Teubner series

Teubner series
Title	Teubner series
Language	German, English
Publisher	B. G. Teubner, Springer
Subject	Mathematical series, analytical methods
Country	Germany

Teubner series is a designation used in mathematical literature for collections of series and expansions published in monographs and textbooks by the Teubner publishing house and successors. The term commonly refers to formal power series, asymptotic expansions, and orthogonal series treatments found in works by European analysts and applied mathematicians. These series techniques have played roles in classical analysis, spectral theory, and mathematical physics through connections to special functions and operator theory.

Definition and formulation

A Teubner series typically denotes a sequence expansion expressed as a formal power series or orthogonal expansion in contexts developed by authors associated with the Teubner imprint, emphasizing rigorous formulation and proof. Authors present series as sums Σ_{n=0}^∞ a_n φ_n where coefficients a_n and basis functions φ_n are specified by boundary conditions or operator spectra, mirroring constructions in the theory of Fourier, Laurent, and Taylor expansions. Foundational texts outline existence and uniqueness in the spirit of results by Bernhard Riemann, Niels Henrik Abel, Joseph Fourier, Augustin-Louis Cauchy, and Karl Weierstrass, and discuss convergence in relation to classical theorems by Georg Cantor, Émile Borel, Henri Poincaré, Richard Dedekind, and Sofia Kovalevskaya.

Convergence criteria and properties

Convergence analysis in these treatments uses criteria inspired by tests and theorems from the European analytic tradition: comparison tests related to work of Augustin-Louis Cauchy and Karl Weierstrass, ratio tests akin to Edward Titchmarsh's expositions, and uniform convergence frameworks associated with Ulisse Dini and Frigyes Riesz. Many Teubner-associated expositions examine absolute, conditional, uniform, and pointwise modes of convergence, invoking summability methods related to Georg Cantor and regularization approaches influenced by John von Neumann and Paul Dirac. Properties such as analytic continuation, radius of convergence, and singularity structure are connected to results by Gaston Darboux, Bernhard Reimann (sic), Karl Pearson (note: Pearson is a statistician often referenced), and classical spectral results of David Hilbert and John von Neumann.

Examples and special cases

Canonical examples appear in monographs treating the Taylor series expansion of elementary functions, orthogonal polynomial series such as Legendre polynomial expansions discussed alongside Chebyshev polynomials, and Fourier-type expansions on domains treated by Joseph Fourier. Special-case treatments include asymptotic series for Bessel functions in the style of Friedrich Bessel, Airy function expansions connected to George Biddell Airy, and eigenfunction series for Sturm–Liouville problems aligned with work by Sturm (Charles-François Sturm) and Joseph Liouville. Other illustrative cases involve elliptic function expansions linked to Niels Henrik Abel and Carl Gustav Jacobi, and generating-function techniques reflecting methods popularized by Srinivasa Ramanujan and G. H. Hardy.

Historical development and contributors

The development of the monographic tradition published by Teubner traces through 19th- and 20th-century European analysis. Key contributors whose works were carried in the Teubner catalog include editors and authors in the lineage of Felix Klein, Hermann Schwarz, Leopold Kronecker, Ernst Zermelo, Hermann Weyl, David Hilbert, Emmy Noether, Paul Ehrenfest, and Ludwig Boltzmann for mathematical physics context. Later contributors and commentators encompassed Constantin Carathéodory, Stefan Banach, Andrey Kolmogorov, Norbert Wiener, Salomon Bochner, Israel Gelfand, Elliott Lieb (Elliott H. Lieb), and Michael Reed in spectral and operator-theoretic expositions. Editors at B. G. Teubner and successor firms curated editions by authors such as Ernst Kreyszig, E. T. Whittaker, and George B. Arfken whose textbooks disseminated these series techniques.

Applications in analysis and physics

Teubner-style expositions have been applied to boundary-value problems in potential theory treated by Pierre-Simon Laplace, spectral decompositions in quantum mechanics following Erwin Schrödinger, perturbation series in quantum field theory influenced by Paul Dirac and Richard Feynman, and scattering theory connected to Ludwig Boltzmann and John von Neumann. In fluid dynamics they inform expansions used by George Gabriel Stokes and Osborne Reynolds; in statistical mechanics they underlie cluster expansions referencing J. Willard Gibbs and Ludwig Boltzmann. Techniques are also deployed in signal analysis traditions traced to André-Marie Ampère and Joseph Fourier, and in numerical analysis frameworks advanced by Carl Friedrich Gauss and John von Neumann.

Generalizations and related series

Generalizations include multivariate power series treated in the spirit of Bernhard Riemann and Henri Poincaré, q-series and basic hypergeometric series linked to Gasper Rahman-style developments and to earlier work by Srinivasa Ramanujan and G. H. Hardy, and distributional expansions influenced by Laurent Schwartz and Paul Dirac. Related constructs appear in orthogonal function theory developed by André Weil, generalized eigenfunction expansions of Marshall Stone, and noncommutative series contexts connected to Alain Connes and Israel Gelfand. Contemporary treatments extend into numerical spectral methods associated with John Boyd and operator-theoretic expansions in the vein of Tosio Kato.

Category:Mathematical series