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This volume contains the proceedings of the Special Session on Several Complex Variables, which was held during the first USA-Uzbekistan Conference on Analysis and Mathematical Physics from May 20-23, 2014, at California State University, Fullerton.This volume covers a wide variety of topics in pluripotential theory, symplectic geometry and almost complex structures, integral formulas, holomorphic extension, and complex dynamics. In particular, the reader will find articles on Lagrangian submanifolds and rational convexity, multidimensional residues, S-parabolic Stein manifolds, Segre varieties, and the theory of quasianalytic functions.

A wandering domain for a diffeomorphism We first prove that the measure (or the capacity) of these wandering domains is exponentially small, with an upper bound of the form The second part of the paper is devoted to the construction of near-integrable Gevrey systems possessing wandering domains, for which the capacity (and thus the measure) can be estimated from below. We suppose

Let f be an analytic function on a simply-connected compact continuum E of the complex z -plane. This might be an interval of the real line, where f might be real analytic. How can we calculate good estimates of the analytic continuation of f to other points z ∈ C ? How can we estimate the locations of real or complex singularities of f ? We review both the theory and the practice of some existing methods for these problems and propose that excellent results can be obtained from the computation of rational approximations of f by the AAA algorithm. In the case of analytic functions of two or more variables, the rational approximations are applied along line segments or other analytic arcs.

Transformation based on downward continuation of potential fields is an important tool in their interpretation – depths of shallowest important sources can be determined by means of stable downward continuation algorithms. We analyse here selected properties of one from these algorithms (based on Tikhonov’s regularization approach) from the scope of two most important discretization parameters – dimensions of the areal coverage of the interpreted field and the sampling interval size. Estimation of the source depth is based on the analysis of computed L -norms for various continuation depths. A typical local minimum of these norms disappears at the source depth. We show on several synthetic bodies (sphere, horizontal cylinder, vertical rod) and also real-world data-sets (results from a magnetic survey for unexploded ordnance detection) that there is a need for relatively large surroundings around the interpreted anomalies. Beside of this also the sampling step plays its important role – grids with finer sampling steps give better interpretation results, when using this downward continuation method. From this point of view, this method is more suitable for the interpretation of objects in near surface and mining geophysics (anomalies from cavities, unexploded ordnance objects and ore bodies). Anomalies should be well developed and separable, and densely sampled. When this is not valid, several algorithms of interpolation and extrapolation (grid padding methods) can improve the interpretation properties of studied downward continuation method.

In the present paper, we establish an analytic continuation of Drinfeld logarithms by using the techniques introduced in Furusho (Tunis J Math 4(3):559–586, 2022) . This result can be seen as an analogue of the analytic continuation of the elliptic integrals of the first kind for Drinfeld modules.

We describe maximal, in a sense made precise, 𝕃-analytic continuations of germs at +∞ of unary functions definable in the o-minimal structure ℝan,exp on the Riemann surface 𝕃 of the logarithm. As one application, we give an upper bound on the logarithmic-exponential complexity of the compositional inverse of an infinitely increasing such germ, in terms of its own logarithmic-exponential complexity and its level. As a second application, we strengthen Wilkie’s theorem on definable complex analytic continuations of germs belonging to the residue field 𝓡poly of the valuation ring of all polynomially bounded definable germs.

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 FK 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.

Among general functions of two variables f ( x , y ), analytic functions f ( z ) with z = x + i y 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.

This note proposes an algorithm for identifying the poles and residues of a meromorphic function from its noisy values on the imaginary axis. The algorithm uses Möbius transform and Prony’s method in the frequency domain. Numerical results are provided to demonstrate the performance of the algorithm.

The paper establishes an analytical extension of two ratios of Lauricella–Saran hypergeometric functions FK 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 FK(a1,a2,1,b2;a1,b2,c3;z) and FK(a1,1,b1,b2;a1,b2,c3;z) to the corresponding branched continued fractions were obtained. To illustrate this, we provide some numerical experiments at the end.