Topic
Closed-form expression
About: Closed-form expression is a research topic. Over the lifetime, 507 publications have been published within this topic receiving 16956 citations. The topic is also known as: analytical expression & closed-form solution.
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Book•
01 Jan 1966
TL;DR: In this article, the authors present a model for vector analysis based on the Calculus of Variations and the Sturm-Liouville theory, which includes the following: Curved Coordinates, Tensors.
Abstract: Vector Analysis. Curved Coordinates, Tensors. Determinants and Matrices. Group Theory. Infinite Series. Functions of a Complex Variable I. Functions of a Complex Variable II. Differential Equations. Sturm-Liouville Theory. Gamma-Factrial Function. Bessel Functions. Legendre Functions. Special Functions. Fourier Series. Integral Transforms. Integral Equations. Calculus of Variations. Nonlinear Methods and Chaos.
7,811 citations
TL;DR: This work proposes the use of a modified version of the correlation coefficient as a performance criterion for the image alignment problem and proposes an efficient approximation that leads to a closed form solution which is of low computational complexity.
Abstract: In this work we propose the use of a modified version of the correlation coefficient as a performance criterion for the image alignment problem. The proposed modification has the desirable characteristic of being invariant with respect to photometric distortions. Since the resulting similarity measure is a nonlinear function of the warp parameters, we develop two iterative schemes for its maximization, one based on the forward additive approach and the second on the inverse compositional method. As it is customary in iterative optimization, in each iteration the nonlinear objective function is approximated by an alternative expression for which the corresponding optimization is simple. In our case we propose an efficient approximation that leads to a closed form solution (per iteration) which is of low computational complexity, the latter property being particularly strong in our inverse version. The proposed schemes are tested against the forward additive Lucas-Kanade and the simultaneous inverse compositional algorithm through simulations. Under noisy conditions and photometric distortions our forward version achieves more accurate alignments and exhibits faster convergence whereas our inverse version has similar performance as the simultaneous inverse compositional algorithm but at a lower computational complexity.
575 citations
Book•
01 Jan 1953
TL;DR: The Transcendental Functions of Whittaker and Watson's celebrated "Modern Analysis" were described in detail in the first volume of Bateman's "Guide to the Functions" as discussed by the authors.
Abstract: The work of which this book is the first volume might be described
as an up-to-date version of Part II. The Transcendental Functions of Whittaker and Watson's celebrated "Modern Analysis". Bateman (who
was a pupil of E. T. Whittaker) planned his "Guide to the Functions" on a gigantic scale. In addition to a detailed account of the properties
of the most important functions, the work was to include the historic
origin and definition of, the basic formulas relating to, and a bibliography
for all special functions ever invented or investigated. These
functions were to be catalogued and classified under twelve different
headings according to their definition by power series, generating functions,
infinite products, repeated differentiations, indefinite integrals, definite integrals, differential equations, difference equations, functional
equations, trigonometric series, series of orthogonal functions, or integral
equations. Tables of definite integrals representing each function and
numerical tables of a few new functions were to form part of the "Guide".
An extensive table of definite integrals and a Guide to numerical tables
of special functions were planned as companion works.
433 citations
TL;DR: In this paper, the authors give a precise definition of the elementary functions and develop the theory of integration of functions of a single varia' Z. They also give an algorithm for determining the elementary integrability of those elementary functions which can be built up (roughly speaking) using only the rational operations, exponentiation and taking logarithms; however, if these exponentiations and logariths can be replaced by adjoining constants and performing algebraic operations, the algorithm, as it is presented here, cannot be applied.
Abstract: This paper deals with the problem of telling whether a given elementary function, in the sense of analysis, has an elementary indefinite integral. In ?1 of this work, we give a precise definition of the elementary functions and develop the theory of integration of functions of a single varia' Z. By using functions of a complex, rather than a real variable, we can limit ourselves to exponentiation, taking logs, and algebraic operations in defining the elementary functions, since sin, tan- 1, etc., can be expressed in terms of these three. Following Ostrowski [9], we use the concept of a differential field. We strengthen the classical Liouville theorem and derive a number of consequences. ?2 uses the terminology of mathematical logic to discuss formulations of the problem of integration in finite terms. ?3 (the major part of this paper) uses the previously developed theory to give an algorithm for determining the elementary integrability of those elementary functions which can be built up (roughly speaking) using only the rational operations, exponentiation and taking logarithms; however, if these exponentiations and logarithms can be replaced by adjoining constants and performing algebraic operations, the algorithm, as it is presented here, cannot be applied. The man who established integration in finite terms as a mathematical discipline was Joseph Liouville (1809-1882), whose work on this subject appeared in the years 1833-1841. The Russian mathematician D. D. Mordoukhay-Boltovskoy (1876-1952) wrote much on this and related matters. The present writer received his introduction to this subject through the book [10] by the American J. F. Ritt
321 citations
TL;DR: In this article, a set of functional equations for the general three point function of exponential fields in quantum Liouville theory on the sphere is considered, and it is shown that the expression proposed by the Zamolodchikovs solves these functional equations.
279 citations