Radial and circular slit maps of unbounded multiply connected circle domains

2008 ◽  
Vol 464 (2095) ◽  
pp. 1719-1737 ◽  
Author(s):  
T.K DeLillo ◽  
T.A Driscoll ◽  
A.R Elcrat ◽  
J.A Pfaltzgraff
Keyword(s):  
Analytic Continuation ◽  
Recent Progress ◽  
Infinite Product ◽  
Reflection Principle ◽  
Numerical Implementation ◽  
Infinite Products ◽  
Multiply Connected ◽  
Slit Domain ◽  
Circular Slit ◽  
Product Mapping

Infinite product formulae for conformally mapping an unbounded multiply connected circle domain to an unbounded canonical radial or circular slit domain, or to domains with both radial and circular slit boundary components are derived and implemented numerically and graphically. The formulae are generated by analytic continuation with the reflection principle. Convergence of the infinite products is proved for domains with sufficiently well-separated boundary components. Some recent progress in the numerical implementation of infinite product mapping formulae is presented.

VANISHING COEFFICIENTS IN QUOTIENTS OF THETA FUNCTIONS OF MODULUS FIVE

2020 ◽  
Vol 102 (3) ◽  
pp. 387-398
Author(s):  
SHANE CHERN ◽  
DAZHAO TANG
Keyword(s):  
Infinite Product ◽  
Theta Functions ◽  
Infinite Products ◽  
General Family

Following recent investigations of vanishing coefficients in infinite products, we show that such instances are very rare when the infinite product is among a family of theta-quotients of modulus five. We also prove that a general family of products of theta functions of modulus five can always be effectively 5-dissected.

scholarly journals Circular Slits Map of Bounded Multiply Connected Regions

2012 ◽  
Vol 2012 ◽  
pp. 1-26 ◽  
Author(s):  
Ali W. K. Sangawi ◽  
Ali H. M. Murid ◽  
M. M. S. Nasser
Keyword(s):  
Integral Equations ◽  
Boundary Integral Equations ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Connected Region ◽  
Numerical Conformal Mapping ◽  
Multiply Connected ◽  
Circular Slit ◽  
Multiply Connected Regions ◽  
Multiply Connected Region

We present a boundary integral equation method for the numerical conformal mapping of bounded multiply connected region onto a circular slit region. The method is based on some uniquely solvable boundary integral equations with adjoint classical, adjoint generalized, and modified Neumann kernels. These boundary integral equations are constructed from a boundary relationship satisfied by a function analytic on a multiply connected region. Some numerical examples are presented to illustrate the efficiency of the presented method.

scholarly journals CONGRUENCE PROPERTIES OF BORCHERDS PRODUCT EXPONENTS

2013 ◽  
Vol 09 (06) ◽  
pp. 1563-1578
Author(s):  
KEENAN MONKS ◽  
SARAH PELUSE ◽  
LYNNELLE YE
Keyword(s):  
Modular Forms ◽  
Logarithmic Derivative ◽  
Infinite Product ◽  
Modular Representation ◽  
Infinite Products ◽  
Analogous Statement ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Congruence Properties ◽  
Invent Math

In his striking 1995 paper, Borcherds [Automorphic forms on Os+2,2(ℝ) and infinite products, Invent. Math.120 (1995) 161–213] found an infinite product expansion for certain modular forms with CM divisors. In particular, this applies to the Hilbert class polynomial of discriminant -d evaluated at the modular j-function. Among a number of powerful generalizations of Borcherds' work, Zagier made an analogous statement for twisted versions of this polynomial. He proves that the exponents of these product expansions, A(n,d), are the coefficients of certain special half-integral weight modular forms. We study the congruence properties of A(n,d) modulo a prime ℓ by relating it to a modular representation of the logarithmic derivative of the Hilbert class polynomial.

scholarly journals Generalization of the Schwarz–Christoffel mapping to multiply connected polygonal domains

2014 ◽  
Vol 470 (2166) ◽  
pp. 20130848 ◽  
Author(s):  
Giovani L. Vasconcelos
Keyword(s):  
Conformal Mapping ◽  
Circular Domain ◽  
Polygonal Domains ◽  
Multiply Connected ◽  
Slit Domain ◽  
Combined Use ◽  
Polygonal Region ◽  
Upper Half Plane ◽  
Indefinite Integral ◽  
Detailed Derivation

A generalization of the Schwarz–Christoffel mapping to multiply connected polygonal domains is obtained by making a combined use of two preimage domains, namely, a rectilinear slit domain and a bounded circular domain. The conformal mapping from the circular domain to the polygonal region is written as an indefinite integral whose integrand consists of a product of powers of the Schottky-Klein prime functions, which is the same irrespective of the preimage slit domain, and a prefactor function that depends on the choice of the rectilinear slit domain. A detailed derivation of the mapping formula is given for the case where the preimage slit domain is the upper half-plane with radial slits. Representation formulae for other canonical slit domains are also obtained but they are more cumbersome in that the prefactor function contains arbitrary parameters in the interior of the circular domain.

scholarly journals A reflection principle for solutions to the Helmholtz equation and an application to the inverse scattering problem

1977 ◽  
Vol 18 (2) ◽  
pp. 125-130 ◽  
Author(s):  
David Colton
Keyword(s):  
Helmholtz Equation ◽  
Analytic Continuation ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Inverse Scattering Problem ◽  
Reflection Principle ◽  
Dimensional Euclidean Space ◽  
Finite Radius ◽  
Spherical Boundary ◽  
Limiting Case

A classical result in potential theory is the Schwarz reflection principle for solutions of Laplace's equation which vanish on a portion of a spherical boundary. The question naturally arises whether or not such a property is also true for solutions of the Helmholtz equation. This has been answered in the affirmative by Diaz and Ludford ([4]; see also [10]) in the limiting case of the plane. It is the purpose of this paper to show that a reflection principle is also valid for spheres of finite radius. As an application of this result we shall study the problem of the analytic continuation of solutions to the Helmholtz equation defined in the exterior of a bounded domain in three-dimensional Euclidean space ℝ3 We shall show that through the use of the reflection principle derived in this paper, this problem can be reduced to the problem of the analytic continuation of an analytic function of two complex variables, which in turn can be performed through a variety of known methods (cf. [7]).

scholarly journals On uniform convergence of some classes of infinite products

1987 ◽  
pp. 107
Author(s):  
K.M. Slepenchuk
Keyword(s):  
Uniform Convergence ◽  
Infinite Product ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Infinite Products ◽  
Necessary And Sufficient

We find necessary and sufficient conditions $\{ \alpha_k(x) \}$ must satisfy for the infinite product$$\prod\limits_{k=1}^{\infty} \bigl[ 1 + \alpha_k(x) u_k(x) \bigr]$$to converge uniformly under the condition that:1) the series $\sum\limits_{k=1}^{\infty} |\Delta u_k(x)|$ converges uniformly; 2) $\sum\limits_{k=1}^{\infty} |\Delta u_k(x)| = O(1)$.

scholarly journals Digit Sums and Infinite Products

2020 ◽  
Author(s):  
Samin Riasat
Keyword(s):  
Closed Form ◽  
Large Class ◽  
Rational Function ◽  
Systematic Approach ◽  
Infinite Product ◽  
Infinite Products

Consider the sequence un defined as follows: un=+1 if the sum of the base b digits of n is even, and un=−1 otherwise, where we take b=2. Recall that the Woods-Robbins infinite product involves a rational function in n and the sequence un. Although several generalizations of the Woods-Robbins product are known in the literature, no other infinite product involving a rational function in n and the sequence un was known in closed form until recently. In this chapter we introduce a systematic approach to these products, which may be generalized to other values of b. We illustrate the approach by evaluating a large class of similar infinite products.

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