = AejΩt+φ can be visualized in the complex plane as a vector of length A that is rotating at a rate of Ω radians per second and has angle φ relative to the real axis at time t = 0. a) The projection of the complex sinusoid onto the real axis is the cosine A cos(Ωt + φ). b) The projection of the complex sinusoid onto the imaginary axis is the cosine A sin(Ωt + φ). AllSignalProcessing.com 3) The output of a linear time invariant system in response to a com- plex sinusoid input is a complex sinusoid of the of the same fre- quency. The system only changes the amplitude and phase of the input complex sinusoid. a) Arbitrary input signals are represented as weighted sums of complex sinusoids of different frequencies. b) Then, the output is a weighted sum of complex sinusoids where the weights have been multiplied by the complex gain of the system. 4) Complex sinusoids also simplify algebraic manipulations. AllSignalProcessing.com 5) Real sinusoids can be expressed as a sum of a positive and negative frequency complex sinusoid. 6) The concept of negative frequency is not physically meaningful for real data and is an artifact of using complex sinusoids.