Search results
Results From The WOW.Com Content Network
Geometrically, the two Laurent series may have non-overlapping annuli of convergence. Two Laurent series with only finitely many negative terms can be multiplied: algebraically, the sums are all finite; geometrically, these have poles at , and inner radius of convergence 0, so they both converge on an overlapping annulus.
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.
For type I, the singularities are (movable) double poles of residue 0, and the solutions all have an infinite number of such poles in the complex plane. The functions with a double pole at have the Laurent series expansion
The residue Res(f, c) of f at c is the coefficient a −1 of (z − c) −1 in the Laurent series expansion of f around c. Various methods exist for calculating this value, and the choice of which method to use depends on the function in question, and on the nature of the singularity.
In 2D Euclidean field theory, the operator product expansion is a Laurent series expansion associated with two operators. In such an expansion, there are finitely many negative powers of the variable, in addition to potentially infinitely many positive powers of the variable.
Its zeros in the left halfplane are all the negative even integers, and the Riemann hypothesis is the conjecture that all other zeros are along Re(z) = 1/2. In a neighbourhood of a point , a nonzero meromorphic function f is the sum of a Laurent series with at most finite principal part (the terms with negative index values):
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.
1 Examples. 2 Convergence. 3 See also. 4 References. ... Laurent series, Fourier series, Liouville-Neumann series, formal power series, and Puiseux series. Convergence