residue


Also found in: Dictionary, Thesaurus, Medical, Legal, Financial, Wikipedia.

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 ?
The spokesman added that excessive pesticide residues in food may arise from the trade not observing Good Agricultural Practice, e.
The tender will be for two years for 1,000 metric tonnes a day with "a capping price of Rs 5,500 per metric tonne has been kept for agro residue based pellets and Rs 6,600 per tonne for pellets or briquettes of torrefied agro residue.
A simple and fast analytical method using High Performance Liquid Chromatography with ultraviolet detection (HPLC-UV) was used to determine residues of commonly used agricultural insecticides (imidacloprid, thiametoxam, profenofos, endosulfan, spinosad and deltamethrin) in multi-floral raw honey of Apis dorsata F.
LO materials are halide-free and were neither the source of the chlorides nor the cause of the residue.
Further, it is observed that during directly returning residues to soil, some quantity of crop residue is pulverized into soil while still a large amount is also placed on top of soil.
Pacific Ag said it is nation's largest builder and operator of supply chains for agricultural residue.
At the last sampling date, after residue recovery, soil samples were air dried.
Compared with the control formulation, as the level of tofu residue increased, dough viscosity and cookie hardness decreased.
Food processing a tool to pesticide residue dissipation A review.
Leveraging a global network of laboratories and food experts, SGS provides a comprehensive range of food safety and quality solution including analytical tests, pesticide residue testing (http://www.