Basic Methods for Computing Special Functions

This paper gives an overview of methods for the numerical evaluation of special functions, that is, the functions that arise in many problems from mathematical physics, engineering, probability theory, and other applied sciences. We consider in detail a selection of basic methods which are frequently used in the numerical evaluation of special functions: converging and asymptotic series, including Chebyshev expansions, linear recurrence relations, and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. We give examples of recent software for special functions where these methods are used. We mention a list of new publications on computational aspects of special functions available on our website

Incremental Fringe Formulation for a Complex Source Point Beam Expansion

An incremental fringe formulation (IFF) for the scattering by large metallic objects illuminated by electromagnetic complex source points (CSPs) is presented. This formulation has two main advantages. First, it improves the accuracy of physical optics (PO) computations by removing spurious scattered field contributions and, at the same time, substituting them with more accurate Incremental Theory of Diffraction field contributions. Second, it reduces the complexity of PO computations because it is applicable to arbitrary illuminating fields represented in terms of a CSP beam expansion. The advantage of using CSPs is mainly due to their beam-like properties: truncation of negligible beams lowers the computational burden in the determination of the solution. Explicit dyadic expressions of incremental fringe coefficients are derived for wedge-shaped configurations. Comparisons between the proposed method, PO and the Method of Moments (MoM) are provided

Propagation models and inversion approaches for subsurface and through-wall imaging

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ITD Double-Edge Diffraction for Complex Source Beam Illumination

A new high-frequency incremental theory of diffraction (ITD) formulation for the double diffraction by metallic wedges when illuminated by complex source points (CSP) is provided. The main motivation is the extension of the class of problems that can be studied using asymptotic (i.e., ray-based and incremental) methods by providing a double diffraction description for CSP, which are considered because they are efficient to analyze electrically large structures. The new formulation provides an accurate asymptotic description of the interaction between two edges in an arbitrary configuration, including slope diffraction contributions. Advantages of the ITD formulation for CSP illumination include avoiding the typical ray-caustic impairments of the GTD/UTD ray techniques and not requiring ray tracing in complex space. Numerical results are presented and compared to a Method-of-Moments analysis to demonstrate the accuracy of the solution

Efficient computation of multiple diffracted short-pulses using ray fields

Numerically efficient results for short pulse propagation in a complex environment is presented in terms of multiple diffraction. As such, the time domain (TD) diffraction propagators are based on discretization of a generic short-pulse in terms of narrow rectangles, and on closed form representations of diffracted fields when the excitation is a narrow rectangular pulse

Symmetry Properties of Spheroidal Functions With Respect to Their Parameter

Spheroidal wave functions depend on a parameter c. Their behavior with respect to changes of sign of c is investigated, and explicit formulas are provided. Sample applications of the resulting symmetry rules are provided for some electromagnetic scattering problems

Recent advances in the Incremental Theory of Diffraction for Complex Source Point illumination

The accurate prediction of the far field radiated or scattered by large structures, such as large reflector antennas, requires efficient techniques for representing the illuminating field. Complex Source Points (CSP) inherently contain information about the source directivity, hence they can be used as the basis function to expand a given, but arbitrary, radiating wave field [1-3], such as the field incident on an antenna or a more general complex structure. As a consequence, a CSP field representation, when combined with the analytic continuation in complex space of typical ray-techniques such as the Geometrical and the Uniform Geometrical Theory of Diffraction (GTD/UTD), may provide a very efficient tool to estimate the fields radiated by large objects [4]. In this framework, an extension of the Incremental Theory of Diffraction (ITD) formulation for the scattering by wedges illuminated by CSP has been introduced [5], which essentially overcomes the typical impairments of the GTD/UTD ray techniques associated with possible ray caustics and with the difficulties of ray tracing in complex space. On the other hand, when dealing with the description of the field radiated by large structures, many of the existing electromagnetic codes resort to a Physical Optics (PO) representation also with an arbitrary incident field. It is however well known that the PO approach does not always produce accurate field predictions [6]. A significative augmentation of the PO field estimate can be achieved by including along the structure's edges a line integration of an incremental fringe field, that acts as a correction term for the field estimate. Several techniques have been published to derive these elementary contributions, leading to Physical Theory of Diffraction (PTD), Elementary Edge Waves/Incremental Length Diffraction Coefficient (EEW/ILDC), and ITD. In this work we discuss some recent advances in the incremental formulation for the field diffracted by edges in perfect electric conductor (PEC) objects when illuminated by a CSP beam expansion, with application to the analysis of large reflector antennas. A fringe formulation of the field diffracted by wedges with PEC faces when illuminated by a single CSP has been recently presented [7]. At each point on the edge the incremental fringe correction term is deduced from tangential canonical problems as the difference between the local ITD diffracted field [5] and an appropriate incremental end-point PO field (IEPO) scattered by the half-lit plane tangent to the edge [8]. The total spurious effects due to the presence of the edge are corrected by adiabatically distributing and integrating the local incremental fringe field coefficients along the line of the edges. This formulation yields more accurate predictions of the radiated field. For configurations in which several metallic edges are present and for grazing aspects of incidence and observation, the correct interactions between the edges in the problem need to be properly accounted for. Hence we introduce correct incremental double-diffraction coefficients for CSP illumination in the first-order fringe formulation [9]. These incremental coefficients have been derived by a proper analytical continuation of their real counterparts [10]. The formulation provides an accurate first-order asymptotic description of the interaction between two edges, which is valid both for skewed separate wedges and for edges joined by a common PEC face. It also includes a double incremental slope diffraction augmentation, which provides the correct dominant high-frequency incremental contribution at grazing aspects of incidence and observation. The total doubly-diffracted field is obtained from a double spatial integration along each of the two edges on which consecutive diffractions occur. In the application to the analysis of large reflector antennas the first-order fringe correction to the PO scattered fields tends to fail in those directions parallel to the aperture plane. Here, the addition of the incremental double diffracted field provides the correct estimation of the radiated field