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Thus when defining formal Laurent series, one requires Laurent series with only finitely many negative terms. Similarly, the sum of two convergent Laurent series need not converge, though it is always defined formally, but the sum of two bounded below Laurent series (or any Laurent series on a punctured disk) has a non-empty annulus of convergence.
A formal power series can be loosely thought of as an object that is like a polynomial, but with infinitely many terms.Alternatively, for those familiar with power series (or Taylor series), one may think of a formal power series as a power series in which we ignore questions of convergence by not assuming that the variable X denotes any numerical value (not even an unknown value).
The formal Laurent series over a finite field: the ring of integers of F q ((T)) is the ring of formal power series F q [[T]]. Its maximal ideal is (T) (i.e. the set of power series whose constant terms are zero) and its residue field is F q. Its normalized valuation is related to the (lower) degree of a formal Laurent series as follows:
Then the valuation ring R consists of rational functions with no pole at x = a, and the completion is the formal Laurent series ring F((x−a)). This can be generalized to the field of Puiseux series K {{ t }} (fractional powers), the Levi-Civita field (its Cauchy completion), and the field of Hahn series , with valuation in all cases returning ...
In this case the algebra of formal power series is the total algebra of the monoid of natural numbers over the underlying term ring. [76] If the underlying term ring is a differential algebra, then the algebra of formal power series is also a differential algebra, with differentiation performed term-by-term.
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 complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula.
For instance as a Laurent series might suggest a series , but it already designates the ordinary formal power series =. In fact most expressions have both an interpretation when negative powers are restricted and when positive powers are restricted (i.e., as formal Laurent series in X −1 ) and their coefficients (at monomials they both ...