Special functions of mathematical physics : a unified introduction with applications
01 Jan 1988-
TL;DR: The theory of classical or thogonal polynomials of a discrete variable on both uniform and non-uniform lattices has been given a coherent presentation, together with its various applications in physics as discussed by the authors.
Abstract: With students of Physics chiefly in mind, we have collected the material on special functions that is most important in mathematical physics and quan tum mechanics We have not attempted to provide the most extensive collec tion possible of information about special functions, but have set ourselves the task of finding an exposition which, based on a unified approach, ensures the possibility of applying the theory in other natural sciences, since it pro vides a simple and effective method for the independent solution of problems that arise in practice in physics, engineering and mathematics For the American edition we have been able to improve a number of proofs; in particular, we have given a new proof of the basic theorem ( 3) This is the fundamental theorem of the book; it has now been extended to cover difference equations of hypergeometric type ( 12, 13) Several sections have been simplified and contain new material We believe that this is the first time that the theory of classical or thogonal polynomials of a discrete variable on both uniform and nonuniform lattices has been given such a coherent presentation, together with its various applications in physics"
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01 Jan 1991
TL;DR: The theory of elliptic integrals was introduced by Abel as discussed by the authors, who proposed a special function to evaluate integrals, which is called integral sine, logarithm, exponential function, probability integral and so on.
Abstract: At first only elementary functions were studied in mathematical analysis. Then new functions were introduced to evaluate integrals. They were named special functions: integral sine, logarithms, the exponential function, the probability integral and so on. Elliptic integrals proved to be the most important. They are connected with rectification of arcs of certain curves. The remarkable idea of Abel to replace these integrals by the corresponding inverse functions led to the creation of the theory of elliptic functions.
1,007 citations
TL;DR: In this article, the Girsanov theorem for the functionals of a fractional Brownian motion using the stochastic calculus of variations was proved for the Ito formula.
Abstract: Since the fractional Brownian motion is not a semi-martingale, the usual Ito calculus cannot be used to define a full stochastic calculus. However, in this work, we obtain the Ito formula, the Ito–Clark representation formula and the Girsanov theorem for the functionals of a fractional Brownian motion using the stochastic calculus of variations.
713 citations
Cites background or methods from "Special functions of mathematical p..."
...The Gauss hypergeometric function F (a;b;c;z) (for details, see [ 11 ]) is dened for any a;b, any z; jzj 1=2, the hypergeometric function is no more continuous in t but we have [ 11 ] :...
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...Using the generating functions of Hermite polynomials (see e.g. [ 11 ]), we see that...
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TL;DR: This work details the observation of non‐Gaussian apparent diffusion coefficient (ADC) profiles in multi‐direction, diffusion‐weighted MR data acquired with easily achievable imaging parameters, and uses it to show that non‐ Gaussian profiles arise consistently in various regions of the human brain.
Abstract: This work details the observation of non-Gaussian apparent diffusion coefficient (ADC) profiles in multi-direction, diffusion-weighted MR data acquired with easily achievable imaging parameters (b 1000 s/mm2). A technique is described for modeling the profile of the ADC over the sphere, which can capture non-Gaussian effects that can occur at, for example, intersections of different tissue types or white matter fiber tracts. When these effects are significant, the common diffusion tensor model is inappropriate, since it is based on the assumption of a simple underlying diffusion process, which can be described by a Gaussian probability density function. A sequence of models of increasing complexity is obtained by truncating the spherical harmonic (SH) expansion of the ADC measurements at several orders. Further, a method is described for selection of the most appropriate of these models, in order to describe the data adequately but without overfitting. The combined procedure is used to classify the profile at each voxel as isotropic, anisotropic Gaussian, or non-Gaussian, each with reference to the underlying probability density function of displacement of water molecules. We use it to show that non-Gaussian profiles arise consistently in various regions of the human brain where complex tissue structure is known to exist, and can be observed in data typical of clinical scanners. The performance of the procedure developed is characterized using synthetic data in order to demonstrate that the observed effects are genuine. This characterization validates the use of our method as an indicator of pathology that affects tissue structure, which will tend to reduce the complexity of the selected model.
556 citations
TL;DR: The exponential generalized beta distribution (EGB) as discussed by the authors is a five-parameter beta distribution which nests the generalized beta and gamma distributions and includes more than thirty distributions as limiting or special cases.
455 citations
Journal Article•
418 citations