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(1) In the theory of numbers, number a is the residue of number b modulo m if the difference a — b is divisible by m (a, b, and m > 0 are integers). For example, the number 24 is the residue of the number 3 modulo 7, since 24 - 3 is divisible by 7. The set of m integers, each of which is the residue of one and only one of the numbers 0, 1, . . . , m—1, is said to be a complete residue system modulo m. For example, the numbers 1, 6, 11, 16, 21, 26 form the complete residue system modulo 6. The number a is called a residue of order n (n≥ 2—an integer) modulo m if there exists an integer x such that the difference xn - a is divisible by m. If this is not the case, then a is called a nonresidue of order n. For example, 2 and 3 are a residue and a nonresidue of the second order (quadratics) modulo 7.
REFERENCEVinogradov, I. M. Osnovy teorii chisel, 7th ed. Moscow, 1965.
A. A. KARATSUBA
(2) In the theory of analytic functions, the residue of a single-valued analytic function f(z) at an isolated singular point Z0 is the coefficient of (z - Z0)—1 in the expansion of f(z) in a series of powers of z - Z0 (Laurent series) valid in a neighborhood of Z0. The notation used is f(z). If Y is a circle with center at Z0 and sufficiently small radius (a circle so small that within it the function f(z) has no singular points other than Z0), then
The importance of residues stems from the following theorem. Let f(Z) be a single-valued function analytic in a domain D, with the exception of isolated singular points, and let G be a simple closed rectifiable curve belonging to domain D together with its interior and not passing through the singular points of f(z); if Z1, . . . , zn are all the singular points of f(z) lying within G, then
Since it is relatively simple to calculate residues, this theorem is an effective means for evaluating integrals.
A. A. GONCHAR