# residue

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## residue

*Law*what is left of an estate after the discharge of debts and distribution of specific gifts

*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## Residue

(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 x^{n} - *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.

### REFERENCE

Vinogradov, 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 Z_{0} is the coefficient of *(z -* Z_{0})^{—1} in the expansion of *f(z)* in a series of powers of *z -* Z_{0} (Laurent series) valid in a neighborhood of Z_{0}. The notation used is *f(z)*. If *Y* is a circle with center at Z_{0} and sufficiently small radius (a circle so small that within it the function f(z) has no singular points other than Z_{0}), 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 Z_{1}, . . . , *z _{n}* 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

## residue

[′rez·ə‚dü]*z*) at an isolated singularity

*z*

_{0}is given by (1/2π

*i*) ∫ ƒ(

*z*)

*dz*along a simple closed curve interior to an annulus about

*z*

_{0}; equivalently, the coefficient of the term (

*z*-

*z*

_{0})

^{-1}in the Laurent series expansion of ƒ(

*z*) about

*z*

_{0}.

*m*of order

*n*, where

*m*and

*n*are integers, is a remainder that results from raising some integer to the

*n*th power and dividing by

*m*.