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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]

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]

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]

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]

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]

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]

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]

We present a method using contour integration to derive definite integrals and their associated infinite sums which can be expressed as a special function. We give a proof of the basic equation and some examples of the method. The advantage of using special functions is their analytic continuation, which widens the range of the parameters of the definite integral over which the formula is valid. We give as examples definite integrals of logarithmic functions times a trigonometric function. In various cases these generalizations evaluate to known mathematical constants, such as Catalan’s constant C and π. [ABSTRACT FROM AUTHOR]

The theoretical development of fractional calculus includes the formulation of different definitions, the extension of properties from standard calculus, and the application of fractional operators to special functions. In two recent papers, incomplete versions of classical fractional operators were formulated in connection with special functions. Here, we develop the theory of incomplete fractional calculus more deeply, investigating further properties of these operators and answering some fundamental questions about how they work. By considering appropriate function spaces, we discover that incomplete fractional calculus may be used to analyse a wider class of functions than classical fractional calculus can consider. By using complex analytic continuation, we formulate definitions for incomplete Riemann-Liouville fractional derivatives, hence extending the incomplete integrals to a fully-fledged model of fractional calculus. Further properties proved here include a rule for incomplete differintegrals of products, and composition properties of incomplete differintegrals with classical calculus operations. [ABSTRACT FROM AUTHOR]

In this work, we introduce and investigate an interesting operator Q based on fractional derivative which is introduced by Owa and Srivastava in [10]. We consider a new technique to prove our results and then, we introduce two subclasses of analytic functions in the open unit disk U concerning with this operator. Some results such as inclusion relations, subordination properties, integral preserving properties and argument estimate are investigated. [ABSTRACT FROM AUTHOR]

Subspaces of the space of analytic functions on a convex domain in the complex plane that are invariant with respect to the differentiation operator are studied. The problem of continuing all functions in an invariant subspace to entire functions is investigated. A simple geometric criterion for the existence of such a continuation is obtained. A criterion for the functions from an invariant subspace to be represented by a series of exponential monomials is also obtained. These monomials are the eigenfunctions and generalized eigenfunctions of the differentiation operator on the invariant subspace. [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]

Using a generalized binomial transform and a novel binomial coefficient identity, we will show that the set of p-recursive sequences is closed under the binomial transform. Using these results, we will derive a new series representation for the dilogarithm function that converges on its domain of analyticity. Finally, we will show that this series representation results in a scheme for numerical evaluation of the dilogarithm function that is accurate, efficient, and stable. [ABSTRACT FROM AUTHOR]

We establish the existence of an internal uniform hydrostatic state of stress inside a coated non-parabolic open elastic inhomogeneity when the surrounding matrix is subjected to a concentrated couple and uniform remote in-plane stresses. The proof of existence is accomplished through analytic continuation and the introduction of a specialized form of the conformal mapping function containing an infinite number of first-order poles. The complex constants appearing in the mapping function can be uniquely determined from a resulting set of non-linear recurrence relations for given material, geometric and loading parameters. The convergence criterion for the series in the mapping function is rigorously established. A simple condition on the remote loading in the matrix is found for given elastic constants of the inhomogeneity and the matrix in order to ensure that the internal stresses inside the non-parabolic inhomogeneity are uniform and hydrostatic. The analysis indicates that the internal uniform hydrostatic stress field, the constant mean stress in the coating and the constant hoop stress along the inhomogeneity-coating interface on the coating side are all independent of the specific open shapes of the two interfaces and are also independent of the existence of the nearby concentrated couple. However, the non-parabolic open shapes of the two interfaces are significantly influenced by the concentrated couple. We also accomplish the design of an uncoated non-parabolic harmonic elastic inhomogeneity in the presence of an arbitrary number of concentrated couples applied in the matrix. [ABSTRACT FROM AUTHOR]

Introduction/purpose: Some properties of the zeta function will be shown as well as its applications in calculus, in particular the “golden nugget formula” for the value of the infinite sum 1 + 2 + 3 +···. Some applications in physics will also be mentioned. Methods: Complex plane integrations and properties of the Gamma function will be used from the definition of the function to its analytic extension. Results: From the original definition of the ζ(s) function valid for s > 1 a meromorphic function is obtained on the whole complex plane with a simple pole in s = 1. Conclusion: The relevance of the zeta function cannot be overstated, ranging from the infinite series to the number theory, regularization in theoretical physics, the Casimir force, and many other fields. [ABSTRACT FROM AUTHOR]

In this note we study the analytic continuation of the Apostol-Vu multiple Fibonacci zeta functions ζ A V F , k ( s 1 , ... , s k , s k + 1 ) = ∑ 1 ≤ m 1 < ... < m k 1 F m 1 s 1 F m 2 s 2 ... F m k s k F m 1 + m 2 + ... + m k s k + 1 , {\zeta _{AVF,k}}\left({{s_1}, \ldots ,{s_k},{s_{k + 1}}} \right) = \sum\limits_{1 \le {m_1} < \ldots < {m_k}} {{1 \over {F_{{m_1}}^{{s_1}}F_{{m_2}}^{{s_2}} \ldots F_{{m_k}}^{{s_k}}F_{{m_1} + {m_2} + \ldots + {m_k}}^{{s_{k + 1}}}}},} where s1,..., sk+1 are complex variables and Fn is the n-th Fibonacci number. We find a complete list of poles and their corresponding residues. [ABSTRACT FROM AUTHOR]

First, we give a formula for the limiting value of the higher Mahler measures for general polynomials of one variable, which refines Biswas and Monico's result. Second, we give an analytic continuation of the zeta Mahler measures for polynomials of one variable to the whole complex plane as meromorphic functions, and we show that all of their poles are simple. [ABSTRACT FROM AUTHOR]

We investigate the elastic field near the tip of an anticrack in a homogeneous decagonal quasicrystalline material subject to plane strain deformations. The phonon and phason stresses exhibit a square root singularity at the anticrack tip. Two real-valued phonon stress intensity factors and two real-valued phason stress intensity factors are introduced to scale four separate modes of deformation. We obtain four analytic functions which completely characterize the induced phonon and phason stresses as well as the displacement field. In particular, we derive a concise yet elegant representation of the anticrack contraction force. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two new subclasses of biunivalent functions analytic in the exterior of the unit disc. The bounds obtained for the zeroth, first and second coefficient improves upon earlier known results. The results are obtained by refining the well-known estimates for the initial coefficients of the Carth 'eodory functions. [ABSTRACT FROM AUTHOR]

Using the technique of analytic continuation, we derive analytical solutions to the problem of a screw dislocation in a three-phase strip. The two traction-free boundaries of the strip are perpendicular to the two perfect interfaces formed by adjacent materials. The screw dislocation can be located anywhere in the composite strip. Elementary expressions for the image force acting on the screw dislocation are obtained. When the strip is extremely thick, our solutions simply recover those in the literature. Our solutions allow us to observe some very interesting and quite unexpected phenomena concerning the mobility and stability of the screw dislocation interacting with the aforementioned interfaces and boundaries. [ABSTRACT FROM AUTHOR]

The analyticity of the entropy and relative entropy rates of continuous-state hidden Markov models is studied here. Using the analytic continuation principle and the stability properties of the optimal filter, the analyticity of these rates is established for analytically parameterized models. The obtained results hold under relatively mild conditions and cover several useful classes of hidden Markov models. These results are relevant for several theoretically and practically important problems arising in statistical inference, system identification and information theory. [ABSTRACT FROM AUTHOR]

The purpose of this article is twofold. First, we introduce the constants ζ k (α , r , q) where α ∈ (0 , 1) and study them along the lines of work done on Euler constant in arithmetic progression γ (r , q) by Briggs, Dilcher, Knopfmacher, Lehmer and some other authors. These constants are used for evaluation of certain integrals involving error term for Dirichlet divisor problem with congruence conditions and also to provide a closed form expression for the value of a class of Dirichlet L-series at any real critical point. In the second half of this paper, we consider the behaviour of the Laurent Stieltjes constants γ k (χ) for a principal character χ. In particular we study a generalization of the "Generalized Euler constants" introduced by Diamond and Ford in 2008. We conclude with a short proof for a closed form expression for the first generalized Stieltjes constant γ 1 (r / q) which was given by Blagouchine in 2015. [ABSTRACT FROM AUTHOR]

Conformal mapping and analytic continuation are employed to prove the existence of an internal uniform electroelastic field inside a non-elliptical piezoelectric inhomogeneity interacting with a screw dislocation. We focus specifically on the case when the piezoelectric matrix surrounding the inhomogeneity is subjected to uniform remote anti-plane mechanical and in-plane electrical loading and a constraint is imposed between the remote loading and the screw dislocation. The constraint can be expressed in a relatively simple decoupled form by utilizing orthogonality relationships between two corresponding eigenvectors. The internal uniform electroelastic field is found to be independent of the presence of the screw dislocation; moreover, it can be expressed in decoupled form. [ABSTRACT FROM AUTHOR]

Abstract For a positive irrational number α , we study the ordinary Dirichlet series ζ α (s) = ∑ n ≥ 1 ⌊ α n ⌋ − s and S α (s) = ∑ n ≥ 1 (⌈ α n ⌉ − ⌈ α (n − 1) ⌉) n − s. We prove relations between them and J α (s) = ∑ n ≥ 1 ({ α n } − 1 2) n − s. Motivated by the previous work of Hardy and Littlewood, Hecke and others regarding J α , we show that ζ α and S α can be continued analytically beyond the imaginary axis except for a simple pole at s = 1. Based on the latter results, we also prove that the series ζ α (s ; β) = ∑ n ≥ 0 (⌊ α n ⌋ + β) − s can be continued analytically beyond the imaginary axis except for a simple pole at s = 1. [ABSTRACT FROM AUTHOR]

Abstract This is a continuation of our previous paper [7] , in which multiple Fibonacci zeta functions of depth 2 have been studied. In this article, we consider more general situation. In particular, we prove the meromorphic continuation of the multiple Lucas zeta functions of depth d : ∑ 0 < n 1 < ⋯ < n d 1 U n 1 s 1 ⋯ U n d s d , where U n is the n -th Lucas number of first kind and ∑ i = j d Re (s i) > 0 for 1 ≤ j ≤ d. We compute a complete list of poles and their residues. We also prove that the multiple Lucas zeta values at negative integer arguments are rational. [ABSTRACT FROM AUTHOR]

Abstract Let r k (n) denote the number of representations of the positive integer n as the sum of k squares. We rigorously prove for the first time a Voronoï summation formula for r k (n) , k ≥ 2 , proved incorrectly by A.I. Popov and later rediscovered by A.P. Guinand, but without proof and without conditions on the functions associated in the transformation. Using this summation formula we establish a new transformation between a series consisting of r k (n) and a product of two Bessel functions, and a series involving r k (n) and the Gaussian hypergeometric function. This transformation can be considered as a massive generalization of well-known results of G.H. Hardy, and of A.L. Dixon and W.L. Ferrar, as well as of a classical result of A.I. Popov that was completely forgotten. An analytic continuation of this transformation yields further useful results that generalize those obtained earlier by Dixon and Ferrar. [ABSTRACT FROM AUTHOR]

Methods employing auxiliary sources are applied to electromagnetic scattering problems by 2-D noncircular scatterers. Specific cases are considered for which the singularities of the analytic continuation of the scattered field into the interior of the scatterer can be analytically determined. Then, it is shown that the amplitudes (currents) of the sources diverge and oscillate when the auxiliary curve does not enclose these singularities. Still, the electric fields computed by means of the divergent and oscillatory currents are convergent and correct. The correctness of the computed fields is validated by comparison with the exact solutions (where possible) as well as with the numerical solutions computed via commercial field simulation software. [ABSTRACT FROM AUTHOR]

In the present study a new modification of Riemann zeta function known as Lucas-balancing zeta function is introduced. The Lucas-balancing zeta function admits its analytic continuation over the whole complex plane except its poles. This series converges to a fixed rational number − 1/2 at negative odd integers. Further, in accordance to Dirichlet L-function, the analytic continuation of Lucas-balancing L-function is also discussed. [ABSTRACT FROM AUTHOR]

In this article, we prove the meromorphic continuation of the multiple Fibonacci zeta function]s of depth 2: . . . where Fn is the n-th Fibonacci number, Re(s1) > 0 and Re(s2) > 0. We compute a complete list of its poles and their residues. We also prove that multiple Fibonacci zeta values at negative integer arguments are rational. [ABSTRACT FROM AUTHOR]

We consider a circular elastic inclusion embedded in an infinite matrix each from a particular class of compressible hyperelastic materials of harmonic type. A concentrated couple is applied either inside the circular inclusion or in the matrix. Closed-form solutions of the corresponding boundary value problems are obtained using complex variable methods, in particular the principle of analytic continuation. Our analysis reveals several interesting conclusions including: the sum σ 11 + σ 22 of the normal stresses inside the inclusion remains constant when the concentrated couple is located in the surrounding matrix; the sum of the normal stresses is zero everywhere in the matrix when the concentrated couple is located inside the inclusion. [ABSTRACT FROM AUTHOR]

Let L be the unique even self-dual lattice of signature ( 25 , 1 ) . The automorphism group Aut ( L ) acts on the hyperbolic space H 25 . We study a Poincaré series E ( z , s ) defined for z in H 25 , convergent for Re ( s ) > 25 , invariant under Aut ( L ) and having singularities along the mirrors of the reflection group of L . We compute the Fourier expansion of E ( z , s ) at a “Leech cusp” and prove that it can be meromorphically continued to Re ( s ) > 25 / 2 . Analytic continuation of Kloosterman sum zeta functions imply that the individual Fourier coefficients of E ( z , s ) have meromorphic continuation to the whole s -plane. [ABSTRACT FROM AUTHOR]

We construct certain Casimir operators and study the spectral properties of their resolvents on L 2 ( Γ \ Sp 2 ( R ) ) . We define non-holomorphic multi-variable Poincaré series of exponential type for symplectic groups and continue them analytically in case of genus two for the small weight four using the above resolvents. We apply our results to describe the holomorphic projection to the weight four holomorphic discrete series in terms of Fourier coefficients by using Sturm's operator. This paper is a study of a prototype for symplectic groups and special orthogonal groups. [ABSTRACT FROM AUTHOR]

We consider the problem of numerical analytic continuation of an analytic function f ( z ) = f ( x + i y ) on a strip domain Ω + = { z = x + i y ∈ C | x ∈ R , 0 < y < y 0 } , where the data is given approximately only on the line y = 0 . This is a severely ill-posed problem. Motivated by the advantage of iterative methods for solving ill-posed problems, we propose a new modified iterative method to solve this problem under both a priori and a posteriori parameter choice rules. Moreover, some sharp error estimates between the exact solution and its approximation are proved. Some interesting numerical examples are conducted for showing that the newly-developed method works well. [ABSTRACT FROM AUTHOR]

A symbolic computation technique is used to derive closed-form expressions for an analytic continuation of the Euler–Zagier zeta function evaluated at negative integers. This continuation was recently proposed by Sadaoui. The approach presented here yields explicit contiguity identities, recurrences on the depth of the zeta values and their generating functions. Moreover, it allows to prove that the resulting multiple zeta values computed at negative integers coincide with those obtained by another analytic continuation technique that uses the Euler–MacLaurin summation formula. [ABSTRACT FROM AUTHOR]

In this work we show how by using a Padé type analytical continuation scheme, based on the Schlessinger point method, it is possible to find higher production thresholds in hadron physical problems. Recently, an extension of this numerical approach to the complex energy plane enabled the calculations of auto-ionization decay resonance poles in atomic and molecular systems. Here we use this so-called Resonances via Padé (RVP) method, to show its convergence beyond the singular point in hadron physical problems. In order to demonstrate the capabilities of the RVP method, two illustrations for the ability to identify singularities and branch points are given. In addition, two applications for hadron physical problems are given. In the first one, we identify the decay thresholds from a numerically calculated spectral function. In the second one, we use experimental data. First, we calculate the resonance pole of the f 0 ( 500 ) or σ meson using the S 0 partial wave amplitude for ππ scattering in very good agreement with the literature. Second, we use data on the cross section ratio R ( s ) for e + e − collisions and discuss the prediction of decay thresholds which proves to be difficult if the data is noisy. [ABSTRACT FROM AUTHOR]

Development of appropriate load flow model of Flexible AC Transmission System (FACTS) devices is an important issue for proper planning, control, and protection of power system. Holomorphic Embedding Load-Flow Method (HELM) is a novel technique for solving load flow nonlinear equations which overcomes the numerical problems faced by traditional iterative techniques. In order to evaluate the effects of FACTS devices in load flow problem by HELM technique, it is necessary to develop HELM modeling of these devices. This paper presents HELM modeling of Thyristor-based FACTS controllers, i.e., Static Var Compensator (SVC), Thyristor Controlled Switched Capacitor (TCSC), Thyristor Controlled Voltage Regulator (TCVR), and Thyristor Controlled Phase Angle Regulator (TCPAR). In this modeling, white germ solution is investigated along with recursive formula and controlling FACTS devices operation bounds. [ABSTRACT FROM AUTHOR]

We study the analytic continuation of Feynman integrals from the kite family, expressed in terms of elliptic generalisations of (multiple) polylogarithms. Expressed in this way, the Feynman integrals are functions of two periods of an elliptic curve. We show that all what is required is just the analytic continuation of these two periods. We present an explicit formula for the two periods for all values of t ∈ R . Furthermore, the nome q of the elliptic curve satisfies over the complete range in t the inequality | q | ≤ 1 , where | q | = 1 is attained only at the singular points t ∈ { m 2 , 9 m 2 , ∞ } . This ensures the convergence of the q -series expansion of the ELi-functions and provides a fast and efficient evaluation of these Feynman integrals. [ABSTRACT FROM AUTHOR]

In a recent paper, Hamidi and Jahangiri [C. R. Acad. Sci. Paris, Ser. I 352 (2014) 17–20] introduced and investigated the class of bi-close-to-convex functions, and determined the estimates for the general Taylor–Maclaurin coefficients of the functions therein. This note mainly aims to point out and correct the errors of the main result in the above-mentioned paper. [ABSTRACT FROM AUTHOR]

An analog of Pólya's theorem on the estimate of the transfinite diameter for a class of multivalued analytic functions with finitely many branch points and of the corresponding class of admissible compact sets located on the associated (with this function) two-sheeted Stahl-Riemann surface is obtained. [ABSTRACT FROM AUTHOR]

We prove the analytic continuation of the resolvent of the Laplacian on asymptotically hyperbolic spaces on differential forms, including high energy estimates in strips. This is achieved by placing the spectral family of the Laplacian within the framework developed, and applied to scalar problems, by the author recently, roughly by extending the problem across the boundary of the compactification of the asymptotically hyperbolic space in a suitable manner. The main novelty is that the non-scalar nature of the operator is dealt with by relating it to a problem on an asymptotically Minkowski space to motivate the choice of the extension across the conformal boundary. [ABSTRACT FROM AUTHOR]

Using Langlands's beyond endoscopy idea, we study the Asai L-function associated to a real quadratic field K/Q. We prove that the Asai L-function associated to a cuspidal automorphic representation over K has analytic continuation to the complex plane with at most a simple pole at s = 1. We then show if the L-function has a pole then the representation is a base change from Q. While this result is known using integral representations from the work of Asai and Flicker, the approach here uses novel analytic number techniques and gives a deeper understanding of the geometric side of the relative trace formula. Moreover, the approach in this paper will make it easier to grasp more complicated functoriality via beyond endoscopy. We also describe a connection of Langlands's technique to the distinguishing of representations via periods. [ABSTRACT FROM AUTHOR]

The anomalous dimension of twist-2 operators of arbitrary spin in planar $$ \mathcal{N} $$ = 4 SYM theory is found at seven loops by using the quantum spectral curve to compute values at fixed spin, and reconstructing the general result using the LLL-algorithm together with modular arithmetic. The result of the analytic continuation to negative spin is presented, and its relation with the recently computed correction to the BFKL and double-logarithmic equation is discussed. [ABSTRACT FROM AUTHOR]

In this article we obtain the meromorphic continuation of multiple zeta functions, together with a complete list of their poles and residues, by means of an elementary and simple translation formula for these multiple zeta functions. The use of matrices to express this translation formula leads, in particular, to a succinct description of the residues of the multiple zeta functions. We conclude our paper by introducing certain interesting weighted variants of multiple zeta functions. They are shown to behave particularly nicely with respect to product formulas and location of poles. [ABSTRACT FROM AUTHOR]

We prove the conjecture of Sfetsos, Siampos and Thompson that suitable analytic continuations of the Poisson–Lie T-duals of the bi-Yang–Baxter sigma models coincide with the recently introduced generalized λ -models. We then generalize this result by showing that the analytic continuation of a generic σ -model of “universal WZW-type” introduced by Tseytlin in 1993 is nothing but the Poisson–Lie T-dual of a generic Poisson–Lie symmetric σ -model introduced by Klimčík and Ševera in 1995. [ABSTRACT FROM AUTHOR]