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A Laurent series is a generalization of the Taylor series, allowing terms with negative exponents; it takes the form = and converges in an annulus. [6] In particular, a Laurent series can be used to examine the behavior of a complex function near a singularity by considering the series expansion on an annulus centered at the singularity.
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. [ 1 ] [ 2 ] A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts. [ 3 ]
In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. [1] The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions.
Taylor series are used to define functions and "operators" in diverse areas of mathematics. In particular, this is true in areas where the classical definitions of functions break down. For example, using Taylor series, one may extend analytic functions to sets of matrices and operators, such as the matrix exponential or matrix logarithm.
A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, .
In a large class of singularly perturbed problems, the domain may be divided into two or more subdomains. In one of these, often the largest, the solution is accurately approximated by an asymptotic series [2] found by treating the problem as a regular perturbation (i.e. by setting a relatively small parameter to zero). The other subdomains ...
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
In mathematics, the Laurent series of a complex function is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied.