DOCSLIB.ORG

Homework 2 Posted, Due October 11. Introduce

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Homework 2 Posted, Due October 11. Introduce Complex exponential and logarithm Monday, September 30, 2013 1:59 PM Homework 2 posted, due October 11. Introduce complex-valued transcendental functions via the complex exponential defined as: where We'll take this for now as just the definition, and observe that it satisfies desired properties. Another tactic to define transcendental functions like exponential is via Taylor series; we'll emphasize this point of view later. We can see that the definition of exponential has the following properties: • Agrees with the real-valued definition of exponential on the real axis (y=0) • • and the complex exponential is analytic on the whole complex plane. The Principle of Analytic Continuation (which we'll discuss later) shows that the definition we gave is the unique one that has these properties. Remark: exp(z)=ez is a single-valued function, meaning that even if z=m/n (a fraction), then we don't interpret as a multivalued number when viewing as a function. The polar representation of the complex exponential is often convenient: ComplexAnalysis Page 1 We note in particular that ez does not vanish anywhere, but can be negative or positive. The usual algebraic operations on exponentials extend to the complex version: How does the complex exponential behave geometrically? ComplexAnalysis Page 2 This mapping is not one-to-one; it has the following periodicity: That is, the complex exponential is one-to-one when restricted to horizontal strips of the form: Let's now turn to defining a complex version of the logarithm, which is supposed to be an inverse function for the exponential. But this will be tricky because we've just seen that the complex exponential is not 1-1. Let's try to define the logarithm simply by standard inverse function considerations. w=log z (with natural logarithm implied despite the notation which might suggest base 10 logarithm) ComplexAnalysis Page 3 So then, what this implies is that the appropriate definition of logarithnm is: This is a multivalued function because arg z is a multivalued function. Note then how one should compute complex logarithms: As is typical for multivalued functions, one can define single-valued versions of them, at a cost… Principal value of the logarithm: Then: (aka the principal branch) While this principal value of the logarithm is a canonical single-valued version of the ComplexAnalysis Page 4 While this principal value of the logarithm is a canonical single-valued version of the multivalued logarithm, there is nothing particularly special or natural about it, and we will see that there are occasions to define other single-valued versions. With any single-valued version of a multivalued function comes the problem that an artificial discontinuity is introduced; here it's along the negative real axis. The multivalued function itself has no discontinuity there, or anywhere other than the origin. To reconcile these single-valued and multi-valued views of complex functions, it is useful to introduce the concept of Riemann surface, which is a topological representation of the multivalued function. The Riemann surface for the logarithm takes the form of a infinite-sheeted spiral surface glued together at the origin. To every Riemann surface can be associated a group, with the group action corresponding to a path that creates a change in the sheet. For the logarithm, the fundamental group (which is related to topology) is just the group of integers under addition. To work concretely with Riemann surfaces, it is helpful to introduce the following associated concepts: • Branch points: Any point in the (extended) complex plane about which arbitrarily close closed paths get mapped to non-closed paths. One can show that branch points are never places at which the function is analytic. In other words, branch points can only happen at ComplexAnalysis Page 5 places at which the function is analytic. In other words, branch points can only happen at "singularities" of functions (where they are not analytic). • Branch of a multivalued function is a single-valued restriction of the definition of the function. The specification of a branch will inevitably lead to the introduction of discontinuities, and these are referred to as branch cuts. Branch cuts generally speaking will be lines or arcs connecting branch points (so long as you think about infinity as being a possible branch point). Let's look at logarithm with these concepts. Branch points at 0 and at At a fundamental level, check this by simply verifying that small loops around these points (on the Riemann sphere). Other points are excluded from being branch points because either noting that log z is analytic elsewhere, or by just checking small loops around other points are mapped always to closed loops. To define a branch analytically, one begins by specifying a branch cut of choice. Not at all a unique choice; choose whatever is convenient, but for efficiency always try to have the branch cut be a path from one branch point to another branch point. So for log z, good choices of branch cuts are any curve that connects 0 and Why not make it straight for good measure? The point of introducing the branch cut is that one can no longer draw loops around the branch points without crossing a branch cut. This is the essential purpose of introducing branch cuts. Here we can define the branch cut in polar coordinates as ComplexAnalysis Page 6 A branch is a single-valued version of the multivalued function that is well-behaved (analytic) away from the branch cut: We can define a branch of log z using our branch cut as follows: with With Log z is simply the branch so defined with One can check that away from the branch points, log z is analytic. Can use Cauchy-Riemann, but a pain. Easier way to show this fact is through the Inverse Function Theorem If f(z) is an analytic function, then any branch of its inverse function g(w)= f-1 (w) is analytic away from branch cuts, branch points, and places where f'(g(w)) = 0. Away from such points, Proof is same as real-variable case, just have to modify. ComplexAnalysis Page 7 Proof is same as real-variable case, just have to modify. ComplexAnalysis Page 8 .
Load more

Recommended publications
  • Branch Points and Cuts in the Complex Plane
    BRANCH POINTS AND CUTS IN THE COMPLEX PLANE Link to: physicspages home page. To leave a comment or report an error, please use the auxiliary blog and include the title or URL of this post in your comment. Post date: 6 July 2021. We’ve looked at contour integration in the complex plane as a technique for evaluating integrals of complex functions and finding infinite integrals of real functions. In some cases, the complex functions that are to be integrated are multi- valued. As a preliminary to the contour integration of such functions, we’ll look at the concepts of branch points and branch cuts here. The stereotypical function that is used to introduce branch cuts in most books is the complex logarithm function logz which is defined so that elogz = z (1) If z is real and positive, this reduces to the familiar real logarithm func- tion. (Here I’m using natural logs, so the real natural log function is usually written as ln. In complex analysis, the term log is usually used, so be careful not to confuse it with base 10 logs.) To generalize it to complex numbers, we write z in modulus-argument form z = reiθ (2) and apply the usual rules for taking a log of products and exponentials: logz = logr + iθ (3) = logr + iargz (4) To see where problems arise, suppose we start with z on the positive real axis and increase θ. Everything is fine until θ approaches 2π. When θ passes 2π, the original complex number z returns to its starting value, as given by 2.
    [Show full text]
  • Riemann Surfaces
    RIEMANN SURFACES AARON LANDESMAN CONTENTS 1. Introduction 2 2. Maps of Riemann Surfaces 4 2.1. Defining the maps 4 2.2. The multiplicity of a map 4 2.3. Ramification Loci of maps 6 2.4. Applications 6 3. Properness 9 3.1. Definition of properness 9 3.2. Basic properties of proper morphisms 9 3.3. Constancy of degree of a map 10 4. Examples of Proper Maps of Riemann Surfaces 13 5. Riemann-Hurwitz 15 5.1. Statement of Riemann-Hurwitz 15 5.2. Applications 15 6. Automorphisms of Riemann Surfaces of genus ≥ 2 18 6.1. Statement of the bound 18 6.2. Proving the bound 18 6.3. We rule out g(Y) > 1 20 6.4. We rule out g(Y) = 1 20 6.5. We rule out g(Y) = 0, n ≥ 5 20 6.6. We rule out g(Y) = 0, n = 4 20 6.7. We rule out g(C0) = 0, n = 3 20 6.8. 21 7. Automorphisms in low genus 0 and 1 22 7.1. Genus 0 22 7.2. Genus 1 22 7.3. Example in Genus 3 23 Appendix A. Proof of Riemann Hurwitz 25 Appendix B. Quotients of Riemann surfaces by automorphisms 29 References 31 1 2 AARON LANDESMAN 1. INTRODUCTION In this course, we’ll discuss the theory of Riemann surfaces. Rie- mann surfaces are a beautiful breeding ground for ideas from many areas of math. In this way they connect seemingly disjoint fields, and also allow one to use tools from different areas of math to study them.
    [Show full text]
  • On the Continuability of Multivalued Analytic Functions to an Analytic Subset*
    Functional Analysis and Its Applications, Vol. 35, No. 1, pp. 51–59, 2001 On the Continuability of Multivalued Analytic Functions to an Analytic Subset* A. G. Khovanskii UDC 517.9 In the paper it is shown that a germ of a many-valued analytic function can be continued analytically along the branching set at least until the topology of this set is changed. This result is needed to construct the many-dimensional topological version of the Galois theory. The proof heavily uses the Whitney stratification. Introduction In the topological version of Galois theory for functions of one variable (see [1–4]) it is proved that the character of location of the Riemann surface of a function over the complex line can prevent the representability of this function by quadratures. This not only explains why many differential equations cannot be solved by quadratures but also gives the sharpest known results on their nonsolvability. I always thought that there is no many-dimensional topological version of Galois theory of full value. The point is that, to construct such a version in the many-variable case, it would be necessary to have information not only on the continuability of germs of functions outside their branching sets but also along these sets, and it seemed that there is nowhere one can obtain this information from. Only in spring of 1999 did I suddenly understand that germs of functions are sometimes automatically continued along the branching set. Therefore, a many-dimensional topological version of the Galois theory does exist. I am going to publish it in forthcoming papers.
    [Show full text]
  • Draftfebruary 16, 2021-- 02:14
    Exactification of Stirling’s Approximation for the Logarithm of the Gamma Function Victor Kowalenko School of Mathematics and Statistics The University of Melbourne Victoria 3010, Australia. February 16, 2021 Abstract Exactification is the process of obtaining exact values of a function from its complete asymptotic expansion. This work studies the complete form of Stirling’s approximation for the logarithm of the gamma function, which consists of standard leading terms plus a remainder term involving an infinite asymptotic series. To obtain values of the function, the divergent remainder must be regularized. Two regularization techniques are introduced: Borel summation and Mellin-Barnes (MB) regularization. The Borel-summed remainder is found to be composed of an infinite convergent sum of exponential integrals and discontinuous logarithmic terms from crossing Stokes sectors and lines, while the MB-regularized remainders possess one MB integral, with similar logarithmic terms. Because MB integrals are valid over overlapping domains of convergence, two MB-regularized asymptotic forms can often be used to evaluate the logarithm of the gamma function. Although the Borel- summed remainder is truncated, albeit at very large values of the sum, it is found that all the remainders when combined with (1) the truncated asymptotic series, (2) the leading terms of Stirling’s approximation and (3) their logarithmic terms yield identical valuesDRAFT that agree with the very high precision results obtained from mathematical software packages.February 16,
    [Show full text]
  • 3 Elementary Functions
    3 Elementary Functions We already know a great deal about polynomials and rational functions: these are analytic on their entire domains. We have thought a little about the square-root function and seen some difficulties. The remaining elementary functions are the exponential, logarithmic and trigonometric functions. 3.1 The Exponential and Logarithmic Functions (§30–32, 34) We have already defined the exponential function exp : C ! C : z 7! ez using Euler’s formula ez := ex cos y + iex sin y (∗) and seen that its real and imaginary parts satisfy the Cauchy–Riemann equations on C, whence exp C d z = z is entire (analytic on ). Indeed recall that dz e e . We have also seen several of the basic properties of the exponential function, we state these and several others for reference. Lemma 3.1. Throughout let z, w 2 C. 1. ez 6= 0. ez 2. ez+w = ezew and ez−w = ew 3. For all n 2 Z, (ez)n = enz. 4. ez is periodic with period 2pi. Indeed more is true: ez = ew () z − w = 2pin for some n 2 Z Proof. Part 1 follows trivially from (∗). To prove 2, recall the multiple-angle formulae for cosine and sine. Part 3 requires an induction using part 2 with z = w. Part 4 is more interesting: certainly ew+2pin = ew by the periodicity of sine and cosine. Now suppose ez = ew where z = x + iy and w = u + iv. Then, by considering the modulus and argument, ( ex = eu exeiy = eueiv =) y = v + 2pin for some n 2 Z We conclude that x = u and so z − w = i(y − v) = 2pin.
    [Show full text]
  • The Monodromy Groups of Schwarzian Equations on Closed
    Annals of Mathematics The Monodromy Groups of Schwarzian Equations on Closed Riemann Surfaces Author(s): Daniel Gallo, Michael Kapovich and Albert Marden Reviewed work(s): Source: Annals of Mathematics, Second Series, Vol. 151, No. 2 (Mar., 2000), pp. 625-704 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/121044 . Accessed: 15/02/2013 18:57 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals of Mathematics. http://www.jstor.org This content downloaded on Fri, 15 Feb 2013 18:57:11 PM All use subject to JSTOR Terms and Conditions Annals of Mathematics, 151 (2000), 625-704 The monodromy groups of Schwarzian equations on closed Riemann surfaces By DANIEL GALLO, MICHAEL KAPOVICH, and ALBERT MARDEN To the memory of Lars V. Ahlfors Abstract Let 0: 7 (R) -* PSL(2, C) be a homomorphism of the fundamental group of an oriented, closed surface R of genus exceeding one. We will establish the following theorem. THEOREM. Necessary and sufficient for 0 to be the monodromy represen- tation associated with a complex projective stucture on R, either unbranched or with a single branch point of order 2, is that 0(7ri(R)) be nonelementary.
    [Show full text]
  • Inverse Trigonometric Functions
    Chapter 2 INVERSE TRIGONOMETRIC FUNCTIONS vMathematics, in general, is fundamentally the science of self-evident things. — FELIX KLEIN v 2.1 Introduction In Chapter 1, we have studied that the inverse of a function f, denoted by f–1, exists if f is one-one and onto. There are many functions which are not one-one, onto or both and hence we can not talk of their inverses. In Class XI, we studied that trigonometric functions are not one-one and onto over their natural domains and ranges and hence their inverses do not exist. In this chapter, we shall study about the restrictions on domains and ranges of trigonometric functions which ensure the existence of their inverses and observe their behaviour through graphical representations. Besides, some elementary properties will also be discussed. The inverse trigonometric functions play an important Aryabhata role in calculus for they serve to define many integrals. (476-550 A. D.) The concepts of inverse trigonometric functions is also used in science and engineering. 2.2 Basic Concepts In Class XI, we have studied trigonometric functions, which are defined as follows: sine function, i.e., sine : R → [– 1, 1] cosine function, i.e., cos : R → [– 1, 1] π tangent function, i.e., tan : R – { x : x = (2n + 1) , n ∈ Z} → R 2 cotangent function, i.e., cot : R – { x : x = nπ, n ∈ Z} → R π secant function, i.e., sec : R – { x : x = (2n + 1) , n ∈ Z} → R – (– 1, 1) 2 cosecant function, i.e., cosec : R – { x : x = nπ, n ∈ Z} → R – (– 1, 1) 2021-22 34 MATHEMATICS We have also learnt in Chapter 1 that if f : X→Y such that f(x) = y is one-one and onto, then we can define a unique function g : Y→X such that g(y) = x, where x ∈ X and y = f(x), y ∈ Y.
    [Show full text]
  • Universit`A Degli Studi Di Perugia the Lambert W Function on Matrices
    Universita` degli Studi di Perugia Facolta` di Scienze Matematiche, Fisiche e Naturali Corso di Laurea Triennale in Informatica The Lambert W function on matrices Candidato Relatore MassimilianoFasi BrunoIannazzo Contents Preface iii 1 The Lambert W function 1 1.1 Definitions............................. 1 1.2 Branches.............................. 2 1.3 Seriesexpansions ......................... 10 1.3.1 Taylor series and the Lagrange Inversion Theorem. 10 1.3.2 Asymptoticexpansions. 13 2 Lambert W function for scalar values 15 2.1 Iterativeroot-findingmethods. 16 2.1.1 Newton’smethod. 17 2.1.2 Halley’smethod . 18 2.1.3 K¨onig’s family of iterative methods . 20 2.2 Computing W ........................... 22 2.2.1 Choiceoftheinitialvalue . 23 2.2.2 Iteration.......................... 26 3 Lambert W function for matrices 29 3.1 Iterativeroot-findingmethods. 29 3.1.1 Newton’smethod. 31 3.2 Computing W ........................... 34 3.2.1 Computing W (A)trougheigenvectors . 34 3.2.2 Computing W (A) trough an iterative method . 36 A Complex numbers 45 A.1 Definitionandrepresentations. 45 B Functions of matrices 47 B.1 Definitions............................. 47 i ii CONTENTS C Source code 51 C.1 mixW(<branch>, <argument>) ................. 51 C.2 blockW(<branch>, <argument>, <guess>) .......... 52 C.3 matW(<branch>, <argument>) ................. 53 Preface Main aim of the present work was learning something about a not- so-widely known special function, that we will formally call Lambert W function. This function has many useful applications, although its presence goes sometimes unrecognised, in mathematics and in physics as well, and we found some of them very curious and amusing. One of the strangest situation in which it comes out is in writing in a simpler form the function .
    [Show full text]
  • Various Mathematical Properties of the Generalized Incomplete Gamma Functions with Applications
    Various Mathematical Properties of the Generalized Incomplete Gamma Functions with Applications by Bader Ahmed Al-Humaidi A Dissertation Presented to the DEANSHIP OF GRADUATE STUDIES In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY IN MATHEMATICS KING FAHD UNIVERSITY OF PETROLEUM & MINERALS DHAHRAN, SAUDI ARABIA May, 2011 \l'.r..41gzi4r-,1*-414444.4-14444:4414r.14p414.1.4.4*4(4,444 4 VARIOUS MATHEMATICAL PROPERTIES OF THE GENERALIZED INCOMPLETE GAMMA FUNCTIONS 4 WITH APPLICATIONS 4 4 4 BY is BADER AHMED AL HUMAIDI A Dissertation Presented to the DEANSHIP OF GRADUATE STUDIES 4 KING FAHD UNIVERSITY OF PETROLEUM & MINERALS DHAHRAN, SAUDI ARABIA fit,. in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY In 4 MATHEMATICS **. 4 MAY 2011 •> 4 ;4`19PV't4 W4.Lc L'Ff KING FAHD UNIVERSITY OF PETROLEUM & MINIRALS DHAHRAN, SAUDI ARABIA DEANSHIP OF GRADUATE STUDIES this dissertation, written by BADER AHMED AL HUMAIDI under the direction of his thesis advisors and approved by his thesis committee, has been presented to and accepted by the Dean of Graduate Studies, in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY IN MATHEMATICS. Dissertation Committee Prof. M. A. Chaudhry Dissertation Committee Chairman Prof. S. M. Zubair Co-Chairman Prof. M. A. EI-Gebeily Member Dr. Hattan Tawfiq Prof. A. H. Bokhari Department Chairman Member Dr. Salam Zummo Prof. M. T. Mustafa 10, sr Dean of Graduate Studies i- .eiPiKOY ..;:.-- Member /.0„ ." vtGRADOos" ► 2-1-6 I" Date To my parents, wife, kids, brothers and sisters iii Acknowledgements First, praise be to ALLAH who helped and guided me to accomplish this work.
    [Show full text]
  • Digitally Continuous Multivalued Functions, Morphological Operations and Thinning Algorithms
    Digitally Continuous Multivalued Functions, Morphological Operations and Thinning Algorithms Carmen Escribano • Antonio Giraldo • María Asunción Sastre Abstract In a recent paper (Escribano et al. in Discrete Introduction Geometry for Computer Imagery 2008. Lecture Notes in Computer Science, vol. 4992, pp. 81-92, 2008) we have in­ The notion of continuous function is a fundamental concept troduced a notion of continuity in digital spaces which ex­ in the study of topological spaces. For dealing with digital tends the usual notion of digital continuity. Our approach, spaces, several approaches to define a reasonable notion of which uses multivalued functions, provides a better frame­ continuous function have been proposed. The first one goes work to define topological notions, like retractions, in a far back to A. Rosenfeld [19] in 1986. He defined continuous more realistic way than by using just single-valued digitally functions in a way similar to that used for continuous maps continuous functions. in W1. It turned out that continuous functions agreed with In this work we develop properties of this family of con­ functions taking 4-adjacent points into 4-adjacent points. He tinuous functions, now concentrating on morphological op­ proved, amongst other results, that a function between digi­ erations and thinning algorithms. We show that our notion of tal spaces is continuous if and only if it takes connected sets continuity provides a suitable framework for the basic oper­ into connected sets. Independently of Rosenfeld, L. Chen ations in mathematical morphology: erosion, dilation, clos­ [5, 6] seems to have developed the same notion of continu­ ing, and opening.
    [Show full text]
  • On the Lambert W Function 329
    On the Lambert W function 329 The text below is the same as that published in Advances in Computational Mathematics, Vol 5 (1996) 329 – 359, except for a minor correction after equation (1.1). The pagination matches the published version until the bibliography. On the Lambert W Function R. M. Corless1,G.H.Gonnet2,D.E.G.Hare3,D.J.Jeffrey1 andD.E.Knuth4 1Department of Applied Mathematics, University of Western Ontario, London, Canada, N6A 5B7 2Institut f¨ur Wissenschaftliches Rechnen, ETH Z¨urich, Switzerland 3Symbolic Computation Group, University of Waterloo, Waterloo, Canada, N2L 3G1 4Department of Computer Science, Stanford University, Stanford, California, USA 94305-2140 Abstract The Lambert W function is defined to be the multivalued inverse of the function w → wew. It has many applications in pure and applied mathematics, some of which are briefly described here. We present a new discussion of the complex branches of W ,an asymptotic expansion valid for all branches, an efficient numerical procedure for evaluating the function to arbitrary precision, and a method for the symbolic integration of expressions containing W . 1. Introduction In 1758, Lambert solved the trinomial equation x = q +xm by giving a series develop- ment for x in powers of q. Later, he extended the series to give powers of x as well [48,49]. In [28], Euler transformed Lambert’s equation into the more symmetrical form xα − xβ =(α − β)vxα+β (1.1) by substituting x−β for x and setting m = α/β and q =(α − β)v. Euler’s version of Lambert’s series solution was thus n 1 2 x =1+nv + 2 n(n + α + β)v + 1 n(n + α +2β)(n +2α + β)v3 6 (1.2) 1 4 + 24 n(n + α +3β)(n +2α +2β)(n +3α + β)v +etc.
    [Show full text]
  • Sequential Discontinuities of Feynman Integrals and the Monodromy Group
    Sequential Discontinuities of Feynman Integrals and the Monodromy Group Jacob L. Bourjaily1,2, Holmfridur Hannesdottir3, Andrew J. McLeod1, Matthew D. Schwartz3, and Cristian Vergu1 1Niels Bohr International Academy and Discovery Center, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark 2Institute for Gravitation and the Cosmos, Department of Physics, Pennsylvania State University, University Park, PA 16892, USA 3Department of Physics, Harvard University, Cambridge, MA 02138, USA July 29, 2020 Abstract We generalize the relation between discontinuities of scattering amplitudes and cut diagrams to cover sequential discontinuities (discontinuities of discontinuities) in arbitrary momentum channels. The new relations are derived using time-ordered per- turbation theory, and hold at phase-space points where all cut momentum channels are simultaneously accessible. As part of this analysis, we explain how to compute se- quential discontinuities as monodromies and explore the use of the monodromy group in characterizing the analytic properties of Feynman integrals. We carry out a number of cross-checks of our new formulas in polylogarithmic examples, in some cases to all loop orders. arXiv:2007.13747v1 [hep-th] 27 Jul 2020 Contents 1 Introduction1 2 Cutting rules: a review4 2.1 Cutkosky, 't Hooft and Veltman . .5 2.2 Time-ordered perturbation theory . .7 3 Discontinuities 10 3.1 Covariant approach . 11 3.2 Discontinuities in TOPT . 14 4 Discontinuities as monodromies 17 4.1 Warm-up: the natural logarithm . 17 4.2 The monodromy group . 21 4.3 Monodromies of propagators . 31 5 Sequential discontinuities 33 5.1 Sequential discontinuities in the same channel . 34 5.2 Sequential discontinuities in different channels .
    [Show full text]