Mathematical Analysis: Functions, Limits, Series, Continued Fractions 1483194361, 9781483194363

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L.A. Lyusternik, A.R. Yanpol'skii (Eds.)

M AT H E M AT I C A L ANALYSIS Functions, Limits, Series, Continued Fractions

MATHEMATICAL ANALYSIS Functions, Limits, Series, Continued Fractions E

d it e d

by

L. A. L Y U S T E R N I K a n d A. R. Y A N P O L ’SK II

T

ra nsla ted

by

D. E. BROW N T

r a n s l a t io n

e d it e d

by

E. S P E N C E

PERGAMON PRESS OXFORD • LONDON • EDINBURGH • NEW YORK PARIS • FRA N K FU RT

PERGAMON PRESS LTD. Headington Hill Hall, Oxford 4 and 5 Fitzroy Square, London W.l| PERGAMON PRESS (SCOTLAND) LTD. 2 and 3 Teviot Place, Edinburgh 1 PER G A M O N P R E S S IN G . 122 East 55th Street, New York 22, N.Y. G A U T H IE R -V IL L A R S ED. 55 Quai des Grands-Augustins, Paris 6 PERG A M O N PR ESS G.m.b.H. Kaiserstrasse 75, Frankfurt am Main*

Copyright © 1965 P e r o a m o n P ress L t d .

First Edition 1965

Library of Congress Catalog Card Number 63-19330

This is an edited translation of the original Russian volume entitled MameMamuHeacuu anajai3 - ) respectively. The set ofnumbersx satisfying the inequalities a ^ x is called a segment (or closed interval), and is written as [a, 6 ]. The sets of points x satisfying the inequalities a ^ x < b,

a < x ^ 6,

are called semi-intervals and are written respectively as [a, b), (a, b ]. The infinite semi-intervals ( —o©, a] and [6, + ©©) are similarly defined. The interval (x—e, x+ e), (6 > 0) is called an e-neighbourhood of the point x. If an element x belongs {does not belong) to the set X, this is written symbolically as x £ X (x £ X or x $ X). If all the elements of a set X are simultaneously elements of a set Y, X is said to be a subset of the set T, and we write symbolically: X c. Y. Otherwise, X is not a subset of Y9 and we write this symbol­ ically as X c Y(or X /? if the class A wholly contains the class B, which does not coincide with Ayand a >- r, where r is any rational number of the class A . Hence only one of the following relationships is possible for any two real numbers a and fl : a = /3, a ft, If sections are performed as described above in the domain of real numbers, it turns out that there always exists for any such section A | A ’ a real number accomplishing the section. This is the essence of Dedekind’s basic theorem. This property of the set of all real numbers is described by saying that it is complete or continuous. We can introduce for real numbers the concepts of the arithmetical operations and laws (addition, multiplication, division by a non-zero number, etc.). For instance, the sum of two real numbers a and is taken to be the real number y = a -f /? which satisfies the relationship y -c a! + b \ where a, a \ b and b9 are all possible rational numbers satisfying the inequalities: a < a < a \ b All the other arithmetical operations can be similarly introduced, while retaining the fundamental properties. Cantor's theory. We take all possible fundamental sequences (see § 3, sec. 2) of rational numbers. A sequence of rational numbers, convergent to a rational limit, is fundamental. At the same time, there exist fundamental sequences of rational numbers which do not have rational limits, as for instance the sequence of decimal approximations {1; 1*4; 1*41;... } of the square root of two. Two infinite sequences {jcn} and are said to be equivalent or confinal if | xn~ y n | tends to zero as n — «>. This means that two equivalent fundamental sequences {xn} and {yn} can only have the same rational limit x as n — ©o. All equivalent fundamental sequences of rational numbers are referred to one class — the equi­ valence class, and the set of all fundamental sequences of rational numbers is divided into equivalence classes. There are two possibilities: either there exists a rational number r, the common limit as n «> of all sequences {jcn} of the same

THE ARITHMETICAL LINEAR CONTINUUM

9

equivalence class X, or there is no such number among all the rational numbers. We say in the first case that the equivalence class X defines the rati­ onal number r; in the second case, we say that the equivalence class X defines an irrational number x (which is also regarded as the limit of the sequences of the class X as n 0), if the corresponding equivalence class X contains a fundamental sequence of positive rational numbers not convergent to zero. The inequality a > where a and /3 are two real numbers, means that a - |3 > 0 . The concepts of fundamental sequence and equivalence class can also be defined on the set Ex of all real numbers. It turns out that all fundamental sequences are convergent on E1 (see § 3, sec. 2), so that any equivalence class on Ex defines a real number — the common limit of sequences convergent to it. No new numbers can be obtained with such a completion of the set Ex; in this sense, the set Ex is complete. Thus the real number set Ex is obtained as a result of completion of the set of rational numbers by the limits of all possible fundamental sequences, of rational numbers. This idea of completion has acquired great value in functional analysis. As regards other irrational number theories, we may mention, in addition to Weierstrass’s, A.N. Kolmogorov’s argument (see ref. 6, p. 269) and the axiomatic construction of the real numbers (see ref. 4, p. 157, and ref. 15, p. 180).

10

MATHEMATICAL ANALYSIS

§ 2. Functions. Sequences 1. Functions of one variable If we are given a set of real numbers X = {*}, and each number x £ X is associated with a corresponding number y , where Y = {y} is the set of all such y, the function y = fix ) is said to be defined on the set X. The set X is called the domain of definition of the function fix), and every number x £ X the argument. The set Y is called the range of the function fix). If the argument x is given, the value y = fix ) of the function is given. For instance, for the function y = x3 the sets X and Y coincide with the real axis E1; for the function y = tan x the set Y is the whole of the real axis while X = E1—M , where M is the set of all numbers of the form y

mt (n = 0, ± 1, ± 2,...);

for the function y = x ! the sets Y and X are the sets of natural numbers; for the function y = E(x) (the integral part of x), X is the real line E1, and Y is the set of natural numbers. For the function -1 0 1

for for for

x < 0, x = 0, 0

the set X is the real line, while the set Y consists of three numbers: - 1, 0 , 1.

Any finite set of numbers {aj (i = 1, 2, can be regarded as a function, given on the finite set of natural numbers X < {1, 2, . .. , n} and associating with each of these numbers i the value of the function /(/) = a{ (i = 1, 2, . . . , n). The concept of a function has been subjected to wide generaliza­ tion. X can be a set of arbitrary elements. A numerical function is said to be given on this set if a number fix) is associated with every element x of the set. We shall consider in Chapter II functions defined on a set of points (or vectors) of /a-dimensional space. The area bounded by a polygon, or its perimeter, can be regarded as functions defined on a set of plane polygons; physical magnitudes, such as the mass of

THE ARITHMETICAL LINEAR CONTINUUM

11

a body, its charge, etc. are defined on the set of corresponding physical bodies, etc. The elements of a set X, on which a function is defined, are occa­ sionally called points. 2.

Upper and lower bounds o f a function

An upper (lower) bound of a function f(x), defined on a set X, is a number M(m) such that f(x) S M(f(x) & m) for all x £ X. If this number exists, the function f(x) is said to be bounded from above (below) on X. A function, bounded from above and below on X, is said to be bounded on X. The least (greatest) of all the upper (lower) bounds of a function f(x) is called the strict upper (lower) bound M*. (m*) of the function and is written as M* = sup f(x) inf xiX

xZX

If there exists an element x 0 (xj) of X, for which f i x o) = sup f(x) xSX

If(xi) = inf f(x)\, \

xSX

)

then sup f(x) = f( x o) xSX

( inf f(x) = /fo )) \x(X

l

is called the absolute maximum {minimum) o f the function f{x) and is written as f i x 0) = sup f{x) = max f{x) (f{x0 = inf f{x) = mmf{x)\. x€X Xtx v Xtx ) In this case, we say that f{x) attains its absolute maximum {minimum) at the point x0 (Xj) Given the finite set ax, a2, . . . , an, we write max {al9 a2, . , . , an)

^min {al9 a2> . . . , an)^

for the maximum {minimum) of the numbers ax, a2, . . . , an . For example, inf — = 0, min sin x = —1, max sin x — x€ (0, «*) X x€E 1 x€Ei m ax{4, 3, 7, 11, 8} = 11, min {4, 3, 2, 10, 17}= 2.

12

MATHEMATICAL ANALYSIS

The following inequalities hold: (1)

sup/(x )+ sup/^x) ^ sup (f(x)+f1(x)), x€X

x€X

inf /(x )+ inf / x(x) x€X

x

€

(1.3)

x € i

X

inf (f(x)+f1(x)).

(1.4)

x£X

If /(x), f f x ) and A(*)+/(*) attain a maximum (minimum) on X> then we have max/(*) + ma x / 1(x) ^ max (/(x)+/,(x)), xtX Xtx Xtx min f(x) + mi n/^x) ^ min (/M + /i(* )). x€X x€X Xtx

(1.3a) (1.4a)

The sign of equality holds in (1.3a) (or in (1.4a)) when f(x) and attain a maximum (m in im u m ) at the same point. For example,

A (x )

max sin x - f max cos x = 2 > max (sin x +cos x ) = 0 :

3.

Fundamental theorems concerning lim its

1°. A sequence can have only one limit. 2°. Tf a sequence has a finite (infinite) limit, it is bounded (nonbounded). 3°. I f a sequence {*n} has a unique limit point jc0, the sequence is convergent and x0 is its limit. Conversely, if a sequence ( x j is convergent to x0, x0 is its unique limit point. 4°. I f x is a limit point of a sequence {an}, there exists a subsequence {anJ , convergent to x. Conversely, if y is the limit o f some subse­ quence {anJ , y is the limit point of the sequence {an}. 5°. Similarly, if x is a limit point o f a set M o f the numerical axis Ei, M contains a sequence {xn} of numbers different from x that con­ verges to x . 6°. On the assumption that lim xn and lim yn exist, we have: 71—► o °

71 —► oo

23

THE ARITHMETICAL LINEAR CONTINUUM

lim (xn ± yn) = lim xn ± lim yny

n—►oo

n—►oo

lim

n—►«©

n—►o©

= lim xn . lim yn, n -voo

n

«©

lim xn lim — = n->oo

n—»■oo

where

lim 7n ’

lim yn ^ 0; 71—► oo

71—► oo

if xn < yn, then lim xn ^ 71—► oo

lim yn . 71—>■ oo

4 . Som e propositions on lim its

1°. //■ lim an = a and all the ai >■ 0 (i = 1, 2, . . . ) , then n— ►oo lim

. . . an = a,

lim a i~^a2~*~

71—► oo

71—► oo

JlP ” =

^

2°. I f A is the greatest o f the numbers

au a2, . . . , an (a4 ^ 0) and Pi > 0 (i = 1, 2, 3, . . . , n), we have lim vVp1a?+ p2a$l+ . . . + pna% = A, 771-* ■ oo

) I

Um pla?+'+ p 2aZl+1+ . . . +pn< +1 _ A [ m-*- =o ^ 1^ r + m m+ • • • + Pn< j 3°. //", for a sequence {an}, lim V^n = 5 71 —► oo

and lim

= a

7i —► oo

#71

both exist, then a — a.

24

MATHEMATICAL ANALYSIS

4°. I f lim ---------£2--------- = 0,

Pi > 0,

n -> « P o + P l + • • • + P n

and, as n — » , jn

a //m/< e?ua/ to s, we have

Jjm soPn 4- S iP n -l + h P n - 2 + » • • + SnPo >-*•« P 0 + P 1 + P 2 + • • • +Pn 5. Upper and lower limits of a sequence The upper {lower) limit o f a sequence {x„} is the strict upper (lower) bound of the set numbers which are limits of the sequence, and is written as lim xB 7l-*oo

/ lim x ^ . \7l-* 00 /

IE ["( 71— ► L 00

1 ''a + a l~ 1I____ 1 II

For example,

lim |"( _ 1)n+ I ] = 7l-*oo L

- 1,

while at the same time, sup n -1, 2,... |_ inf [ n-l, 2, ... 1 Every bounded sequence has an upper and a lower limit. If a sequence is convergent, its limit coincides with its upper and lower limits; if the upper and lower limits are the same, the sequence is convergent to their common value. Given any sequence that has an upper and a lower limit, we can readily form monotonic sequences convergent respectively to the upper and lower limits: a monotonically non-increasing {xBn} to the lower, and a monotonically non-decreasing {x*} to the upper:

THE ARITHMETICAL LINEAR CONTINUUM

25

* n = SUp {*n , Xn +1, Xn +2, . . .} ,

= inf {*n, xn+1, xn+2, ...} , lim xn = lim x*,

n—►oo

n —v oo

lim xn = lim x*n. n —►«©

i f K ) is a monotonically non-decreasing (non-increasing) sequence and a = lim xn, then a:* = *n, x*n = a (x* = a, x*n = x*).

6.

Uniform ly distributed sequences

Let the sequence {xn} lie on the segment [a, 6], Let iVn (a,/9) denote the number of the points x k(k = 1, 2 , . . . , ti) which lie in the interval (a, /S) c [a, 6]. If the limit lim Nn(*> £) n /i— ►«

a = /3—a

exists, then no matter what the interval (a, £) c [a, 6], the sequence is said to be uniformly distributed on the segment [a, b]. E xample 18. The sequence {yn} of Example 11 (see § 2, sec. 7) is uniformly distributed on [0, 1]. Example 19. The sequence {xn}, where xn = an* —[an?],

a > 0, 0 < a < 1

(n = 1, 2, ...),

([x] is the integral part of x) is uniformly distributed on [0, 1]. Example 20. The sequence {xn}, where xn = a(\nnY —[a(lnn) 0, a > 1

(n = 1, 2, ...),

is uniformly distributed on the segment [0, 1]. Uniformly distributed sequences have applications in numerical integration. Obviously, all interior points of the segment [a, b] are limit points for {xn}, and moreover, a as lim xn9

b = lim xn.

n-*o©

n-^oo

Given any function /(x) continuous on [a, b], and a sequence {xn}‘

26

MATHEMATICAL ANALYSIS

uniformly distributed on [a, Z>], the following relationship holds: 1

n

I f 6

f(x)dx. Wi-1 b 0}a Conversely, if this equation is satisfied for all functions continuous on [a, b \ then the sequence {xn} is uniformly distributed. 7 . Recurrent sequences

A sequence {xn} is said to be given by a recurrence formula if the first few terms are given and a formula is known, with the aid of which xn is expressible in terms of the preceding terms: Xn = /(**_!, 2> • • • 5 -^n—p)> P — 1 (P = 1*2, . . .). For instance, xa = 1, x2 = 2, xn = x = xn_2 (72 = 3, 4, . . . ) . The sequence itself is sometimes described as recurrent. The simplest example of a recurrent sequence is an iterative sequence {*„}: Xn = / ( * n - l ) Iterative, and in general recurrent, sequences are of great value in approximation methods — for example, in the method of successive approximations and Newton’s method. (a) The method o f successive approximations for solving the equa­ tion x = /(x), where /(x) is a continuous function, leads to an iter­ ative sequence {xA}: **+1 = f { xk} (k = 0, 1, 2, ...). where some arbitrary number is taken as x0. Here, if the sequence {xfe} is convergent to x, x proves to be a solution of the equation: X = f(x). (b) Newton's method (or method o f tangents) for finding the roots of the equation /(x) = 0, where /(x) is a differentiable function, also leads to an iterative sequence {xk}: ~

__ f (X k)Xk ~ f ( X k)

Xk+i —

f\X k)

( V — c\ 1 0

v^c — u i , z , . .

where some number is taken as x0; here, if {xk} is convergent, then lim x h = x is the required root, i.e. /(x) s 0. ft— > 00

THE ARITHMETICAL LINEAR CONTINUUM

27

(See above, for a sufficient condition for the convergence of the sequence {xn} indicated in method (a).). 8. The symbols o(a„) and 0{an) A variable that takes a certain sequence {an} of values is called a variant. For instance, the variable term of any progression is a variant. If an and /3n are given variants, while their ratio an//?„ tends to zero as « —oo ^ lim an//?n = Oj, we say thatan (/?„) is an infinitesimal 0infinitely large quantity) with respect to /3n ( 0 (C is a constant), we say that /3n has a rale o / decrease not faster than a n or that an has a rate o f increase not faster than (ln, and we write symbolically: «n = t w

­

in particular, if lim an//9n = C # 0, then a n = O (/?„) or n— ►oo Ai = 0 ( 4 For example, n = 0(yjn2~hi). The equation «n = 0 (1 )

implies that the sequence {an} is bounded, i.e. that |an | ^ C for all n. T h e o r e m 3. Given an arbitrary sequence { X n } = { * * } o f the following sequences: Xx = {aj, xj, x\. • • • > Arn> • • •}. X2 = {x\, x\, x\. . ♦ . , . . •}> Y71 ♦. Xn = {a?, a", x", ' **>-vm,

28

MATHEMATICAL ANALYSIS

there exists a numerical sequence X — {xj,} = {xlt x2, . . . , xh, .. increasing faster (decreasing faster) than any o f the sequences {*£}. For example, if Xn = {«”*} = {n1, n2, n2, . . . , nm, . . the sequence X = {w!} = {1!, 2!, 3!, . . . , m \ , . . . } increases at a faster rate than any sequence {Xn}: lira nn/m! = 0 for any n.

9 . Lim it o f a function

Let the function f(x) be defined on some set X. We say that the number A is the limit of fix) as x -*• x0: A = lim f(x), X-+XO

or that f(x) tends to A as x — x0) if, given any sequence {xn} € X, convergent to jc0 ( lim xn = x 0), the sequence {/(*„)} is convergent to A. Or, in other words, the number A is the limit of the function J{x) as x — x0: A —

lim

fix)

Iflx) — A

as

x — x0),

x-+Xo, x £ X

if, given any positive number e, there exists d>- 0 such that \flx)—A |< < s for aU x € X such that 0 -e | x —x0| -), ( —« , 0], [ 0, + «>), ( — 0), ( 0, + » ) , etc. Let fix) be defined on an interval (x0, a). The number A =/(x 0+ 0) is called the limit o f thefunction fix) from the right at the point x0= x 0

THE ARITHMETICAL LINEAR CONTINUUM

29

if, given any sequence {x„} of (x0, a), convergent to x0, / ( x j is con­ vergent to A. We can similarly define at the point x — x 0 the limit from the left f( x 0—0) = B of fix), defined in an interval (A, x0). For example, we have for the function y = £(x) (see § 2, sec. 1) and for the integral argument* = n: E(n —0) = n —l, E(n + 0) = n. When the argument x„ = 0, the limits from the left and right of fix) are written respectively as / ( —0) and /(+ 0 ). For example, if f(x) = = sign x (see § 2, sec. 1), th en /( —0) = —1 ,/(+ 0 ) = +1. A function fix), defined at a point * = x 0, is continuous there from the right (left) if f( x 0+ 0) [/(x0—0)] exists, equal to /(x 0). 11. Continoous and discontinuous functions

If f i x o- 0 ) = /(x 0+ 0) = /(x 0),

( 1. 12)

the function fix ) is said to be continuous at the point * = x 0. Iff(x) is not defined in the interval (A, x0) or (x0, a), the left- (or right-) hand term in (1.12) is ignored. (See also p. 59). Iffix), defined on the set X, is continuous at every point * € X, it is said to be continuous on the set X. For instance, f(x ) = V(x) is continuous for all * m 0. Otherwise, f(x) is described as discontinuous. We say that (f(x) has a discontinuity o f the first kind or jump at the point * = x0 if f(x 0—0) and f(x04- 0) exist, but (1.12) is not satisfied. In all other cases of a discontinuous function, the point * = *0 is called a point of discont­ inuity o f the second kind. For instance, the function for for

| x | -= 1, |x i s 1

has a discontinuity of the first kind at the points = —1 and x2 = + 1. Certain commonly encountered functions y = f(x) are equal, at a point of discontinuity of the first kind x = x0, to the arithmetic mean value ^ _ /(* o~ 0) + /(* o+ 0)

30

MATHEMATICAL ANALYSIS

For example, in the case of f{x) = sign x (see § 2, sec. 1):

We say that f(x) is uniformly continuous on the set X if, given any e > 0 , there exists a 6 = 6(e) => 0 such that, for any pair of points x \ x" £ X for which f(x) has a meaning, \ x '—x"\ -= 6 implies |fix ') — —f(x") | < e. A function f{x), defined and continuous on a bounded segment [a, 6], is uniformly continuous on this segment (Cantor’s theorem). 12. Functional sequences

An important role is played in analysis by functional sequences {/n(x)} (n = 1, 2, ...) , defined on some set X of the numerical axis Ev Various definitions can be given of passage to the limit for such sequences. It is natural to start from passage to the limit at each point, when {/„(*)} becomes a numerical sequence for any fixed x £ X . If the sequence {/„(*)} is convergent as n — /(*) or lim /„(*) = f(x) x ^ X . 71

E x a m ple 21.

T 71 lim arc tan nx = — sign x = < 0 n-*- +»»

for

x «= 0,

for

x = 0,

71

for x > 0. 1 This example shows that the limit of continuous functions may be a discontinuous function. As a result of a double passage to the limit, functions can be obtained in the limit that have even more complicated discontinuities, for instance, lim lim (cos 2nm! x)n = ^(x), m —►oo n — o©

where %(x) is Dirichlefs function: z(*) -

1 0

{

if if

x is rational x is irrational.

31

THE ARITHMETICAL LINEAR CONTINUUM

Remark. Together with functional sequences, we often encounter in analysis sequences of numbers, dependent on functions (functionals) For instance, the mean values of functions are defined by the limits of such sequences. If the function f(x) is integrable on [a, b] and = /(a-Hd*),

1 71 1 rb fix) dx lim - Y f hn = ~ n b -a a is the arithmetic mean of fix ) on [a, b]\ V/i«/a» •••/«» = exp j fZ~a Ja/W dx

}

is the geometric mean of f(x) on [a9 b]9 and

is the harmonic mean of f(x) on [a9b]. The following also holds: Theorem 4. I f functions fix ) and gfx) are continuous and positive on [a9 b]9 then (1)

(2)

lim " oo \

[ b g ix ) [fix )]n dx = m ax f{x). *€ [a, bj 'J «

6 g i x ) [ / ( * ) ] n+1 d x lim ^ Cl n —►oo j \ ( * ) tfix)]n dx

=

m ax f{x ). x€ [a, b]

13. Uniform convergence o f functions

The concept of a uniformly convergent sequence of functions {/n(*)} plays an exceptionally important role in analysis. Definition. A sequence o f functions {/nM}, defined on a set X a El9 is said to converge uniformly as n -* » to the function /(*), also defined on the set X, i f given any positive number e, an integer

32

MATHEMATICAL ANALYSIS

N = N(s) can be found, independent o fx € X, such that, for all n s- N: !/„(*)-/(*) I < «. (U 3) If {/ n(;c)} is convergent as k — ~ at every point x € X, but does not satisfy the uniformity condition (1.13), we say that {/„(*)} converges non-uniformly to f(x) on the set X. With non-uniform convergence, the number N depends not only on the choice of e, but also on the number x £ X: N = N(e, x). Theorem 5. I f all the functions f n(x) o f a sequence {/„(*)}, uni­ formly convergent to f(x) on X as n — are continuous, the limit function f{x) is also continuous on X. It follows from this that, if the limit function is discontinuous, the convergence as n - « of the sequence ( /n(x)} is non-uniform. E xample 22. The sequence {/„(*)} = {xn} is uniformly convergent as n — oo to f(x) = 0 on any segment [0, q], 0 < q -= 1; on the segment [0, 1] this sequence converges non-uniformly to the function ,, „ /W

f 0 = ( l

for

0 s* < l,

for

, - l .

Cauchy’s Test. A necessary and sufficient condition for a sequence o f functions f n(x), defined on a set X c Ex%to be uniformly convergent asn — oo to a function fix ) is that, given any e > 0, there exists an N, depending only on e, such that | / n(x)—f m(x)\ < e for all x £ X, provided only that n N and m =» N.

THE ARITHMETICAL LINEAR CONTINUUM

33

Geometrical interpretation of uniform convergence. Let f n(x) (rc = = 1,2,...) be continuous in [a, b], and let the sequence {/n(x)} be uniformly convergent as n — N (see Fig. 1), i.e. they are contained in the strip between the curves y = f(x) —e and y = fix) + 6. 14. Convergence in the mean A functional sequence {/n(*)} is convergent in the mean on the segment [a, b] c Ex as n -*■ oo to the function f(x) if, given any e > 0, there is a number N such that, for all n > JV, we have J 6 [A ( x ) - f( x ) fd x < « (it is assumed that this integral exists). Use is often made of convergence in the mean in various branches of analysis (for instance, in the approximation methods of analysis). Convergence, defined by the norm o f a space (see Chapter II, §1, sec. 2), is a generalization of convergence in the mean.15 15. The symbols o(x) and 0(x) Given two functions x(t) and y(t), defined on a set X, if their ratio x(t)/y{t) tends to zero as t a £ X ( lim x(t)/y(t) = 0), we say that t->a

x(t) [y(0] is an infinitesimal (iinfinitely large quantity) with respect to y(t) [x(t)] and we write symbolically: x(t) = o(y(t)). For example, t2 = o(sin t) as t — 0, tn = o(e*) as t oo for any n > 0. If x(t) and y(t) are infinitesimals as t — a, and x(t) = o(y(t)), we say that *(/) (y{t)) is an infinitesimal of higher (lower) order with respect to y(t) (*(/)), or: *(/) (j(r)) decreases faster (more slowly) than y (t) (x(t)). Ifjc(r),y(0are infinitely large quantities as t — a, andx(/)= o(y(/)), we say that x(/) (y(t)) increases more slowly (faster) than y (t) (x(t)).

34

MATHEMATICAL ANALYSIS

If | x(t) | m C | y(t) | (C is a positive constant), we say that x(t) does not decrease at a faster rate than y (t) or that *(/) does not increase at a faster rate than y{t), and we write symbolically: x(t) = 0(y(t)). For instance, t = 0(t sin(l //)) and t = 0(tan 2t) as t -*• 0, e1= o{e2*) and ex — o(e*/