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Chapter 1 The Real and Complex N u m b e r Systems

1.1 Introduction

1.2 The field axioms

1.3 The order axioms . -

1.4 Geometric representation of real numbers 1.5 Intervals

1.6 Integers

1.7 The unique factorization theorem for integers

1.8 Rational numbers

1.9 Irrational numbers

1.10 Upper bounds, maximum element, least upper bound

( supremum)

1.11 The completeness axiom

1.12 Some properties of the supremum

1.13 Properties of the integers deduced from the completen

1.14 The Archimedean property of the real-number system

1.15 Rational numbers with finite decimal representation

1.16 Finite decimal approximations to real numbers

1.17 Infinite decimal representation of real numbers

1.18 Absolute values and the triangle inequality 1.19 The Cauchy-Schwarz inequality

1.20 Plus and minus infinity and the extended real number

1.21 Complex numbers

1.22 Geometric representation of complex numbers 1.23 The imaginary unit 1.24 Absolute value of a complex number 1.25 Impossibility of ordering the complex numbers 1.26 Complex exponentials

1.27 Further prope rties, of complex exponentials

1.28 The argument of a complex number 1.29 Integral powers and roots of complex numbers 1.30 Complex logarithms

1.31 Complex powers •

1.32 Complex sines and cosines

1.33 Infinity and the extended complex plane C* Exercises

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.

4.57 In a m etr ic space (S, d), le t y 4 an d 5be dis jo in t c losed subse ts

exist dis jo in t open subse ts U and V of S such that A £ C / and

#(X) =/ A ( X ~ ) ~ / B ( 'X ~ ) > in the n o t a t i o n o f Exercise 4.56, an d c o n s i<r j(0,+00).

Discontinuities

4.58 Locate an d classify the d i s c o n t i n u i t i e s of the f u n c t i o n s / d e f i n e d

i n g e q u a t i o n s :

a) /(A) = ( s in A) /A if x *0,/(0) = 0.

b) f(x} = e1/x if A -*0,/(0) = 0.

c) /(A)-el/x + sin ( I /A ) if A - ^ 0, /(O) = 0.

d) /(A) = 1/0 - e1'*) ifA jt 0,/(0) = 0.

4.59 Loc a te the po in ts in R2 a t wh ich each o f th e func t i ons i n E x

t i n u o u s .

Monotonic functions

4.60 Let/be defined i n the o p e n i n t e r v a l (a, b) an d assume tha t f o r

o f (a, b) there exists a 1-ball B(x) in which/is increas ing . Prove t

f u n c t i o n t h r o u g h o u t (a, b}.4.61 Let/be c o n t i n u o u s on a c o m p a c t i n t e r v a l [a, b] and assum e t

l o c a l m a x i m u m o r a l o c a l m i n i m u m a t any i n t e r io r p o in t . (Se

Exercise 4.25.) Pro ve that/must b e m o n o t o n i c o n [a, b].

4.62 If/is one- to-one an d c o n t i n u o u s o n [a, b], prove that/must o n [a, b]. That is , prove that every t o p o l o g i c a l m a p p i n g o f [a, b] om u s t be s t r i c t ly monoton ic .

4.63 Let/be an i nc reas ing f u n c t i o n d e f i n e d on [a, b] and let x:,.

the in ter ior such tha t a < xl < x2 < • • • < xn < b.

a) S h o w t h a t £JU [/(A, + ) -/(A,-)] </(£-) -/(a+ ).

b) Deduce f rom part (a) that the set of discon t inu i t i e s of/is

c ) Prove that/has p o i n t s o f c o n t i n u i t y in every open s u b i n t

4.64 Give an e x a m p l e o f a f u n c t i o n / defined an d s t r i c t ly increas ing

that/"1 i s n o t c o n t i n u o u s o n / ( > S ) .

















































































































































































o see a e converse s no rue, cons er e exam p e

n = i n

This alternating series converges, by Theorem 8.16,but it absolutely.

Theorem 8.19. Let ]Tan be a given series with real-valued terms

n ~ n , _ 1 2

Pn — ^ ' "̂ 9 ~Then:

i) If^_an is conditionally convergent, both ^pn and^gn diver

ii) JfY\an\ both ^pn and^gn converge and we have

oo

an — 2-^1 Pn ~~ L-Jn=l

NOTE. p n = an and qn = 0 if an > 0, whereas gn =—an and p

. We have an =pn - gn, \an\ pn + qn. To prove (i), converges and 2|an| diverges. If £#„ converges, then ~£pn

Theorem 8.8), since pn =an -f qn. Similarly, if ^pn conver

converges. Hence, if either ^L,pn or ^qn converges, both mu

deduce that ^\an\, since \a n\ pn + qn. This contradiction prove

To prove (ii), we simply use (4) in conjunction with Theor

8.9 REAL AND IMAGINARY PARTS OF A COMPLEX SE

Let ^cn be a series with complex terms and write cn — an + i

are real. The series ^an and ^bn are called, respectively, the

parts of 2cn- In situations involving complex series, it is often the real and imag inary parts separately. O f course, convergen

~£bn implies convergence of £cn. Conversely, convergence o

vergence of both ^>n and ^°n- The same remarks hold for ab

























































































































































































































































,- ,-

EXERCISES

Jacobians

13.1 Let / be the complex-valued function defined for each comequation/(z) = 1/z. Show that //(z) = — z|~4. Show that/is one-t

f~l explicitly.

13.2 Let f = (/1;/2, /3)be the vector-valued function denned (for ever

in R3 for which x± + x2 + x3 ^ — T ) as follows:

xk

X

+ x2 + x3}~4

= 1, 2, 3

Show that fhow that /f(^l5 x2, x 3 ~ ) = (1 +

compute f"1 explicitly.

13.3 Let f = (/l5 ...,/„) be a vector-valued function defined in

on R", and let /f(x) denote the Jacobian determinant. Let g1,...,

functions denned on R1 and having continuous derivatives g{, ...

/*0i(*i)> • • • .#nCO]> k = 1, 2, . . . , n, and put h = (A l3 . . . , A n). Sh

13.4 a) If x(r, 9) = r cos 9, y(r, 6} = r sin B , show that

d(x, y)=r

8(r, 0)

b) If x(r, 9, < / > ) = r cos 9 sin < / > , y(r, 6, $ ) = r sin 9 sin $, z =

d(x, > > z) 2 . ._L_LZi_^ = - r 2 sm ^.3(r, 5, ^)

13.5 a) State conditions on / and g which will ensure that the equ

y — g(u, v) can be solved fo r u and u in a neighborhood of (

tions are u = F(x, 7), v = G(x , y), and if / = d(f, g ~ ) / d ( u , v

d F_= 18g_

dx J dv

8F Idf

Jdv'

8G

8 x

ldg_

Jdu

















































































































































l5 xz, x3)

There are similar expressions for the other eight derivatives Dkgt.

13.7 Let/ = u + z u be a complex-valued function satisfying the

u e C' and v e C' on the open disk A = {z : \z\ l};/is continuous

A = {z:\z\ 1}; u(x, y} = x and u(x, y) = y whenever x2 + y2 J f(z ) > OifzeA. Let B = f(A) denote the image of A under /an

a) If J^is an open subset of A, then/(JO is an open subset 'o

b) B is an open disk of radius 1.

c) For each point U Q + iv 0 in B, there is only a finite number

that/(z) = w0 + w0.

Extremum problems

13.8 Find and classify the extreme values (if any) of the functions de

equations:

a) f(x, y} = y~ + x~y + x4;

b) f(x, y) = x2 + y2 + x + y + xy,

c) f(x, y) = (x- I)4 + (x- y)\) f(x, y)=y2 - x\9

x~ — 4y = 0 . Solve this problem using Lagrange's method an

Lagrange's method.

13.10 Solve the following geometric problems by Lagrange's meth

a) Find the shortest distance from the point ( < 2 1 ; a2, a 3 ~ ) in equation is b-^x^ + b2x2 + b3x3 + b0 = 0.

b) Find the point on the line of intersection of the two plane

a^x^ + a2x2 + a3x3 + a0 = 0

andb±x{ + b2x2 + b3x3 + bQ — 0

which is nearest the origin.





















































































































































































































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