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| Taylor series | |
|---|---|
| Name | Taylor series |
| Field | Mathematics |
| Introduced by | Brook Taylor |
| First appeared | 1715 |
| Related | Maclaurin series, power series, analytic function |
Taylor series A Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. It generalizes the Maclaurin series concept and is fundamental in the study of Brook Taylor, Leonhard Euler, Joseph-Louis Lagrange, Augustin-Louis Cauchy, and Karl Weierstrass. Taylor series link classical results in Calculus, Complex analysis, and the development of methods used by Isaac Newton, Srinivasa Ramanujan, and Pafnuty Chebyshev.
Given a function f that is infinitely differentiable at a point a, the series is formed using derivatives f^{(n)}(a) and powers of (x−a). The formal expression involves factorials n! and leads to the power series centered at a, often written alongside the Maclaurin series when a = 0. Notation commonly references expansions used by Augustin-Louis Cauchy in Cours d'analyse and the formulations later refined by Bernhard Riemann and Gottfried Wilhelm Leibniz-era developments. The coefficients are determined by local data at a point, a perspective developed in the work of Joseph-Louis Lagrange and used by Niels Henrik Abel.
Convergence of the series to the function depends on analytic continuation and radius of convergence determined by singularities in the complex plane studied by Bernhard Riemann and Karl Weierstrass. The remainder after N terms can be bounded using forms attributed to Joseph-Louis Lagrange (Lagrange form of the remainder) and integral forms linked to Augustin-Louis Cauchy. Radius of convergence is often computed via the Cauchy–Hadamard formula, a tool whose lineage includes Enrico Betti and Jacques Hadamard. For entire functions like those investigated by Sofia Kovalevskaya and Henri Poincaré, the series converges everywhere; for meromorphic functions studied by Bernhard Riemann and George Gabriel Stokes, poles restrict convergence. Analytic continuation techniques used by Riemann and Felix Klein extend domains where convergence yields function values.
Standard examples include expansions for exponential, trigonometric, and logarithmic functions. The exponential expansion used by Leonhard Euler, the sine and cosine expansions associated with Joseph Fourier and Jean le Rond d'Alembert, and the logarithm expansion applied in work by John Napier and Brook Taylor appear universally in texts by Augustin-Louis Cauchy and Carl Friedrich Gauss. Polynomial approximations like those studied by Pafnuty Chebyshev and Andrey Kolmogorov approximate functions on intervals while classical series such as the binomial series feature in treatments by Isaac Newton and Abraham de Moivre. Special function expansions—Bessel, hypergeometric, and gamma—trace to Friedrich Bessel, Carl Gustav Jacobi, Felix Klein, and Ernst Kummer.
Termwise differentiation and integration of power series are justified under uniform convergence on compact sets, principles formalized by Augustin-Louis Cauchy and later clarified by Karl Weierstrass. Multiplication and composition of series invoke convolution of coefficients, techniques used in the theories developed by Niels Henrik Abel and Bernhard Riemann. Analytic continuation, monodromy, and singularity analysis—central to the work of Riemann, Henri Poincaré, and Felix Klein—govern where series manipulations preserve equality with original functions. The uniqueness of power series coefficients at a point is an outcome leveraged by Émile Picard and André Weil in higher-dimensional generalizations.
Taylor expansions underpin numerical methods originating with Isaac Newton and Joseph-Louis Lagrange, including root-finding and finite-difference schemes used by Carl Friedrich Gauss and Arthur Cayley. In differential equations, series solutions appear in the studies of Sofia Kovalevskaya, Jean Baptiste Joseph Fourier, and George Gabriel Stokes for heat, wave, and potential problems. Approximation theory and spectral methods developed by Pafnuty Chebyshev, David Hilbert, and Norbert Wiener exploit polynomial truncations. In physics and engineering, perturbation expansions used in Paul Dirac-era quantum mechanics and asymptotic analyses by Donald Knuth-era numerical analysts rely on these series, while signal processing techniques owe lineage to approaches by Joseph Fourier and Harry Nyquist.
Generalizations include multivariable Taylor polynomials studied by Carl Gustav Jacob Jacobi and Sofia Kovalevskaya, formal power series appearing in algebraic geometry as in the work of Alexander Grothendieck, and non-commutative formal expansions used in operator theory by John von Neumann and Israel Gelfand. The concept extends to analytic manifolds investigated by Henri Cartan and André Weil and to p-adic analytic functions explored by Kurt Hensel and Alexander Ostrowski. Asymptotic series and the resurgence theory developed by Écalle and applied in semiclassical analysis link to the work of Henri Poincaré and Vladimir Arnold.