"In mathematics, a power series (in one variable) is an infinite series of the form..."
Techniques for expressing functions as power series or other types of series.
Introduction to Sequences and Series: A general overview of the fundamental concepts of sequences and series, including definitions, terms, notations, and basic properties.
Convergence and Divergence: A study of the conditions that determine whether a series converges (i.e., approaches a finite limit) or diverges (i.e., does not have a limit). This includes tests such as the ratio, root, integral, comparison, and alternating series tests.
Power Series: An investigation of series that can be expressed as a sum of power functions of a variable, called power series. The properties and applications of power series, including radius and interval of convergence, differentiation and integration, and Taylor and Maclaurin series.
Fourier Series: A discussion of the representation of periodic functions as infinite series of sines and cosines, known as Fourier series. The convergence and properties of Fourier series, the Fourier transform, and Fourier analysis.
Laurent Series: A description of series expansions of functions that have singularities or poles, known as Laurent series. The properties and applications of Laurent series in complex analysis, such as residue calculus and contour integration.
Generating Functions: A study of series expansions of functions that encode combinatorial or arithmetic data, called generating functions. The types and properties of generating functions, including ordinary generating functions, exponential generating functions, and Dirichlet generating functions.
Asymptotic Series: An examination of series expansions that approximate functions or integrals in the limit of large or small parameters, known as asymptotic series. The techniques and applications of asymptotic expansions, such as the method of steepest descent, Laplace transform, and WKB approximation.
Zeta and L-Functions: An overview of certain classes of series expansions of number-theoretic functions, such as the Riemann zeta function, Dirichlet L-functions, and modular forms. Their properties, connections to algebraic geometry, and open problems in number theory.
Applications in Physics and Engineering: A survey of examples of series expansions in physical and engineering problems, such as solving differential equations, approximating solutions to boundary value problems and eigenvalue problems, and analyzing stochastic processes. The use of series expansions in simulations, numerical methods, and experimental data analysis.
Computer Algebra and Symbolic Computation: An introduction to the use of computer algebra systems, such as Mathematica or Maple, for manipulating and computing with series expansions. The syntax, functions, and examples of computer algebra commands for series expansions and related topics.
Power series: A series of the form ∑an(x-c)n, where c and an are constants and x is a variable.
Taylor series: A power series that represents a function as an infinite sum of terms computed from the function's values and derivatives at a single point.
Laurent series: A series that includes both positive and negative powers of the variable x, with coefficients that may be complex valued.
Fourier series: A series that decomposes a function into a finite or infinite sum of sinusoidal functions.
Dirichlet series: A series consisting of infinite terms that arise in number theory.
Hilbert series: A generating function that counts the dimensions of the vector spaces of a graded object.
Puiseux series: A type of power series in which the exponents may be rational or irrational.
Poincaré-Lelong series: A power series that describes the distribution of a holomorphic function’s zeros in a domain.
Jacobi theta functions: A family of functions that satisfy certain useful identities and are used in several branches of mathematics and physics.
Euler-Maclaurin series: A method for approximating a sum of functions using integrals of derivatives.
Heine series: A series of rational functions that arise in the study of orthogonal polynomials.
Gamma series: A series that relates the gamma function to other special functions.
Zeta series: A series that relates the Riemann zeta function to other special functions.
Bessel series: A series that represents solutions to differential equations of mathematical physics.
Legendre series: A series that represents solutions to differential equations of mathematical physics.
Chebyshev series: A series that approximates a function by a finite or infinite sum of Chebyshev polynomials.
Laurent-Puiseux series: A generalization of Laurent series, allowing for fractional powers of the variable x.
Stieltjes series: A series used in the study of continued fractions and continued fractions with complex coefficients.
Mellin series: A different way of evaluating complex integrals that is particularly useful in number theory.
Hermite series: A series that represents solutions to differential equations of mathematical physics.
Laguerre series: A series that represents solutions to differential equations of mathematical physics.
"an represents the coefficient of the nth term..."
"Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions."
"In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function."
"In many situations, c (the center of the series) is equal to zero, for instance when considering a Maclaurin series."
"In such cases, the power series takes the simpler form..."
"Beyond their role in mathematical analysis, power series also occur in combinatorics as generating functions..."
"...generating functions (a kind of formal power series)..."
"...and in electronic engineering (under the name of the Z-transform)."
"The familiar decimal notation for real numbers can also be viewed as an example of a power series..."
"...with integer coefficients..."
"...but with the argument x fixed at 1⁄10."
"In number theory, the concept of p-adic numbers is also closely related to that of a power series."
"Borel's theorem implies that every power series is the Taylor series of some smooth function."
"When considering a Maclaurin series, the power series takes the simpler form..."
"Power series also occur in combinatorics as generating functions..."
"...and in electronic engineering (under the name of the Z-transform)."
"Borel's theorem implies that every power series is the Taylor series of some smooth function."
"In number theory, the concept of p-adic numbers is also closely related to that of a power series."
"Power series arise as Taylor series of infinitely differentiable functions."