# reduced residue system

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## 1—10 of 161 matching pages

##### 1: 27.2 Functions

##### 2: 13.28 Physical Applications

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###### §13.28(i) Exact Solutions of the Wave Equation

►The reduced wave equation ${\nabla}^{2}w={k}^{2}w$ in paraboloidal coordinates, $x=2\sqrt{\xi \eta}\mathrm{cos}\varphi $, $y=2\sqrt{\xi \eta}\mathrm{sin}\varphi $, $z=\xi -\eta $, can be solved via separation of variables $w={f}_{1}(\xi ){f}_{2}(\eta ){\mathrm{e}}^{\mathrm{i}p\varphi}$, where … ►###### §13.28(iii) Other Applications

►For dynamics of many-body systems see Meden and Schönhammer (1992); for tomography see D’Ariano et al. (1994); for generalized coherent states see Barut and Girardello (1971); for relativistic cosmology see Crisóstomo et al. (2004).##### 3: 27.15 Chinese Remainder Theorem

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►The Chinese remainder theorem states that a system of congruences $x\equiv {a}_{1}\phantom{\rule{0.949em}{0ex}}(mod{m}_{1}),\mathrm{\dots},x\equiv {a}_{k}\phantom{\rule{0.949em}{0ex}}(mod{m}_{k})$, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod $m$), where $m$ is the product of the moduli.
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►By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod ${m}_{1}$), (mod ${m}_{2}$), (mod ${m}_{3}$), and (mod ${m}_{4}$), respectively.
Because each residue has no more than five digits, the arithmetic can be performed efficiently on these residues with respect to each of the moduli, yielding answers ${a}_{1}\phantom{\rule{0.949em}{0ex}}(mod{m}_{1})$, ${a}_{2}\phantom{\rule{0.949em}{0ex}}(mod{m}_{2})$, ${a}_{3}\phantom{\rule{0.949em}{0ex}}(mod{m}_{3})$, and ${a}_{4}\phantom{\rule{0.949em}{0ex}}(mod{m}_{4})$, where each ${a}_{j}$ has no more than five digits.
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##### 4: Bibliography H

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Sums with cylindrical functions that reduce to the probability function and to related functions.
Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).
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Numerical Tools for the Study of Finite Gap Solutions of Integrable Systems.
Ph.D. Thesis, Technischen Universität Berlin.
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Bernoulli numbers and polynomials via residues.
J. Number Theory 76 (2), pp. 178–193.
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##### 5: 28.32 Mathematical Applications

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###### §28.32(i) Elliptical Coordinates and an Integral Relationship

… ► ►###### §28.32(ii) Paraboloidal Coordinates

►The general paraboloidal coordinate system is linked with Cartesian coordinates via …is separated in this system, each of the separated equations can be reduced to the Whittaker–Hill equation (28.31.1), in which $A,B$ are*separation constants*. …##### 6: 1.10 Functions of a Complex Variable

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►Analytic continuation is a powerful aid in establishing transformations or functional equations for complex variables, because it enables the problem to be reduced to: (a) deriving the transformation (or functional equation) with real variables; followed by (b) finding the domain on which the transformed function is analytic.
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►The coefficient ${a}_{-1}$ of ${(z-{z}_{0})}^{-1}$ in the Laurent series for $f(z)$ is called the

*residue*of $f(z)$ at ${z}_{0}$, and denoted by ${res}_{z={z}_{0}}[f(z)]$, $\underset{z={z}_{0}}{res}[f(z)]$, or (when there is no ambiguity) $res[f(z)]$. … ►###### §1.10(iv) Residue Theorem

… ►Suppose that …##### 7: 28.33 Physical Applications

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►with $W(x,y,t)={\mathrm{e}}^{\mathrm{i}\omega t}V(x,y)$, reduces to (28.32.2) with ${k}^{2}={\omega}^{2}\rho /\tau $.
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►If the parameters of a physical system vary periodically with time, then the question of stability arises, for example, a mathematical pendulum whose length varies as $\mathrm{cos}\left(2\omega t\right)$.
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McLachlan (1947, Chapters XVI–XIX) for applications of the wave equation to vibrational systems, electrical and thermal diffusion, electromagnetic wave guides, elliptical cylinders in viscous fluids, and diffraction of sound and electromagnetic waves.

Daymond (1955) for vibrating systems.

McLachlan (1947, Chapter XV) for amplitude distortion in moving-coil loud-speakers, frequency modulation, dynamical systems, and vibration of stretched strings.

##### 8: Bibliography R

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Diffraction of plane radio waves by a parabolic cylinder. Calculation of shadows behind hills.
Bell System Tech. J. 33, pp. 417–504.
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Elliptic hypergeometric series on root systems.
Adv. Math. 181 (2), pp. 417–447.
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##### 9: 15.13 Zeros

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►If $a$, $b$, $c$, $c-a$, or $c-b\in \{0,-1,-2,\mathrm{\dots}\}$, then $F(a,b;c;z)$ is not defined, or reduces to a polynomial, or reduces to ${(1-z)}^{c-a-b}$ times a polynomial.
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