Topic
Elliptic function
About: Elliptic function is a research topic. Over the lifetime, 4204 publications have been published within this topic receiving 78561 citations.
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Book•
01 Jun 1985
TL;DR: In this paper, the Laurent series is used for expanding functions in Taylor series, and the calculus of residues is used to expand functions in Laurent series volumes II, III, and IV.
Abstract: Volume I, Part 1: Basic Concepts: I.1 Introduction I.2 Complex numbers I.3 Sets and functions. Limits and continuity I.4 Connectedness. Curves and domains I.5. Infinity and stereographic projection I.6 Homeomorphisms Part 2: Differentiation. Elementary Functions: I.7 Differentiation and the Cauchy-Riemann equations I.8 Geometric interpretation of the derivative. Conformal mapping I.9 Elementary entire functions I.10 Elementary meromorphic functions I.11 Elementary multiple-valued functions Part 3: Integration. Power Series: I.12 Rectifiable curves. Complex integrals I.13 Cauchy's integral theorem I.14 Cauchy's integral and related topics I.15 Uniform convergence. Infinite products I.16 Power series: rudiments I.17 Power series: ramifications I.18 Methods for expanding functions in Taylor series Volume II, Part 1: Laurent Series. Calculus of Residues: II.1 Laurent's series. Isolated singular points II.2 The calculus of residues and its applications II.3 Inverse and implicit functions II.4 Univalent functions Part 2: Harmonic and Subharmonic Functions: II.5 Basic properties of harmonic functions II.6 Applications to fluid dynamics II.7 Subharmonic functions II.8 The Poisson-Jensen formula and related topics Part 3: Entire and Meromorphic Functions: II.9 Basic properties of entire functions II.10 Infinite product and partial fraction expansions Volume III, Part 1: Conformal Mapping. Approximation Theory: III.1 Conformal mapping: rudiments III.2 Conformal mapping: ramifications III.3 Approximation by rational functions and polynomials Part 2: Periodic and Elliptic Functions: III.4 Periodic meromorphic functions III.5 Elliptic functions: Weierstrass' theory III.6 Elliptic functions: Jacobi's theory Part 3: Riemann Surfaces. Analytic Continuation: III.7 Riemann surfaces III.8 Analytic continuation III.9 The symmetry principle and its applications Bibliography Index.
1,426 citations
TL;DR: In this article, a Jacobi elliptic function expansion method was proposed to construct the exact periodic solutions of nonlinear wave equations, which includes some shock wave solutions and solitary wave solutions.
1,231 citations
Book•
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 paper, the authors assume that P operates on half-densities rather than functions and show that P is a positive elliptic self-adjoint pseudodifferential operator of order m>0 on a compact boundaryless C ∞ manifold.
Abstract: Let X be a compact boundaryless C ∞ manifold and let P be a positive elliptic self-adjoint pseudodifferential operator of order m>0 on X. For technical reasons we will assume that P operates on half-densities rather than functions.
928 citations
TL;DR: In this article, the authors obtained the best possible estimates for the remainder term in the asymptotic formula for the spectral function of an arbitrary elliptic (pseudo-)differential operator.
Abstract: In this paper we shall obtain the best possible estimates for the remainder term in the asymptotic formula for the spectral function of an arbitrary elliptic (pseudo-)differential operator. This is achieved by means of a complete description of the singularities of the Fourier transform of the spectral function for low frequencies.
918 citations