Analytic Continuation Formulae Research Articles

Conformal Mapping of a Z-Shaped Domain

For the problem of conformal mapping of a half-plane onto a Z-shaped domain with arbitrary geometry, an efficient method is developed for finding parameters of the Schwarz–Christoffel integral, i.e., the preimages of the vertices (prevertices) and the pre-integral multiplier. Special attention is given to the situation of crowding prevertices, in which case conventional integration methods face significant difficulties. For this purpose, the concept of a cluster is introduced, its center is determined, and all integrand binomials with prevertices from this cluster are expanded into a fast-convergent series by applying a unified scheme. Next, the arising integrals are reduced to single or double series in terms of Gauss hypergeometric functions F(a, b, c, q). The fast convergence of the resulting expansions is ensured by applying formulas for analytic continuation of F(a, b, c, q) to a neighborhood of the point q = 1 and using numerically stable recurrence relations. The constructed expansions are also fairly efficient for choosing initial approximations for prevertices in Newton’s iteration method. By using the leading terms of these expansions, the approximations for the prevertices are expressed in explicit form in terms of elementary functions, and the subsequent iterations ensure the fast convergence of the algorithm. After finding the parameters in the integral, the desired mapping is constructed as a combination of power series expansions at prevertices, regular expansions at the preimage of the center of symmetry, a Laurent series in a semi-annulus, and special series near the preimages of the vertical segments. Numerical results demonstrate the high efficiency of the developed method, especially in the case of strong crowding of prevertices.

Formulas for Computing Euler-Type Integrals and Their Application to the Problem of Constructing a Conformal Mapping of Polygons

This paper deals with Euler-type integrals and the closely related Lauricella function , which is a hypergeometric function of many complex variables. For new analytic continuation formulas are found that represent it in the form of Horn hypergeometric series exponentially converging in corresponding subdomains of, including near hyperplanes of the form, . The continuation formulas and identities for found in this paper make up an effective apparatus for computing this function and Euler-type integrals expressed in terms of it in the entire complex space , including complicated cases when the variables form one or several groups of closely spaced neighbors. The results are used to compute parameters of the Schwarz–Christoffel integral in the case of crowding and to construct conformal mappings of polygons.

Formulas for Analytic Continuation of Horn Functions of Two Variables

Formulas for Analytic Continuation of Horn Functions of Two Variables

Analytic continuation of the Kampé de Fériet function and the general double Horn series

For the Kampé de Fériet function, such analytic continuation formulas are obtained that allow one to represent this function as exponentially converging hypergeometric series in the complement to the convergence domain of the original series. This result has been obtained using the found formulas for the analytic continuation of the Horn series in two variables.

Analytic Continuation Formulas for the Hypergeometric Functions in Three Variables of Second Order

Analytic Continuation Formulas for the Hypergeometric Functions in Three Variables of Second Order

Dirichlet twists ofGLn-automorphicL-functions and hyper-Kloosterman Dirichlet series

We calculate mean values of $\operatorname{GL}_n$-automorphic $L$-functions twisted by primitive even Dirichlet characters of prime-power conductor, at arbitrary points within the critical strip, by derivation of special Voronoi summation formulae. Our calculation is novel in that the twisted sum can be expressed in terms of the average itself, and also that it sees the derivation of various new summation formulae in the setting of prime-power modulus. One consequence, as we explain, is to show the analytic continuation and additive summation formulae for hyper-Kloosterman Dirichlet series associated to $\operatorname{GL}_n$-automorphic $L$-functions.

Uniform asymptotic expansions for the Whittaker functions M κ , μ ( z ) and W κ , μ ( z ) with μ large

Uniform asymptotic expansions are derived for Whittaker’s confluent hypergeometric functions M κ , μ ( z ) and W κ , μ ( z ) , as well as the numerically satisfactory companion function W − κ , μ ( z e − π i ) . The expansions are uniformly valid for μ → ∞ , 0 ≤ κ / μ ≤ 1 − δ < 1 and 0 ≤ arg ⁡ ( z ) ≤ π . By using appropriate connection and analytic continuation formulae, these expansions can be extended to all unbounded non-zero complex z . The approximations come from recent asymptotic expansions involving elementary functions and Airy functions, and explicit error bounds are either provided or available.

Analytic continuation of Lauricella's function F D (N) for variables close to unit near hyperplanes {z j = z l }

For the Lauricella hypergeometric function with an arbitrary number of variables , we construct formulas for analytic continuation into the vicinity of hyperplanes and their intersections providing that all variables are close to unit. Such formulas represent the function near the point as linear combinations of N–multiple hypergeometric series that are solutions of the same system of partial differential equations as . Such series are the N–dimensional analog of the Kummer solutions known for the Gauss equation. The constructed analytical continuation formulas allow one to effectively calculate the function outside the unit polydisk.

Horn's hypergeometric functions with three variables

The work is devoted to obtaining explicit formulas for analytic continuation of quite general hypergeometric series depending on three variables and belonging to the Horn class. We have derived such continuation formulas with respect to one variable for an arbitrary series belonging to the class under consideration. In addition, on the base of several examples we have demonstrated the use of the obtained formulas for construction of analytic continuation with respect to all three variables. The obtained continuation formulas give representations for a triple hypergeometric series outside the domain of its convergence in the form of a combination of other hypergeometric series that also belong to the Horn class. These series satisfy to the same system of partial differential equations as the initial series. The results of the paper are useful for qualitative analysis and calculation of triple hypergeometric functions.

Analytic continuation of Lauricella's functions , and

ABSTRACT We construct formulas for analytic continuation of the Lauricella functions , and into the exterior of the domains, where they are initially defined by N-multiple hypergeometric series. To construct the analytic continuation, we use a new method, which allows to obtain continuation formulas for arbitrary hypergeometric series belonging to Horn's class, which, in particular, contains Lauricella's functions. The obtained formulas give effective apparatus for qualitative analysis and computation of these functions. Thus the results of the paper useful for numerous applications, where Lauricella's functions appear.

Exact WKB analysis of \u2102\u21191 holomorphic blocks

We study holomorphic blocks in the three dimensional mathcal{N} = 2 gauge theory that describes the ℂℙ1 model. We apply exact WKB methods to analyze the line operator identities associated to the holomorphic blocks and derive the analytic continuation formulae of the blocks as the twisted mass and FI parameter are varied. The main technical result we utilize is the connection formula for the 1\U0001d7191q-hypergeometric function. We show in detail how the q-Borel resummation methods reproduce the results obtained previously by using block-integral methods.

Analytic continuation of the Lauricella function with arbitrary number of variables

ABSTRACTThe Lauricella function , which is a generalized hypergeometric function of N variables, and a corresponding system of partial differential equations are considered. For an arbitrary N, we give a complete collection of analytic continuation formulas of . This formulas give representation of the Lauricella function outside the polydisk in the form of a linear combination of other generalized hypergeometric series that are solutions of the same system of partial differential equations, which is also satisfied by the function . The obtained hypergeometric series are N-dimensional analogues of the Kummer solutions well known in the theory of the classical hypergeometric Gauss equation. The obtained analytic continuation formulas provide an effective algorithm for computation of the Lauricella function .

Analytic continuation formulas and Jacobi-type relations for Lauricella function

An approach for constructing a complete system of formulas for the analytic continuation of the Lauricella generalized hypergeometric function FD(N) with any N beyond the boundary of the unit polydisk is proposed. The approach is exposed in detail for the continuation of the function under consideration in neighborhoods of points whose all N components equal 1 or ∞. For the Lauricella function, differential relations being analogues of Jacobi’s formula for the Gaussian hypergeometric function are also presented. The results can be applied to solve the crowding problem for the Schwarz–Christoffel integral and to the theory of the Riemann–Hilbert problem.

On square-integrability of solutions of the stationary Schrödinger equation for the quantum harmonic oscillator in two dimensional constant curvature spaces

In this work, one provides a justification of the condition that is usually imposed on the parameters of the hypergeometric equation, related to the solutions of the stationary Schrödinger equation for the harmonic oscillator in two-dimensional constant curvature spaces, in order to determine the solutions which are square-integrable. One proves that in case of negative curvature, it is a necessary condition of square integrability and in case of positive curvature, a necessary condition of regularity. The proof is based on the analytic continuation formulas for the hypergeometric function. It is observed also that the same is true in case of a slightly more general potential than the one for harmonic oscillator.

Correspondence between the one-loop three-point vertex and the Y and Δ electric resistor networks

Different mathematical methods have been applied to obtain the analytic result for the massless triangle Feynman diagram yielding a sum of four linearly independent (LI) hypergeometric functions of two variables F4. This result is not physically acceptable when it is embedded in higher loops, because all four hypergeometric functions in the triangle result have the same region of convergence and further integration means going outside those regions of convergence. We could go outside those regions by using the well-known analytic continuation formulas obeyed by the F4, but there are at least two ways we can do this. Which is the correct one? Whichever continuation one uses, it reduces a number of F4 from four to three. This reduction in the number of hypergeometric functions can be understood by taking into account the fundamental physical constraint imposed by the conservation of momenta flowing along the three legs of the diagram. With this, the number of overall LI functions that enter the most general solution must reduce accordingly. It remains to determine which set of three LI solutions needs to be taken. To determine the exact structure and content of the analytic solution for the three-point function that can be embedded in higher loops, we use the analogy that exists between Feynman diagrams and electric circuit networks, in which the electric current flowing in the network plays the role of the momentum flowing in the lines of a Feynman diagram. This analogy is employed to define exactly which three out of the four hypergeometric functions are relevant to the analytic solution for the Feynman diagram. The analogy is built based on the equivalence between electric resistance circuit networks of types Y and Δ in which flows a conserved current. The equivalence is established via the theorem of minimum energy dissipation within circuits having these structures.

Cumulant expansion and analytic continuation in Monte Carlo simulation of classical Lennard-Jones clusters

The Monte Carlo (MC) estimates of thermal averages are usually functions of system control parameters λ, such as temperature, volume, and interaction couplings. Given the MC average at a set of prescribed control parameters λ{0}, the problem of analytic continuation of the MC data to λ values in the neighborhood of λ{0} is considered in both classic and quantum domains. The key result is the theorem that links the differential properties of thermal averages to the higher order cumulants. The theorem and analytic continuation formulas expressed via higher order cumulants are numerically tested on the classical Lennard-Jones cluster system of N=13, 55, and 147 neon particles.

On the analyticity of Laguerre series

The transformation of a Laguerre series f(z) = ∑∞n=0λ(α)nL(α)n(z) to a power series f(z) = ∑∞n=0γnzn is discussed. Since many nonanalytic functions can be expanded in terms of generalized Laguerre polynomials, success is not guaranteed and such a transformation can easily lead to a mathematically meaningless expansion containing power series coefficients that are infinite in magnitude. Simple sufficient conditions based on the decay rates and sign patterns of the Laguerre series coefficients λ(α)n as n → ∞ can be formulated which guarantee that the resulting power series represents an analytic function. The transformation produces a mathematically meaningful result if the coefficients λ(α)n either decay exponentially or factorially as n → ∞. The situation is much more complicated—but also much more interesting—if the λ(α)n decay only algebraically as n → ∞. If the λ(α)n ultimately have the same sign, the series expansions for the power series coefficients diverge, and the corresponding function is not analytic at the origin. If the λ(α)n ultimately have strictly alternating signs, the series expansions for the power series coefficients still diverge, but are summable to something finite, and the resulting power series represents an analytic function. If algebraically decaying and ultimately alternating Laguerre series coefficients λ(α)n possess sufficiently simple explicit analytical expressions, the summation of the divergent series for the power series coefficients can often be accomplished with the help of analytic continuation formulae for hypergeometric series p+1Fp, but if the λ(α)n have a complicated structure or if only their numerical values are available, numerical summation techniques have to be employed. It is shown that certain nonlinear sequence transformations—in particular the so-called delta transformation (Weniger 1989 Comput. Phys. Rep. 10 189–371 (equation (8.4-4)))—are able to sum the divergent series occurring in this context effectively. As a physical application of the results of this paper, the legitimacy of the rearrangement of certain one-range addition theorems for Slater-type functions (Guseinov 1980 Phy. Rev. A 22 369–71, Guseinov 2001 Int. J. Quantum Chem. 81 126–29, Guseinov 2002 Int. J. Quantum Chem. 90 114–8) is investigated.

A contour integral representation for the dual five-point function and a symmetry of the genus-4 surface in

The invention of the "dual resonance model" N-point functions BN motivated the development of current string theory. The simplest of these models, the four-point function B4, is the classical Euler Beta function. Many standard methods of complex analysis in a single variable have been applied to elucidate the properties of the Euler Beta function, leading, for example, to analytic continuation formulas such as the contour-integral representation obtained by Pochhammer in 1890. Here we explore the geometry underlying the dual five-point function B5, the simplest generalization of the Euler Beta function. Analyzing the B5 integrand leads to a polyhedral structure for the five-crosscap surface, embedded in RP5, that has 12 pentagonal faces and a symmetry group of order 120 in PGL(6). We find a Pochhammer-like representation for B5 that is a contour integral along a surface of genus five. The symmetric embedding of the five-crosscap surface in RP5 is doubly covered by a symmetric embedding of the surface of genus four in R6 that has a polyhedral structure with 24 pentagonal faces and a symmetry group of order 240 in O(6). The methods appear generalizable to all N, and the resulting structures seem to be related to associahedra in arbitrary dimensions.

The analytic continuation of the Gaussian hypergeometric function 2F1( a, b; c; z) for arbitrary parameters

Our objective is to provide a complete table of analytic condinuation formulas for the Gaussian hypergeometric function 2 F 1( a, b; c; z) which allow its fast and accurate computation for arbitrary values of z and of the parameters a, b, c. To this end we distinguish 12 basis sets of the two-dimensional space of the solutions of the hypergeometric differential equation. Representing 2 F 1 in each of them yields 12 analytic continuation formulas. Each two of them are series in one of the arguments z, z/(z−1), (1−z), (1−1/z), 1/z, 1/(1−z) , respectively, such that any given argument z, with the exception of two single points in the complex plane, lies in the convergence domain of at least one of them. We present rapidly converging series representations of 2 F 1 for all possible constellations of parameters. For minimizing the effort for the derivation of these series we have extensively used the symmetry group of the hypergeometric equation, which is shown to be isomorphic to the cubic group O h .

Matrix elements of vertex operators of the deformed W A n -algebra and the Harish-Chandra solutions to Macdonald's difference equations

In this paper we prove that certain matrix elements of vertex operators of the deformed W A n -algebra satisfy Macdonald's difference equations and form a natural (n + 1)!-dimensional space of solutions. These solutions are the analogues of the Harish-Chandra solutions of the radial parts of the Laplace-Casimir operators on noncompact Riemannian symmetric spaces G/K with prescribed asymptotic behavior. We obtain formulas for analytic continuation of our Harish-Chandra type solutions as a consequence of braiding properties (obtained earlier by Y. Asai, M. Jimbo, T. Miwa, and Y. Pugay) of certain vertex operators of the deformed W A n -algebra.