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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. -
Probabilistic Proofs of Some Beta-Function Identities
1 2 Journal of Integer Sequences, Vol. 22 (2019), 3 Article 19.6.6 47 6 23 11 Probabilistic Proofs of Some Beta-Function Identities Palaniappan Vellaisamy Department of Mathematics Indian Institute of Technology Bombay Powai, Mumbai-400076 India [email protected] Aklilu Zeleke Lyman Briggs College & Department of Statistics and Probability Michigan State University East Lansing, MI 48825 USA [email protected] Abstract Using a probabilistic approach, we derive some interesting identities involving beta functions. These results generalize certain well-known combinatorial identities involv- ing binomial coefficients and gamma functions. 1 Introduction There are several interesting combinatorial identities involving binomial coefficients, gamma functions, and hypergeometric functions (see, for example, Riordan [9], Bagdasaryan [1], 1 Vellaisamy [15], and the references therein). One of these is the following famous identity that involves the convolution of the central binomial coefficients: n 2k 2n − 2k =4n. (1) k n − k k X=0 In recent years, researchers have provided several proofs of (1). A proof that uses generating functions can be found in Stanley [12]. Combinatorial proofs can also be found, for example, in Sved [13], De Angelis [3], and Miki´c[6]. A related and an interesting identity for the alternating convolution of central binomial coefficients is n n n 2k 2n − 2k 2 n , if n is even; (−1)k = 2 (2) k n − k k=0 (0, if n is odd. X Nagy [7], Spivey [11], and Miki´c[6] discussed the combinatorial proofs of the above identity. Recently, there has been considerable interest in finding simple probabilistic proofs for com- binatorial identities (see Vellaisamy and Zeleke [14] and the references therein). -
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. -
Standard Distributions
Appendix A Standard Distributions A.1 Standard Univariate Discrete Distributions I. Binomial Distribution B(n, p) has the probability mass function n f(x)= px (x =0, 1,...,n). x The mean is np and variance np(1 − p). II. The Negative Binomial Distribution NB(r, p) arises as the distribution of X ≡{number of failures until the r-th success} in independent trials each with probability of success p. Thus its probability mass function is r + x − 1 r x fr(x)= p (1 − p) (x =0, 1, ··· , ···). x Let Xi denote the number of failures between the (i − 1)-th and i-th successes (i =2, 3,...,r), and let X1 be the number of failures before the first success. Then X1,X2,...,Xr and r independent random variables each having the distribution NB(1,p) with probability mass function x f1(x)=p(1 − p) (x =0, 1, 2,...). Also, X = X1 + X2 + ···+ Xr. Hence ∞ ∞ x x−1 E(X)=rEX1 = r p x(1 − p) = rp(1 − p) x(1 − p) x=0 x=1 ∞ ∞ d d = rp(1 − p) − (1 − p)x = −rp(1 − p) (1 − p)x x=1 dp dp x=1 © Springer-Verlag New York 2016 343 R. Bhattacharya et al., A Course in Mathematical Statistics and Large Sample Theory, Springer Texts in Statistics, DOI 10.1007/978-1-4939-4032-5 344 A Standard Distributions ∞ d d 1 = −rp(1 − p) (1 − p)x = −rp(1 − p) dp dp 1 − (1 − p) x=0 =p r(1 − p) = . (A.1) p Also, one may calculate var(X) using (Exercise A.1) var(X)=rvar(X1). -
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. -
Extended K-Gamma and K-Beta Functions of Matrix Arguments
mathematics Article Extended k-Gamma and k-Beta Functions of Matrix Arguments Ghazi S. Khammash 1, Praveen Agarwal 2,3,4 and Junesang Choi 5,* 1 Department of Mathematics, Al-Aqsa University, Gaza Strip 79779, Palestine; [email protected] 2 Department of Mathematics, Anand International College of Engineering, Jaipur 303012, India; [email protected] 3 Institute of Mathematics and Mathematical Modeling, Almaty 050010, Kazakhstan 4 Harish-Chandra Research Institute Chhatnag Road, Jhunsi, Prayagraj (Allahabad) 211019, India 5 Department of Mathematics, Dongguk University, Gyeongju 38066, Korea * Correspondence: [email protected] Received: 17 August 2020; Accepted: 30 September 2020; Published: 6 October 2020 Abstract: Various k-special functions such as k-gamma function, k-beta function and k-hypergeometric functions have been introduced and investigated. Recently, the k-gamma function of a matrix argument and k-beta function of matrix arguments have been presented and studied. In this paper, we aim to introduce an extended k-gamma function of a matrix argument and an extended k-beta function of matrix arguments and investigate some of their properties such as functional relations, inequality, integral formula, and integral representations. Also an application of the extended k-beta function of matrix arguments to statistics is considered. Keywords: k-gamma function; k-beta function; gamma function of a matrix argument; beta function of matrix arguments; extended gamma function of a matrix argument; extended beta function of matrix arguments; k-gamma function of a matrix argument; k-beta function of matrix arguments; extended k-gamma function of a matrix argument; extended k-beta function of matrix arguments 1. -
TYPE BETA FUNCTIONS of SECOND KIND 135 Operators
Adv. Oper. Theory 1 (2016), no. 1, 134–146 http://doi.org/10.22034/aot.1609.1011 ISSN: 2538-225X (electronic) http://aot-math.org (p, q)-TYPE BETA FUNCTIONS OF SECOND KIND ALI ARAL 1∗ and VIJAY GUPTA2 Communicated by A. Kaminska Abstract. In the present article, we propose the (p, q)-variant of beta func- tion of second kind and establish a relation between the generalized beta and gamma functions using some identities of the post-quantum calculus. As an ap- plication, we also propose the (p, q)-Baskakov–Durrmeyer operators, estimate moments and establish some direct results. 1. Introduction The quantum calculus (q-calculus) in the field of approximation theory was discussed widely in the last two decades. Several generalizations to the q variants were recently presented in the book [3]. Further there is possibility of extension of q-calculus to post-quantum calculus, namely the (p, q)-calculus. Actually such extension of quantum calculus can not be obtained directly by substitution of q by q/p in q-calculus. But there is a link between q-calculus and (p, q)-calculus. The q calculus may be obtained by substituting p = 1 in (p, q)-calculus. We mentioned some previous results in this direction. Recently, Gupta [8] introduced (p, q) gen- uine Bernstein–Durrmeyer operators and established some direct results. (p, q) generalization of Sz´asz–Mirakyan operators was defined in [1]. Also authors inves- tigated a Durrmeyer type modifications of the Bernstein operators in [9]. We can also mention other papers as Bernstein operators [10], Bernstein–Stancu oper- ators [11]. -
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 . -
Integral Formula for the Bessel Function of the First Kind
INTEGRAL FORMULA FOR THE BESSEL FUNCTION OF THE FIRST KIND ENRICO DE MICHELI ABSTRACT. In this paper, we prove a new integral representation for the Bessel function of the first kind Jµ (z), which holds for any µ,z C. ∈ 1. INTRODUCTION In this note, we prove the following integral representation of the Bessel function of the first kind Jµ (z), which holds for unrestricted complex values of the order µ: z µ π 1 izcosθ 1 iθ C C (1.1) Jµ (z)= e γ∗ µ, 2 ize dθ (µ ;z ), 2π 2 π ∈ ∈ Z− where γ∗ denotes the Tricomi version of the (lower) incomplete gamma function. Lower and upper incomplete gamma functions arise from the decomposition of the Euler integral for the gamma function: w t µ 1 (1.2a) γ(µ,w)= e− t − dt (Re µ > 0), 0 Z ∞ t µ 1 (1.2b) Γ(µ,w)= e− t − dt ( argw < π). w | | Z Analytical continuation with respect to both parameters can be based either on the integrals in (1.2) or on series expansions of γ(µ,w), e.g. [4, Eq. 9.2(4)]: ∞ ( 1)n wµ+n (1.3) γ(µ,w)= ∑ − , n=0 n! µ + n which shows that γ(µ,w) has simple poles at µ = 0, 1, 2,... and, in general, is a multi- valued (when µ is not a positive integer) function with− a− branch point at w = 0. Similarly, even Γ(µ,w) is in general multi-valued but it is an entire function of µ. Inconveniences re- lated to poles and multi-valuedness of γ can be circumvented by introducing what Tricomi arXiv:2012.02887v1 [math.CA] 4 Dec 2020 called the fundamental function of the incomplete gamma theory [11, §2, p. -
CHAPTER 4 Elementary Functions Dr. Pulak Sahoo
CHAPTER 4 Elementary Functions BY Dr. Pulak Sahoo Assistant Professor Department of Mathematics University Of Kalyani West Bengal, India E-mail : sahoopulak1@gmail:com 1 Module-3: Multivalued Functions-I 1 Introduction We recall that a function w = f(z) is called multivalued function if for all or some z of the domain, we find more than one value of w. Thus, a function f is said to be single valued if f satisfies f(z) = f[z(r; θ)] = f[z(r; θ + 2π)]: Otherwise, f is called as a multivalued function. We know that a multivalued function can be considered as a collection of single valued functions. For analytical properties of a multivalued function, we just consider those domains in which the functions are single valued. Branch By branch of a multivalued function f(z) defined on a domain D1 we mean a single valued function g(z) which is analytic in some sub-domain D ⊂ D1 at each point of which g(z) is one of the values of f(z). Branch Points and Branch Lines A point z = α is called a branch point of a multivalued function f(z) if the branches of f(z) are interchanged when z describes a closed path about α. We consider the function w = z1=2. Suppose that we allow z to make a complete circuit around the origin in the anticlockwise sense starting from the point P. Thus we p p have z = reiθ; w = reiθ=2; so that at P, θ = β and w = reiβ=2: After a complete p p circuit back to P, θ = β + 2π and w = rei(β+2π)=2 = − reiβ=2: Thus we have not achieved the same value of w with which we started. -
Riemann Surfaces
Riemann Surfaces See Arfken & Weber section 6.7 (mapping) or [1][sections 106-108] for more information. 1 Definition A Riemann surface is a generalization of the complex plane to a surface of more than one sheet such that a multiple-valued function on the original complex plane has only one value at each point of the surface. Briefly, the notion of a Riemann surface is important whenever considering functions with branch cuts. Perhaps the easiest example is the Riemann surface for log z. Write z = r exp(iθ), then log z = (log r)+ iθ. However, the θ in that expression is not uniquely defined – it is only defined up to a multiple of 2π. As you walk around the origin of the complex plane, the function log z comes back to itself up to additions of 2πi. We can construct a cover of the complex plane on which log z is single-valued by taking a helix over the complex plane – going 360◦ about the origin takes from you from sheet of the cover to another sheet. Put another way, every time you go through the branch cut, you go to a different sheet. We can construct such a cover as follows. (Our discussion here is verbatim from [1][section 106].) Consider the complex z plane, with the origin deleted, as a sheet R0 which is cut along the positive half of the real axis. On that sheet, let θ range from 0 to 2π. Let a second sheet R1 be cut in the same way and placed in front of the sheet R0. -
The Beta Maxwell Distribution Grace Ebunoluwa Amusan [email protected]
Marshall University Marshall Digital Scholar Theses, Dissertations and Capstones 1-1-2010 The Beta Maxwell Distribution Grace Ebunoluwa Amusan [email protected] Follow this and additional works at: http://mds.marshall.edu/etd Part of the Statistics and Probability Commons Recommended Citation Amusan, Grace Ebunoluwa, "The Beta Maxwell Distribution" (2010). Theses, Dissertations and Capstones. Paper 5. This Thesis is brought to you for free and open access by Marshall Digital Scholar. It has been accepted for inclusion in Theses, Dissertations and Capstones by an authorized administrator of Marshall Digital Scholar. For more information, please contact [email protected]. The Beta Maxwell Distribution Grace Ebunoluwa Amusan Thesis submitted to the Graduate College of Marshall University in partial fulfillment of the requirements for the degree of Master of Arts in Mathematics Alfred Akinsete, Ph.D., Chair Basant Karna, Ph.D. Yulia Dementieva, Ph.D. Department of Mathematics Huntington, West Virginia 2010 Keywords: Beta distribution, Maxwell distribution, moments, parameter estimation Copyright 2010 by Grace Ebunoluwa Amusan The Beta Maxwell Distribution Grace Ebunoluwa Amusan ABSTRACT In this work we considered a general class of distributions generated from the logit of the beta random variable. We looked at various works that have been done and discussed some of the results that were ob- tained. Special cases of this class include the beta-normal distribution, the beta-exponential distribution, the beta-Gumbell distribution, the beta-Weibull distribution, the beta-Pareto distribution and the beta- Rayleigh distribution. We looked at the probability distribution func- tions of each of these distributions and also looked at some of their properties.