By Ali H. Nayfeh

The Wiley Classics Library comprises chosen books that experience develop into famous classics of their respective fields. With those new unabridged and cheap variants, Wiley hopes to increase the lifetime of those very important works through making them on hand to destiny generations of mathematicians and scientists.

presently to be had within the Series:

T. W. Anderson

The Statistical research of Time Series

T. S. Arthanari & Yadolah Dodge

Mathematical Programming in Statistics

Emil Artin

Geometric Algebra

Norman T. J. Bailey

the weather of Stochastic Processes

with purposes to the common Sciences

Robert G. Bartle

the weather of Integration and

Lebesgue Measure

George E. P. field & Norman R. Draper

Evolutionary Operation: A Statistical strategy for procedure Improvement

George E. P. field & George C. Tiao

Bayesian Inference in Statistical Analysis

R. W. Carter

Finite teams of Lie sort: Conjugacy Classes

and complicated Characters

R. W. Carter

easy teams of Lie Type

William G. Cochran & Gertrude M. Cox

Experimental Designs, moment Edition

Richard Courant

Differential and essential Calculus, quantity I

RIchard Courant

Differential and necessary Calculus, quantity II

Richard Courant & D. Hilbert

tools of Mathematical Physics, quantity I

Richard Courant & D. Hilbert

equipment of Mathematical Physics, quantity II

D. R. Cox

making plans of Experiments

Harold S. M. Coxeter

creation to Geometry, moment Edition

Charles W. Curtis & Irving Reiner

illustration concept of Finite teams and

Associative Algebras

Charles W. Curtis & Irving Reiner

equipment of illustration Theory

with functions to Finite Groups

and Orders, quantity I

Charles W. Curtis & Irving Reiner

equipment of illustration Theory

with functions to Finite Groups

and Orders, quantity II

Cuthbert Daniel

becoming Equations to information: desktop research of

Multifactor information, moment Edition

Bruno de Finetti

thought of chance, quantity I

Bruno de Finetti

thought of likelihood, quantity 2

W. Edwards Deming

pattern layout in enterprise examine

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**Sample text**

58) r2 Here again difficulty with the straightforward expansion arose because of the infinite domain. 59) satisfies the surface boundary conditions, but it does not satisfy the boundary condition at infinity. Thus, this expansion breaks down for large r and this breakdown is called Whitehead's paradox because Whitehead (1 889) was the first to obtain this solution, although by iteration, and the first to point out its nonuniformity. 2. 1. 2) where e is a small positive number. 2. 5) Y:, Y, = - Y L It can be seen that at any level of approximation n, ynV1is known, hence yn for any n is given by a first-order differential equation.

N,. = = n = constant, n,, = 0). 3. 26) where a is assumed to be small so that sin a m a, and b = r2/rl with r2 the radius of the rigid inclusion. 30). Hence two boundary conditions have to be dropped. 31). 32) is not valid for all r in [ b , 11. 5 by using the method of matched asymptotic expansions. 3. Type Change of a Partial Differential Equation Since the boundary and initial conditions required to make a well-posed problem depend on the type of the partial differential equation, difficulties might arise if the original equation changes, say, from elliptic to parabolic or hyperbolic as the small parameter vanishes.

Conditions under which these operations are justified are discussed by van der Corput (1956), ErdClyi (1956), and de Bruijn (1958). Addition and subtraction are justified in general. 3) EXERCISES Iff&; c) and q,,(e) are integrable functions of E, 19 then Multiplication is not justified in general because in the formal product of Ia,(z)p,,(~) and Zb,,(z)p,,(~)all products q,,(c)p,(e) occur, and it is generally not possible to arrange them so as to obtain an asymptotic sequence. In other words, multiplication is justified if the result is an asymptotic expansion.