residue

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residue

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

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 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.

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 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 Residuef(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

residue

[′rez·ə‚dü]
(chemical engineering)
The substance left after distilling off all but the heaviest components from crude oil in petroleum refinery operations. Also known as bottoms; residuum.
Solids deposited onto the filter medium during filtration. Also known as cake; discharged solids.
(geology)
The in-place accumulation of rock debris which remains after weathering has removed all but the least soluble constituent.
(mathematics)
The residue of a complex function ƒ(z) at an isolated singularity z0 is given by (1/2π i) ∫ ƒ(z) dz along a simple closed curve interior to an annulus about z0; equivalently, the coefficient of the term (z-z0)-1 in the Laurent series expansion of ƒ(z) about z0.
In general, a coset of an ideal in a ring.
A residue of 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.
References in periodicals archive ?
Never-before-circulated bills are free of drug residue, so Connor used them as a standard against which he would compare the circulated bills.
Drug residue on the exterior, or vapors seeping from the interior, can be detected to signal inspectors that an enclosure needs further scrutiny.
Chicken and eggs could contain too many drug residues which have been fed to the birds and may affect consumers, according to a new report.
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The product is a spray that detects drug residue on flat surfaces.
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Even so, the agency adheres to a strict safety margin in its drug residue testing, according to Hubbert in Washington, D.
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It would appear that the sensitivity of NGRC drug residue tests has increased manyfold and drug residues are now detectable for more than one or two weeks.
The latest accounting for 2001 sampling has been released by the Food and Drug Administration's National Milk Drug Residue program.
Specifically, he said, "All slaughter cattle marketed through or to Equity must be accompanied by a signed producer certificate, which indicates that to the best of the producer's knowledge, the cattle have not been fed any feed containing protein derived from mammalian tissues - meat or bone meal - and that none of the cattle have an illegal level of drug residue.
If drug residue is present, a color change will occur within 10 seconds.