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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.
The principal part at = of a function = = ()is the portion of the Laurent series consisting of terms with negative degree. [1] That is, = is the principal part of at .If the Laurent series has an inner radius of convergence of , then () has an essential singularity at if and only if the principal part is an infinite sum.
A Laurent polynomial over may be viewed as a Laurent series in which only finitely many coefficients are non-zero. The ring of Laurent polynomials R [ X , X − 1 ] {\displaystyle R\left[X,X^{-1}\right]} is an extension of the polynomial ring R [ X ] {\displaystyle R[X]} obtained by "inverting X {\displaystyle X} ".
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 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.
Pierre Alphonse Laurent (18 July 1813 – 2 September 1854) was a French mathematician, engineer, and Military Officer best known for discovering the Laurent series, an expansion of a function into an infinite power series, generalizing the Taylor series expansion. He was born in Paris, France.
In mathematics, the regular part of a Laurent series consists of the series of terms with positive powers. [1] That is, if = ...