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Power series

About: Power series is a research topic. Over the lifetime, 7169 publications have been published within this topic receiving 118970 citations.


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Book ChapterDOI
01 Jan 2000
TL;DR: In the 1890s, Henri Poincare took upon himself the task of gleaning as much information from the DEs describing the whole solar system as was possible and the result was the invention of one of the most powerful branches of mathematics (topology) and the realization that the qualitative analysis of (nonlinear) DEs could be very useful.
Abstract: A variety of techniques including the Frobenius method of infinite power series could solve almost all linear DEs of physical interest. However, some very fundamental questions such as the stability of the solar system led to DEs that were not linear, and for such DEs no analytic (including series representation) solution existed. In the 1890s, Henri Poincare, the great French mathematician, took upon himself the task of gleaning as much information from the DEs describing the whole solar system as was possible. The result was the invention of one of the most powerful branches of mathematics (topology) and the realization that the qualitative analysis of (nonlinear) DEs could be very useful.

698 citations

Book
01 Jan 1988
TL;DR: In this article, two equations of state for the properties of steam, which are in the form of power series in pressure and density, are developed from the HGK84 formulation.
Abstract: Two equations of state for the properties of steam, which are in the form of power series in pressure and density, are developed from the HGK84 formulation. These equations are of high accuracy in the equilibrium region where extensive measurements exist. They also accurately represent the extrapolated data in the metastable region between the vapor saturation and spinodal lines. The accuracy of the representations as a function of the number of terms of the series is presented. Their greatest utility is their use for high accuracy calculations that involve small to moderate departures from ideal‐gas behavior. Conversion relationships for the second through the tenth coefficients of the pressure and density series, which apply to the corresponding virial coefficients, are presented. The pressure and density expansions are advantageous for efficient numerical calculations of water vapor properties in the equilibrium and metastable regions.

657 citations

Journal ArticleDOI
TL;DR: In this paper, the authors considered the realization problem of two-dimensional linear filters from a system theoretic point of view, where the input-output behavior of such a system is defined by formal power series in two variables, and a Nerode state space is constructed.
Abstract: The realization problem of two-dimensional linear filters is approached from a system theoretic point of view. The input-output behavior of such a system is defined by formal power series in two variables, and a Nerode state space is constructed. This state space is, in general, infinite dimensional. If the power series is rational, the dynamics of the filter is described by updating equations on finite-dimensional local state space. The notions of local reachability and observability are defined in a natural way and an algorithm for obtaining a reachable and observable realization is given. In general, local reachability and observability do not imply the minimality of the realization.

648 citations

Journal ArticleDOI
Gregory H. Wannier1
TL;DR: In this article, the authors constructed wave functions which satisfy the Schr\"odinger equation for a potential which is a sum of a periodic and a uniform field term, in the form of power series in the field strength.
Abstract: Following up an earlier communication, wave functions are constructed which satisfy the Schr\"odinger equation for a potential which is a sum of a periodic and a uniform field term. The wave functions are Houston modifications of Bloch type functions; the Bloch functions form an orthogonal set whose members are fully determined except for phase. The theory exhibits them in the form of power series in the field strength; the unmodified Bloch band functions form the zero order term of that series. The solutions themselves do not allow for a Zener effect, but the fact that they are only given as power series in $E$ may imply that there is a remainder term causing interband transitions; it would have to be asymptotically smaller than any power of $E$. Instead of constructing time dependent solutions of the Schr\"odinger equation one can take the time independent functions to construct an effective Hamiltonian for electrons in one band; it has the form (16). Certain indeterminacies are attached to this form of representation; it is shown, however, that final physical answers are unique. The study furnishes an incidental proof that k-space is a finite space consisting in its entirety of what is customarily called the first Brillouin zone. An appendix treats the case of degenerate bands; such bands have singularities in k-space even in the absence of a field. The difficulty is circumvented by working with a set which is not yet diagonalized but free of singularities; these intermediate functions can be continued as power series in $E$ in the same way as nondegenerate band functions.

593 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023126
2022296
2021264
2020317
2019281
2018227