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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.
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 area of the blue region converges on the Euler–Mascheroni constant, which is the 0th Stieltjes constant.. In mathematics, the Stieltjes constants are the numbers that occur in the Laurent series expansion of the Riemann zeta function:
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.
3.3.4 Laurent series expansion. 3.4 Bochner–Martinelli formula (Cauchy's integral formula II) 3.5 Identity theorem. 3.6 Biholomorphism.
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.
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.