2. Contour Integration
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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. -
Section 8.8: Improper Integrals
Section 8.8: Improper Integrals One of the main applications of integrals is to compute the areas under curves, as you know. A geometric question. But there are some geometric questions which we do not yet know how to do by calculus, even though they appear to have the same form. Consider the curve y = 1=x2. We can ask, what is the area of the region under the curve and right of the line x = 1? We have no reason to believe this area is finite, but let's ask. Now no integral will compute this{we have to integrate over a bounded interval. Nonetheless, we don't want to throw up our hands. So note that b 2 b Z (1=x )dx = ( 1=x) 1 = 1 1=b: 1 − j − In other words, as b gets larger and larger, the area under the curve and above [1; b] gets larger and larger; but note that it gets closer and closer to 1. Thus, our intuition tells us that the area of the region we're interested in is exactly 1. More formally: lim 1 1=b = 1: b − !1 We can rewrite that as b 2 lim Z (1=x )dx: b !1 1 Indeed, in general, if we want to compute the area under y = f(x) and right of the line x = a, we are computing b lim Z f(x)dx: b !1 a ASK: Does this limit always exist? Give some situations where it does not exist. They'll give something that blows up. -
An Application of the Dirichlet Integrals to the Summation of Series
Some applications of the Dirichlet integrals to the summation of series and the evaluation of integrals involving the Riemann zeta function Donal F. Connon [email protected] 1 December 2012 Abstract Using the Dirichlet integrals, which are employed in the theory of Fourier series, this paper develops a useful method for the summation of series and the evaluation of integrals. CONTENTS Page 1. Introduction 1 2. Application of Dirichlet’s integrals 5 3. Some applications of the Poisson summation formula: 3.1 Digamma function 11 3.2 Log gamma function 13 3.3 Logarithm function 14 3.4 An integral due to Ramanujan 16 3.5 Stieltjes constants 36 3.6 Connections with other summation formulae 40 3.7 Hurwitz zeta function 48 4. Some integrals involving cot x 62 5. Open access to our own work 68 1. Introduction In two earlier papers [19] and [20] we considered the following identity (which is easily verified by multiplying the numerator and the denominator by the complex conjugate (1− e−ix ) ) 1 1 isin x 1 i (1.1) = − = + cot(x / 2) 1− eix 2 2 1− cos x 2 2 Using the basic geometric series identity 1 N y N +1 =∑ yn + 1− y n=0 1− y we obtain N 1 inx (1.1a) ix =∑e + RN () x 1− e n=0 1 iN()+ x iN(1)+ x 2 ei1 iN(1)+ x e where RxN () == e[]1 + i cot( x / 2) = 12− exix 2sin(/2) Separating the real and imaginary parts of (1.1a) produces the following two identities (the first of which is called Lagrange’s trigonometric identity and contains the Dirichlet kernel DxN () which is employed in the theory of Fourier series [55, p.49]) 1 N sin(N+ 1/ 2) x (1.2) = ∑cosnx − 2 n=0 2sin(x / 2) 1 N cos(N+ 1/ 2) x (1.3) cot(x / 2)=∑ sin nx + 2 n=0 2sin(x / 2) (and these equations may obviously be generalised by substituting αx in place of x . -
Math 1B, Lecture 11: Improper Integrals I
Math 1B, lecture 11: Improper integrals I Nathan Pflueger 30 September 2011 1 Introduction An improper integral is an integral that is unbounded in one of two ways: either the domain of integration is infinite, or the function approaches infinity in the domain (or both). Conceptually, they are no different from ordinary integrals (and indeed, for many students including myself, it is not at all clear at first why they have a special name and are treated as a separate topic). Like any other integral, an improper integral computes the area underneath a curve. The difference is foundational: if integrals are defined in terms of Riemann sums which divide the domain of integration into n equal pieces, what are we to make of an integral with an infinite domain of integration? The purpose of these next two lecture will be to settle on the appropriate foundation for speaking of improper integrals. Today we consider integrals with infinite domains of integration; next time we will consider functions which approach infinity in the domain of integration. An important observation is that not all improper integrals can be considered to have meaningful values. These will be said to diverge, while the improper integrals with coherent values will be said to converge. Much of our work will consist in understanding which improper integrals converge or diverge. Indeed, in many cases for this course, we will only ask the question: \does this improper integral converge or diverge?" The reading for today is the first part of Gottlieb x29:4 (up to the top of page 907). -
Arxiv:2007.02955V4 [Quant-Ph] 7 Jan 2021
Harvesting correlations in Schwarzschild and collapsing shell spacetimes Erickson Tjoa,a,b,1 Robert B. Manna,c aDepartment of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada bInstitute for Quantum Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada cPerimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada E-mail: [email protected], [email protected] Abstract: We study the harvesting of correlations by two Unruh-DeWitt static detectors from the vacuum state of a massless scalar field in a background Vaidya spacetime con- sisting of a collapsing null shell that forms a Schwarzschild black hole (hereafter Vaidya spacetime for brevity), and we compare the results with those associated with the three pre- ferred vacua (Boulware, Unruh, Hartle-Hawking-Israel vacua) of the eternal Schwarzschild black hole spacetime. To do this we make use of the explicit Wightman functions for a massless scalar field available in (1+1)-dimensional models of the collapsing spacetime and Schwarzschild spacetimes, and the detectors couple to the proper time derivative of the field. First we find that, with respect to the harvesting protocol, the Unruh vacuum agrees very well with the Vaidya vacuum near the horizon even for finite-time interactions. Sec- ond, all four vacua have different capacities for creating correlations between the detectors, with the Vaidya vacuum interpolating between the Unruh vacuum near the horizon and the Boulware vacuum far from the horizon. Third, we show that the black hole horizon inhibits any correlations, not just entanglement. Finally, we show that the efficiency of the harvesting protocol depend strongly on the signalling ability of the detectors, which is highly non-trivial in presence of curvature. -
Calculus Terminology
AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential -
Lecture 2 Complex Analysis
Complex Analysis Lecture 2 Complex Analysis A Complex Numbers and Complex Variables In this chapter we give a short discussion of complex numbers and the theory of a function of a complex variable. Before we get to complex numbers, let us first say a few words about real numbers. All real numbers have meanings in the real world. Ever since the beginning of civilization, people have found great use of real positive integers such as 2 and 30, which came up in, as an example, the sentence “my neighbor has two pigs, and I have thirty chickens”. The concept of negative real integers, say ?5, is a bit more difficult, but found its use when a person owed another person five ounces of silver. It was also natural to extend the concept of integers to numbers which are not integers. For example, when six persons share equally a melon, the number describing the fraction of melon each of them has is not an integer but the rational number 1/6. The need for other real numbers was found as mathematicians pondered the length of the circumference of a perfectly circular hole with unit radius. This length is a real number which can be expressed neither as an integer nor as a ratio of two integers. This real number is denoted as Z, and is called an irrational number. Another example of irrational number is e. Each of the real numbers, be it integral, rational or irrational, can be geometrically represented by a point on an infinite line, and vice versa. -
Dynamics of Contour Integration
Vision Research 41 (2001) 1023–1037 www.elsevier.com/locate/visres Dynamics of contour integration Robert F. Hess *, William H.A. Beaudot, Kathy T. Mullen Department of Ophthalmology, McGill Vision Research, McGill Uni6ersity, 687 Pine A6enue West (H4-14), Montre´al, Que´bec, Canada H3A 1A1 Received 3 May 2000; received in revised form 6 October 2000 Abstract To determine the dynamics of contour integration the temporal properties of the individual contour elements were varied as well as those of the contour they form. A temporal version of a contour integration paradigm (Field, D. J., Hayes, A., & Hess, R. F. (1993) Vision Research, 33, 173–193) was used to assess these two temporal dynamics as a function of the contrast of individual elements and the curvature of the contour. The results show that the dynamics of contour integration are good when the contrast of the individual elements is modulated in time (10–30 Hz), but are poor when contour linking per se is temporally modulated (1–12 Hz). The dynamics of contour linking is not dependent on the absolute contrast of the linking elements, so long as they are visible, but does vary with the curvature of the contour. For straight contours, temporal resolution is around 6–12 Hz but falls to around 1–2 Hz for curved contours. © 2001 Elsevier Science Ltd. All rights reserved. Keywords: Contour integration; Temporal sensitivity; Dynamics; Curvature; Contrast 1. Introduction columns are interconnected across disparate regions of cortex through long-range connections. There is now There is a growing awareness of the importance of neurophysiological evidence that these network interac- network interactions within the visual cortex as op- tions are context-dependent (e.g. -
Lightning Overview of Contour Integration Wednesday, ÕÉ October Óþõõ Physics ÕÕÕ
Lightning Overview of Contour Integration Wednesday, ÕÉ October óþÕÕ Physics ÕÕÕ e prettiest math I know, and extremely useful. Õ. Overview is handout summarizes key results in the theory of functions of a complex variable, leading to the inverse Laplace transform. I make no pretense at rigor, but I’ll do my best to ensure that I don’t say anything false. e main points are: Õ. e Cauchy-Riemann conditions, which must be satised by a dierentiable function f z of a complex variable z. ó. f z dz þ if f z is continuous and dierentiable everywhere inside the closed path.( ) ì. If∮ f (z) blows= up inside( ) a closed contour in the complex plane, then the value of f z dz is óπi times the sum of the residues of all the poles inside the contour. ( ) ∞ ikx ¦. e Fourier representation of the delta function is −∞ e dk óπδ x . ¢. ∮ e( inverse) Laplace transform equation. ∫ = ( ) ó. Cauchy-Riemann For the derivative of a function of a complex variable to exist, the value of the derivative must be independent of the manner in which it is taken. If z x iy is a complex variable, which we may wish to represent on the x y plane, and f z u x, y iυ x, y , then the d f = + value of should be the same at a point z whether we approach the point along a line parallel dz ( ) = ( ) + ( ) to the x axis or a line parallel to the y axis. erefore, d f ∂u ∂υ i (along x) dz ∂x ∂y d f = ∂u+ ∂υ ∂υ ∂u i i (along y) dz ∂ iy ∂ iy ∂y ∂y = + = − Comparing real and imaginary parts,( we) obtain( the) Cauchy-Riemann equations: ∂u ∂υ ∂υ ∂u and (Õ) ∂x ∂y ∂x ∂y = = − ì. -
Math 259: Introduction to Analytic Number Theory the Contour Integral Formula for Ψ(X)
Math 259: Introduction to Analytic Number Theory The contour integral formula for (x) We now have several examples of Dirichlet series, that is, series of the form1 1 F (s) = a n s (1) X n − n=1 from which we want to extract information about the growth of n<x an as x . The key to this is a contour integral. We regard F (s) asP a function of!1 a complex variable s = σ + it. For real y > 0 we have seen already that s σ 2 y− = y− . Thus if the sum (1) converges absolutely for some real σ0, then itj convergesj uniformly and absolutely to an analytic function on the half-plane Re(s) σ0; and if the sum converges absolutely for all real s > σ0, then it ≥ converges absolutely to an analytic function on the half-plane Re(s) > σ0. Now for y > 0 and c > 0 we have 1; if y > 1; 1 c+i ds 8 1 ys = 1 (2) Z <> 2 ; if y = 1; 2πi c i s − 1 0; if y < 1; :> in the following sense: the contour of integration is the vertical line Re(s) = c, and since the integral is then not absolutely convergent it is regarded as a principal value: c+i c+iT 1 Z f(s) ds := lim Z f(s) ds: c i T c iT − 1 !1 − Thus interpreted, (2) is an easy exercise in contour integration for y = 1, and an elementary manipulation of log s for y = 1. So we expect that if (1)6 converges absolutely in Re(s) > σ0 then c+i 1 1 s ds an = Z x F (s) (3) X 2πi c i s n<x − 1 for any c > σ0, using the principal value of the integral and adding ax=2 to the sum if x happens to be an integer. -
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. -
9.1 Contour Integral
THE CHINESE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH2230A (First term, 2015{2016) Complex Variables and Applications Notes 9 Contour Integration 9.1 Contour Integral Previously, we have learned how to find the harmonic conjugate of a given harmonic function u(x; y). In other words, we would like to find v(x; y) such that u+iv defines an analytic function. The process is to use the Cauchy-Riemann Equations successively. For example, we start with vy = ux = this is known as u is given; and obtain Z v = ux dy + '(x) : 0 Then, we use vx = −uy to get information about ' (x). Next, integrating wrt x will get the result. Observing from both analytically and geometrically, we see that the process corresponds to an integration along the vertical and then horizontal line in the picture below. Sometimes, in order to avoid holes in the domain Ω where u is defined, we may need to take more vertical and horizontal lines (dotted in picture). Nevertheless, one may expect the general theory should be related to integration along a curve. 9.1.1 Contours and Integrals Let us start with the building block of a contour. A smooth path γ in Ω ⊂ C is a continuous function γ :[a; b] ! Ω from an interval with γ(t) = x(t) + iy(t) such that x0(t) and y0(t) exist and are continuous for all t 2 (a; b). The function γ is called a parametrization of the path. It determines a \direction" for the path. We denote γ0(t) = x0(t) + iy0(t).