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Wormholes are considered both from the Wheeler deWitt equation, as well as from the field equations in the Euclidean background of Robertson Walker mini-superspace in R2 gravity. Quantum wormhole satisfies Hawking Page wormhole boundary condition in the Euclidean background of mini-superspace, however, in the Lorentzian background wave functional turns to the usual oscillatory function. The Euclidean field equations for κ = 0,±1 lead to the wormhole configuration, as well as oscillating universe in Euclidean time τ. The oscillating universe in Euclidean time τ transforms to an expanding universe only for κ = 1 in Lorentz time t under analytic continuation τ = it and asymptotically leads to an exponential solution. An Euclidean wormhole in the very early era evolves to an oscillating universe in τ, thereafter crossing deSitter radius transition to an inflationary era is evident at later epoch only for κ = 1. [ABSTRACT FROM AUTHOR]

AbstractWe study the plane strain problem associated with an elliptical thermoelectrically insulated and compressible liquid inclusion embedded in an infinite homogeneous and isotropic nonlinearly coupled thermoelectric matrix under uniform remote electric current density and energy flux. With the aid of conformal mapping and analytic continuation, an exact closed-form solution to the problem is derived. In particular, an elementary expression for the internal uniform hydrostatic tension within the compressible liquid inclusion is obtained. Our result shows that the internal uniform hydrostatic tension is unaffected by the remote energy flux and is proportional to the square of the two components of the remote electric current density vector. Also presented is the hoop stress distribution along the elliptical interface on the matrix side. [ABSTRACT FROM AUTHOR]

We use Muskhelishvili's complex variable formulation to study the interaction problem associated with a circular incompressible liquid inclusion embedded in an infinite isotropic elastic matrix subjected to the action of an edge dislocation at an arbitrary position. A closed-form solution to the problem is derived largely with the aid of analytic continuation. We obtain, in explicit form, expressions for the internal uniform hydrostatic stresses, nonuniform strains and nonuniform rigid body rotation within the liquid inclusion; the hoop stress along the liquid-solid interface on the matrix side and the image force acting on the edge dislocation. We observe that (1) the internal strains and rigid body rotation within the liquid inclusion are independent of the elastic property of the matrix; (2) the internal hydrostatic stress field within the liquid inclusion is unaffected by Poisson's ratio of the matrix and is proportional to the shear modulus of the matrix; and (3) an unstable equilibrium position always exists for a climbing dislocation. [ABSTRACT FROM AUTHOR]

When considering initial stress field in geomaterial, nonzero resultant of shallow tunnel excavation exists, which produces logarithmic items in complex potentials, and would further lead to a unique displacement singularity at infinity to violate geo-engineering fact in real world. The mechanical and mathematical reasons of such a unique displacement singularity in the existing mechanical models are elaborated, and a new mechanical model is subsequently proposed to eliminate this singularity by constraining far-field ground surface displacement, and the original unbalanced resultant problem is converted into an equilibrium one with mixed boundary conditions. To solve stress and displacement in the new model, the analytic continuation is applied to transform the mixed boundary conditions into a homogenerous Riemann-Hilbert problem with extra constraints, which is then solved using an approximate and iterative method with good numerical stability. The Lanczos filtering is applied to the stress and displacement solution to reduce the Gibbs phenomena caused by abrupt change of the boundary conditions along ground surface. Several numerical cases are conducted to verify the proposed mechanical model and the results strongly validate that the proposed mechanical model successfully eliminates the displacement singularity caused by unbalanced resultant with good convergence and accuracy to obtain stress and displacement for shallow tunnel excavation. A parametric investigation is subsequently conducted to study the influence of tunnel depth, lateral coefficient, and free surface range on stress and displacement distribution in geomaterial. • A new mechanical model is proposed to obtain reasonable stress and displacement for shallow tunnel excavation. • The displacement singularity caused by unbalanced resultant of shallow tunnel excavation is eliminated. • Solution convergence is guaranteed using an approximate and iterative method. [ABSTRACT FROM AUTHOR]

In this paper, we consider the representation and extension of the analytic functions of three variables by special families of functions, namely branched continued fractions. In particular, we establish new symmetric domains of the analytical continuation of Lauricella–Saran's hypergeometric function  F K  with certain conditions on real and complex parameters using their branched continued fraction representations. We use a technique that extends the convergence, which is already known for a small domain, to a larger domain to obtain domains of convergence of branched continued fractions and the PC method to prove that they are also domains of analytical continuation. In addition, we discuss some applicable special cases and vital remarks. [ABSTRACT FROM AUTHOR]

We first study the plane strain problem associated with an incompressible liquid inclusion having an n-fold axis of symmetry which is embedded in an infinite isotropic elastic matrix subjected to uniform remote hydrostatic stresses. A closed-form solution is derived using Muskhelishvili's complex variable formulation, a four-term conformal mapping function and the application of analytic continuation. The pair of analytic functions characterizing the elastic field in the matrix is completely determined in elementary closed-form. Explicit expressions are obtained and graphically illustrated for the internal uniform hydrostatic stresses within the liquid inclusion and the hoop stress along the liquid–solid interface on the matrix side. The closed-form solution for a linearly compressible liquid inclusion having an n-fold axis of symmetry is also obtained. [ABSTRACT FROM AUTHOR]

In this article, we investigate the problem of computing the $ * $ ∗ -product of domains in $ \mathbb {C}^N $ C N . Assuming that $ 0\in G\subset \mathbb {C}^N $ 0 ∈ G ⊂ C N is an arbitrary Runge domain and $ 0\in D\subset \mathbb {C}^N $ 0 ∈ D ⊂ C N is a bounded, smooth and linearly convex domain (or a non-decreasing union of such ones), we establish a geometric relation between $ D*G $ D ∗ G and another domain in $ \mathbb {C}^N $ C N which is 'extremal' (in an appropriate sense) with respect to a special coefficient multiplier dependent only on the dimension N. Next, for N = 2, we derive a characterization of the latter domain expressed in terms of planar geometry. These two results, when combined together, give a formula which allows to calculate $ D*G $ D ∗ G for two-dimensional domains D and G satisfying the outlined assumptions. [ABSTRACT FROM AUTHOR]

We recall the complex structure on the generalized loop spaces Wk,2(S, X), where S is a compact real manifold with boundary and X is a complex manifold, and prove a Hartogs-type extension theorem for holomorphic maps from certain domains in generalized loop spaces. [ABSTRACT FROM AUTHOR]

We consider the problem of restoring a two‐dimensional convex source of gravity based on values of its potential function on a union of half‐planes. With that aim we prove a Paley–Wiener type theorem for functions of exponential type in a sector. In particular, we find the maximal set of analytic continuation whose complement is convex inside a sector. Our result serves as a sectorial analog of G. Polya's indicator theorem. It corrects, simplifies, and extends a result of M. Morimoto. It also improves a related result of M. Dzhrbashyan and A. Avetisyan. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the concept of an analytic prolongation of a local Riemannian metric. We propose a generalization of the notion of completeness realized as an analytic prolongation of an arbitrary Riemannian metric. Various Riemannian metrics are studied, primarily those related to the structure of the Lie algebra 픤 of all Killing vector fields for a local metric. We introduce the notion of a quasi-complete manifold, which possesses the property of prolongability of all local isometries to isometries of the whole manifold. A classification of pseudo-complete manifolds of small dimensions is obtained. We present conditions for the Lie algebra of all Killing vector fields 픤 and its stationary subalgebra 픥 of a locally homogeneous pseudo-Riemannian manifold under which a locally homogeneous manifold can be analytically prolonged to a homogeneous manifold. [ABSTRACT FROM AUTHOR]

In this paper, we present some analytical solutions of wormhole in four-dimensional Robertson–Walker Euclidean background considering Einstein–Gauss–Bonnet dilaton interaction with a k -essence field for k = 0. The solutions are obtained assuming some restrictions on coupling function with a form of potential or with the ratio of kinetic to potential energy of the dilaton field. Euclidean wormhole, in one case, evolves through early transient inflationary era and after graceful exit from inflation, it asymptotically yields a radiation dominated era or a matter dominated era. In another solution, initial Euclidean wormhole is accompanied by oscillation of scale factor in Euclidean time at some late era which asymptotically yields exponential expansion. The violation of the null energy condition can be avoided asymptotically. The potential is also found to decay sharply. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The paper establishes an analytical extension of two ratios of Lauricella–Saran hypergeometric functions F K with some parameter values to the corresponding branched continued fractions in their domain of convergence. The PC method used here is based on the correspondence between a formal triple power series and a branched continued fraction. As additional results, analytical extensions of the Lauricella–Saran hypergeometric functions F K (a 1 , a 2 , 1 , b 2 ; a 1 , b 2 , c 3 ; z) and F K (a 1 , 1 , b 1 , b 2 ; a 1 , b 2 , c 3 ; z) to the corresponding branched continued fractions were obtained. To illustrate this, we provide some numerical experiments at the end. [ABSTRACT FROM AUTHOR]

We propose an effective method for the solution of the plane problem of an edge dislocation in the vicinity of a circular inhomogeneity with Steigmann–Ogden interface. Using analytic continuation, the pair of analytic functions defined in the infinite matrix surrounding the inhomogeneity can be expressed in terms of the pair of analytic functions defined inside the circular inhomogeneity. Once the two analytic functions defined in the circular inhomogeneity are expanded in Taylor series with unknown complex coefficients, the Steigmann–Ogden interface condition can be written explicitly in complex form. Consequently, all of the complex coefficients appearing in the Taylor series can be uniquely determined so that the two pairs of analytic functions are then completely determined. An explicit and general expression of the image force acting on the edge dislocation is derived using the Peach–Koehler formula. [ABSTRACT FROM AUTHOR]

The paper considers the issue of constructing basises in the solution space of the system of partial differential equations, which is satisfied by the Lauricella hypergeometric function F D (N) (a ; b , c ; z) , depending on N complex variables (z 1 , ... , z N) =: z and having complex parameters (a 1 , ... , a N) =: a , b, c. For an arbitrary number N of variables, we have obtained explicit representations for such basis functions in the vicinity of points (0 , ... , 0) and (∞ , ... , ∞) in terms of the Horn type hypergeometric series in N variables. For some of these functions we have obtained formulas of analytic continuation. The found continuation formulas are important for calculating the solution of the Riemann – Hilbert problem with piecewise constant coefficients and studying its geometrical meaning. Besides, these formulas are effective for solving the parameters problem for the Schwarz – Christoffel integral and calculating conformal mapping of complex-shaped polygons. [ABSTRACT FROM AUTHOR]

The electronic, magnetic and optical properties of the double perovskites Ca2NiOsO6 and Fe-doped derivative were calculated using the full potential linearized augmented plane wave technique through the GGA + U with PBE exchange correlation functionals. The calculations show that both of the systems are half-metallic with Fe-doped system as a ferromagnet, whereas the undoped system shows the ferrimagnetic ordering. Additionally, the study is extended for calculating the Mott parameters through dynamical mean field theory (DMFT) with the maximum entropy model (MEM). It is found that the MIT model parameters (U, β) for Ca2NiOsO6 and Ca2Fe0.50Ni0.50OsO6 systems are (5.7eV, 6.0(eV)−1) and (6.0eV, 6.0(eV)−1), respectively. Furthermore, the calculations agree with optical Drude peak analysis. The optical conductivity, reflectivity, absorptivity, ELOSS function, dielectric function refractive index and sum-rule are also explored in relation to the photoinduced behaviors of the materials. [ABSTRACT FROM AUTHOR]

In this article, we give a characterization of nontrivial zeros of the Riemann zeta function using two real integrals. Using this characterization we can provide simple proof of the fact that the Riemann zeta function has no nontrivial real zeros. We also establish that this function takes negative values on the real axis within the critical region. [ABSTRACT FROM AUTHOR]

Finite-part integration is a recent method of evaluating a convergent integral in terms of the finite-parts of divergent integrals deliberately induced from the convergent integral itself [E. A. Galapon, The problem of missing terms in term by term integration involving divergent integrals, Proc. R. Soc. A 473 (2017) 20160567]. Within the context of finite-part integration of the Stieltjes transform of functions with logarithmic growths at the origin, the relationship is established between the analytic continuation of the Mellin transform and the finite-part of the resulting divergent integral when the Mellin integral is extended beyond its strip of analyticity. It is settled that the analytic continuation and the finite-part integral coincide at the regular points of the analytic continuation. To establish the connection between the two at the isolated singularities of the analytic continuation, the concept of regularized limit is introduced to replace the usual concept of limit due to Cauchy when the later leads to a division by zero. It is then shown that the regularized limit of the analytic continuation at its isolated singularities equals the finite-part integrals at the singularities themselves. The treatment gives the exact evaluation of the Stieltjes transform in terms of finite-part integrals and yields the dominant asymptotic behavior of the transform for arbitrarily small values of the parameter in the presence of arbitrary logarithmic singularities at the origin. [ABSTRACT FROM AUTHOR]

We study some of the analytic properties of a Dirichlet series defined by the sequence of the Tribonacci numbers, such as its analytic continuation to a meromorphic function of the whole plane and some of its special values. These results are generalisations of earlier properties already established for the Fibonacci Dirichlet series and other Dirichlet series defined by degree 2 recurrence sequences. [ABSTRACT FROM AUTHOR]

In this paper, we will evaluate integrals that define the conditional expectation, variance and characteristic function of stochastic processes with respect to fractional Brownian motion (fBm) for all relevant Hurst indices, i.e. H ∈ (0 , 1). Particularly, the fractional Ornstein–Uhlenbeck (fOU) process gives rise to highly nontrivial integration formulas that need careful analysis when considering the whole range of Hurst indices. We will show that the classical technique of analytic continuation, from complex analysis, provides a way of extending the domain of validity of an integral from H ∈ (1 / 2 , 1) to the larger domain H ∈ (0 , 1). Numerical experiments for different Hurst indices confirm the robustness and efficiency of the integral formulations presented. Moreover, we provide accurate and highly efficient financial option pricing results for processes that are related to the fOU process, with the help of Fourier cosine expansions. [ABSTRACT FROM AUTHOR]

We show that all local holomorphic solutions of matrix soliton equations of parabolic type admit an analytic continuation to globally meromorphic functions of a spatial variable. As examples, we consider the matrix Korteweg–de Vries equation and the matrix modified Korteweg–de Vries equation, as well as various versions of the matrix nonlinear Schrödinger equation. [ABSTRACT FROM AUTHOR]

In this paper, we present a general method, based on the techniques of analytic continuation and conformal mapping, for the analytic solution of Eshelby's problem concerned with a two-dimensional inclusion of arbitrary shape in an infinite homogeneous and isotropic nonlinearly coupled thermoelectric plane. The inclusion is subjected to a prescribed uniform electric current-free thermoelectric potential gradient and a uniform energy flux-free temperature gradient. The corresponding boundary value problem is studied in both the physical and image planes. The closed-form general solution is found to be exact provided that the associated mapping function contains only a finite number of terms. Elementary expressions for the internal electric current density and energy flux in the physical plane are obtained. Examples of elliptical, hypotrochoidal and rectangular inclusions are presented to demonstrate the solution method. Interestingly, in this case Eshelby's uniformity property is found invalid for an elliptical inclusion. [ABSTRACT FROM AUTHOR]

We present a simple algebraic method for the analytic continuation of harmonic sums with integer real or purely imaginary indices near negative and positive integers. We provide a MATHEMATICA code for exact expansion of harmonic sums in a small parameter near these integers. As an application, we consider the analytic continuation of the anomalous dimension of twist-1 operators in the ABJM model, which contains nested harmonic sums with purely imaginary indices. We found that in the BFKL-like limit the result has the same single-logarithmic behavior as in = 4 SYM and QCD, however, we did not find a general expression for the "BFKL Pomeron" eigenvalue in this model. For the slope function, we found full agreement with the expansion of the known general result and give predictions for the first three perturbative terms in the expansion of the next-to-slope function. The proposed method of analytic continuation can also be used for other generalization of nested harmonic sums. [ABSTRACT FROM AUTHOR]

We report multipronged progress on the stochastic averaging approach to numerical analytic continuation of imaginary-time correlation functions computed by quantum Monte Carlo simulations. After reviewing the conventional maximum-entropy approach and established stochastic analytic continuation methods, we present several new developments in which the configurational entropy of the sampled spectrum plays a key role. Parametrizing the spectrum as a large number of δ -functions in continuous frequency space, an exact calculation of the entropy lends support to a simple goodness-of-fit criterion for the optimal sampling temperature. We also compare spectra sampled in continuous frequency with those from amplitudes sampled on a fixed frequency grid. Insights into the functional form of the entropy in different cases allow us to demonstrate equivalence in a generalized thermodynamic limit (large number of degrees of freedom) of the average spectrum and the maximum-entropy solution, with different parametrizations corresponding to different forms of the entropy in the prior probability. These results revise prevailing notions of the maximum-entropy method and its relationship to stochastic analytic continuation. In further developments of the sampling approach, we explore various adjustable (optimized) constraints that allow sharp low-temperature spectral features to be resolved, in particular at the lower frequency edge. The constraints, e.g., the location of the edge or the spectral weight of a quasi-particle peak, are optimized using a statistical criterion based on entropy minimization under the condition of optimal fit. We show with several examples that this method can correctly reproduce both narrow and broad quasi-particle peaks. We next introduce a parametrization for more intricate spectral functions with sharp edges, e.g., power-law singularities. We present tests with synthetic data as well as with real simulation data for the spin-1/2 Heisenberg chain, where a divergent edge of the dynamic structure factor is due to deconfined spinon excitations. Our results demonstrate that distortions of sharp edges or quasi-particle peaks, which arise with other analytic continuation methods, propagate and cause artificial spectral features also at higher energies. The constrained sampling methods overcome this problem and allow analytic continuation of spectra with sharp edge features at unprecedented fidelity. We present results for S = 1 / 2 Heisenberg 2-leg and 3-leg ladders to illustrate the ability of the methods to resolve spectral features arising from both elementary and composite excitations. Finally, we also propose how the methods developed here could be used as "pre processors" for analytic continuation by machine learning. Edge singularities and narrow quasi-particle peaks being ubiquitous in quantum many-body systems, we expect the new methods to be broadly useful and take numerical analytic continuation to a new quantitative level in many applications. [ABSTRACT FROM AUTHOR]

We present novel reconstruction and stability analysis methodologies for two-dimensional, multi-coil MRI, based on analytic continuation ideas. We show that the 2-D, limited-data MRI inverse problem, whereby the missing parts of 퐤 -space (Fourier space) are lines parallel to either k 1 or k 2 (i.e., the 퐤 -space axis), can be reduced to a set of 1-D Fredholm type inverse problems. The Fredholm equations are then solved to recover the 2-D image on 1-D line profiles ("slice-by-slice" imaging). The technique is tested on a range of medical in vivo images (e.g., brain, spine, cardiac), and phantom data. Our method is shown to offer optimal performance, in terms of structural similarity, when compared against similar methods from the literature, and when the 퐤 -space data is sub-sampled at random so as to simulate motion corruption. In addition, we present a Singular Value Decomposition (SVD) and stability analysis of the Fredholm operators, and compare the stability properties of different 퐤 -space sub-sampling schemes (e.g., random vs uniform accelerated sampling). [ABSTRACT FROM AUTHOR]

Traveling waves in woodpile chains are typically nanoptera, which are composed of a central solitary wave and exponentially small oscillations. These oscillations have been studied using exponential asymptotic methods, which typically require an explicit form for the leading‐order behavior. For many nonlinear systems, such as granular woodpile chains, it is not possible to calculate the leading‐order solution explicitly. We show that accurate asymptotic approximations can be obtained using numerical approximation in place of the exact leading‐order behavior. We calculate the oscillation behavior for Toda woodpile chains, and compare the results to exponential asymptotics based on previous methods from the literature: long‐wave approximation and tanh‐fitting. We then use numerical analytic continuation methods based on Padé approximants and the adaptive Antoulas–Anderson (AAA) method. These methods are shown to produce accurate predictions of the amplitude of the oscillations and the mass ratios for which the oscillations vanish. Exponential asymptotics using an AAA approximation for the leading‐order behavior is then applied to study granular woodpile chains, including chains with Hertzian interactions—this method is able to calculate behavior that could not be accurately approximated in previous studies. [ABSTRACT FROM AUTHOR]

The paper is devoted to the piece-wise analytic case of Meijer's G function G p , p m , n . While the problem of its analytic continuation was solved in principle by Meijer and Braaksma we show that in the 'balanced' case m + n = p the formulas take a particularly simple form. We derive explicit expressions for the values of these analytic continuations on the banks of the branch cuts. It is further demonstrated that particular cases of this type of G function having integer parameter differences satisfy identities similar to the Miller–Paris transformations of the generalized hypergeometric function. Finally, we give a presumably new integral evaluation involving G p , p m , n function with m = n and apply it for summing a series involving digamma function and related to the power series coefficients of the product of two generalized hypergeometric functions with shifted parameters. [ABSTRACT FROM AUTHOR]

Using the Ramanujan summation method, we derive some unusual formulas for a class of Euler sums (including divergent Euler sums) similar to the classical relations due to Euler. [ABSTRACT FROM AUTHOR]

Given a circular triangle T having a zero angle at a point at infinity and two equal nonzero angles, it is shown that a conformal mapping of T onto a half-plane can be continued to a semi-infinite strip by applying the Riemann–Schwarz symmetry principle. The problem of analytic continuation of such a mapping arises as an auxiliary problem in constructing a conformal mapping of an L-shaped domain onto a half-plane. [ABSTRACT FROM AUTHOR]

For the Lauricella function , which is a hypergeometric function of several complex variables , analytic continuation formulas are constructed that correspond to the intersection of an arbitrary number of singular hyperplanes of the form , , These formulas give an expression for the considered function in the form of linear combinations of Horn hypergeometric series in variables satisfying the same system of partial differential equations as the original series defining in the unit polydisk. By applying these formulas, the function and Euler-type integrals expressed in terms of can be efficiently computed (with the help of exponentially convergent series) in the entire complex space in the complicated cases when the variables form one or several groups of "very close" quantities. This situation is referred to as crowding, with the term taken from works concerned with conformal maps. [ABSTRACT FROM AUTHOR]

We formulate and prove Cutkosky’s Theorem regarding the discontinuity of Feynman integrals in the massive one-loop case up to the involved intersection index. This is done by applying the techniques to treat singular integrals developed in Fotiadi et al. (Topology 4(2):159–191, 1965). We write one-loop integrals as an integral of a holomorphic family of holomorphic forms over a compact cycle. Then, we determine at which points simple pinches occur and explicitly compute a representative of the corresponding vanishing sphere. This also yields an algorithm to compute the Landau surface of a one-loop graph without explicitly solving the Landau equations. We also discuss the bubble, triangle and box graph in detail. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, some progress has been made in solving the problem of calculating the parameters of the Schwarz–Christoffel integral realizing a conformal mapping of a canonical domain onto a polygon. It is shown that an effective solution of this problem can be found by applying the formulas of analytic continuation of the Lauricella function , which is a hypergeometric function of complex variables. Several new formulas for such a continuation of the function are presented that are oriented to the calculation of the parameters of the Schwarz–Christoffel integral in the "crowding" situation. An example of solving the parameter problem for a complicated polygon is given. [ABSTRACT FROM AUTHOR]

The Jost functions, constructed by fitting available partial cross-sections for the elastic p7Be scattering with J π = 3 + , 2 − , are analytically continued to complex energies, where the resonances are located as their zeros. In addition to the resonance energies and widths, the residues of the S -matrix at the corresponding poles, as well as the Asymptotic Normalization Constants (ANC) are determined. The fitting is done using the semi-analytic representation of the Jost function with proper analytic structure, defined on the Riemann surface whose topology involves not only the square-root but also the logarithmic branching caused by the Coulomb interaction. [ABSTRACT FROM AUTHOR]

Among general functions of two variables f(x, y), analytic functions f(z) with z = x + iy form a very important special case. One consequence of analyticity turns out to be that 2-D finite difference (FD) formulas can be made remarkably accurate already for small stencil sizes. This article discusses some key properties of such complex plane FD formulas. Application areas include numerical differentiation, interpolation, contour integration, and analytic continuation. [ABSTRACT FROM AUTHOR]

We use complex variable methods to derive an analytical solution to the problem in plane elasticity associated with a circular elastic inhomogeneity with an eccentric interphase layer when the matrix is subjected to uniform remote in-plane stresses and the interphase layer undergoes uniform in-plane eigenstrains. The complex coefficients appearing in all three pairs of analytic functions characterizing the elastic fields in the composite are uniquely determined by solving two decoupled sets of linear algebraic equations obtained by enforcing the continuity conditions of tractions and displacements across the two perfect circular interfaces with the aid of analytic continuation. A simple analytical solution is also derived when the circular inhomogeneity becomes a traction-free hole and the interphase layer and the matrix have equal shear modulus but distinct Poisson's ratios. The non-uniform mean stress inside the circular inhomogeneity and the hoop stress along the edge of the circular hole are calculated and illustrated graphically. [ABSTRACT FROM AUTHOR]

The Horn hypergeometric series of two variables and corresponding systems of partial differential equations are considered. A method is proposed for deriving formulas for analytic continuation of arbitrary series of this type in the form of linear combinations of other solutions of the system of equations satisfied by the original series. As an example, formulas for continuation of two series from the well-known Horn list are constructed. [ABSTRACT FROM AUTHOR]

It is shown that any real analytic solution can be continued to a globally meromorphic function of the spatial variable at each fixed value of the time variable . In the totally focusing case, it is also shown that any local real analytic solution can be extended to a real analytic function on a maximum strip that is parallel to the axis and depends on the solution. [ABSTRACT FROM AUTHOR]

Let 픤 be the Lie algebra of all Killing vector fields on a locally homogeneous, analytic Riemannian manifold M, 픥 be a stationary subalgebra of 픤, G be the simply connected group generated by the algebra 픤, H be the subgroup of G generated by the subalgebra 픥, 픷 be the center of the algebra g, r be its radical, and [픤; 픤] be its commutator subgroup. If dim (픥 ∩ (픷 + [픤, 픤])) = dim (픥 ∩ [픤, 픤]), then H is closed in G. If for any semisimple subalgebra 픭 ⊂ 픤 satisfying the condition 픭 + 픯 = 픤, the relation (픭 + 픷) ∩ 픥 = 픭 ∩ 픥 holds, then H is closed in G. We also examine the analytic continuation of the given local, analytic Riemannian manifold. [ABSTRACT FROM AUTHOR]

In this paper, we present some wormhole configurations using analytic and numerical solutions of the Euclidean field equations considering Einstein–Gauss–Bonnet dilaton coupled interaction in the Robertson–Walker metric. Analytic solutions invoking slow-roll type approximation for some potentials yield identical wormholes in the early era. Eventually, in later era, the universe enters to an expanding era much faster than usual inflation accompanied by oscillations with analytic continuation τ = i t. The numerical solutions of wormholes in terms of the scale factor a (τ) are almost identical for some standard potentials of the dilaton field, except for the evolution of potentials. The Hubble parameter and deceleration parameter obtained by curve fit of a (τ) of the numerical solution of wormhole yield an inflationary era away from the throat of the wormhole using τ = i t. A sharp decay of the potential is also observed. [ABSTRACT FROM AUTHOR]

We prove a conjecture of Matsusaka on the analytic continuation of hyperbolic Eisenstein series in weight 2 on the full modular group \mathrm {SL}_{2}(\mathbb {Z}). [ABSTRACT FROM AUTHOR]

For the Lauricella hypergeometric function F D (N) with an arbitrary number of variables z 1 , ... , z N , we construct formulas for analytic continuation into the vicinity of hyperplanes { z j = z l } and their intersections providing that all variables are close to unit. Such formulas represent the function F D (N) near the point (1 , ... , 1) as linear combinations of N–multiple hypergeometric series that are solutions of the same system of partial differential equations as F D (N) . 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 F D (N) outside the unit polydisk. [ABSTRACT FROM AUTHOR]

We consider the Lauricella hypergeometric function F D (N) , depending on N ≥ 2 variables z 1 , ... , z N , and obtain formulas for its analytic continuation into the vicinity of a singular set which is an intersection of the hyperplanes { z j = z l }. It is assumed that all N variables are large in modulo. This formulas represent the function F D (N) outside of the unit polydisk in the form of linear combinations of other N-multiple hypergeometric series that are solutions of the same system of partial differential equations as F D (N) . The derived hypergeometric series are N-dimensional analogues of the Kummer solutions that are well known in the theory of the classical hypergeometric Gauss equation. [ABSTRACT FROM AUTHOR]

Via a unified geometric approach, certain generalized trigonometric functions with two parameters are analytically extended to maximal domains on which they are univalent. Some consequences are deduced concerning radius of convergence for the Maclaurin series, commutation with rotation, continuation beyond the domain of univalence, and periodicity. [ABSTRACT FROM AUTHOR]

Many queueing systems can be modelled as two-dimensional random walks with reflective boundaries, discrete, continuous or mixed. Stationary probabilities are one of the most sought after statistical quantities in queueing analysis. However, explicit expressions are only available for a very limited number of models. Therefore, tail asymptotic properties become more important, since they provide insightful information on the structure of the tail probabilities, and often lead to approximations, performance bounds, algorithms, among possible other applications. In this survey, we provide key ideas of a kernel method, developed from the classical kernel method in analytic combinatorics, for studying so-called exact tail asymptotic properties in stationary probabilities for this type of random walk. [ABSTRACT FROM AUTHOR]

We explicitly express the spectral determinant of Friedrichs Dirichlet Laplacians on the 2-dimensional hyperbolic (Gaussian curvature − 1) cones in terms of the cone angle and the geodesic radius of the boundary. [ABSTRACT FROM AUTHOR]

Closed expressions using the Lerch function for a definite integral are derived and evaluated. Some of these closed expressions are given in Gradshteyn and Ryzhik. Some special cases of the integral are derived and discussed. The majority of the results in this work are new. [ABSTRACT FROM AUTHOR]

An analogue of Brylinski's knot beta function is defined for a compactly supported (Schwartz) distribution T on d -dimensional Euclidean space. This is a holomorphic function on a right half-plane. If T is a (uniform) double-layer on a compact smooth hypersurface, then the beta function has an analytic continuation to the complex plane as a meromorphic function, and the residues are integrals of invariants of the second fundamental form. The first few residues are computed when d = 2 and d = 3. [ABSTRACT FROM AUTHOR]

A new static equilibrium mechanical model on noncircular shallow tunnelling considering initial stress field is proposed in this paper by constraining far-field ground surface, and a mixed boundary value problem thus forms with elimination of displacement singularity at infinity. By applying analytic continuation, the mixed boundaries are transformed to a homogeneous Riemann–Hilbert problem, which is subsequently solved via an efficient and accurate iterative method with boundary conditions of static equilibrium, displacement single-valuedness, and traction along tunnel periphery. The Lanczos filtering technique is used in the final stress and displacement solution to reduce the Gibbs phenomena caused by the constrained far-field ground surface to obtain more reasonable results. Several numerical cases are conducted to intensively verify the proposed solution by examining the utilization of Lanczos filtering, the elimination of displacement singularity, and solution convergence of constraining arc and truncation number, and comparisons with existing solutions, and all the results are in good agreements. Finally, more numerical cases are conducted to investigate the stress and deformation distribution along ground surface and tunnel periphery considering different tunnel geometries, lateral coefficients, and tunnel depths. Further discussions on the defects of the proposed solution are also conducted for objectivity. [Display omitted] • A new mechanical model is proposed for noncircular shallow tunnelling. • Far-field ground surface is constrained to equilibrate unbalanced resultant due to tunnelling. • Displacement singularity at infinity caused by unbalanced resultant is eliminated. • The solution simultaneously gives correct stress and reasonable displacement. [ABSTRACT FROM AUTHOR]