(1 + bn )(1 + an ) = 1,

or otherwise, show that ∞ (1 + aj ) = 0.

j=1

(vi) Suppose aj > 0 and ∞ aj converges. If we write Rn = ∞ j=n+1 aj ,

j=1

∞

show, by considering an appropriate in¬nite product, that j=1 aj /Rj di-

verges.

Exercise K.79. [5.4, T, ‘ ] (i) Suppose that ± > 0. By expanding into

geometric series and multiplying those series show that

(1 ’ 2’± )’1 (1 ’ 3’± )’1 = n’± ,

n∈S(2,3)

where S(2, 3) is the set of strictly positive integers with only 2 and 3 as factors

and we sum over the set S(2, 3) in order of increasing size of the elements.

(ii) Find a similar expression for N (1 ’ p± )’1 where p1 , p2 , . . . , pN

j

j=1

are the ¬rst N primes. If ± > 1 use dominated convergence (Lemma 5.3.3)

to show that

∞

1 1

=

1 ’ p’± n±

n=1

p∈P

where P is the set of primes and we sum over P in order of increasing size

of the elements.

(iii) By considering carefully what happens as ± ’ 1 from above, show

that

1

’∞

1 ’ p’1

p∈P, p¤N

474 A COMPANION TO ANALYSIS

as N ’ ∞. Deduce that

1

diverges.

p

p∈P

(iv) Part (iii) and its proof are due to Euler. It shows that, not merely are

there an in¬nity of primes, but that, in some sense, they are quite common.

Show that if M (n) is the number of primes between 2n and 2n+1 then the

sequence 2’βn M (n) is unbounded whenever β < 1.

Exercise K.80. [5.4, P, ‘] Suppose that an is real and 0 ¤ an < 1 for all

n ≥ 1. Suppose further that an ’ 0 as n ’ ∞. Let the numbers Pn be

de¬ned by taking P0 = 1 and

if Pn’1 ¤ 1,

(1 + an )Pn’1

Pn =

(1 ’ an )Pn’1 if Pn’1 > 1,

for all n ≥ 1. Show that Pn tends to a limit l, say.

What, if anything, can you say about l if ∞ an diverges? What, if

n=1

anything, can you say about l in general? Prove your answers.

n j

(1 + z 2 )

Exercise K.81. [5.4, P] By considering the partial products j=1

and using the dominated convergence theorem, show that

∞ ∞

2j

z k = (1 ’ z)’1

(1 + z ) =

j=1 k=0

for all z ∈ C with |z| < 1.

Exercise K.82. (Vieta™s formula.) [5.5, T] (See Exercise K.78 for the

de¬nition of an in¬nite product if you really need it.) Show that if x ∈ R

and x = 0 then

n

’n

cos(2’j x) = 2’n sin x.

sin(2 x)

j=0

Deduce that

∞

sin x

cos(2’j x) = .

x

j=0

Does the result hold if x ∈ C and x = 0?

475

Please send corrections however trivial to twk@dpmms.cam.ac.uk

By setting x = π/2 obtain Vieta™s formula (probably the ¬rst published

in¬nite product)

2 1 11 1 11 11 1

= + + + ...

π 2 22 2 22 22 2

∞

2

un where u2 = (1 + un )/2

or (in a more reasonable notation) = n+1

n=1

π

and u1 = 2’1/2 .

Exercise K.83. [5.6, P, G] We say that a function f : A ’ B is injective

if f (a) = f (a ) implies a = a . We say that a function f : A ’ B is surjective

if, given b ∈ B, we can ¬nd an a ∈ A such that f (a) = b. If f is both injective

and surjective we say that f is bijective.

(a) Consider the functions f : A ’ B, g : B ’ C and their composition

g —¦ f : A ’ C given by g —¦ f (a) = g(f (a)). Prove the following results.

(i) If f and g are surjective, then so is g —¦ f .

(ii) If f and g are injective, then so is g —¦ f .

(iii) If g —¦ f is injective, then so is f .

(iv) If g —¦ f is surjective, then so is g.

(b) Give an example where g —¦ f is injective and surjective but f is not

surjective and g is not injective.

(c) If any of your proofs of parts (i) to (iv) of (a) involved contradiction,

reprove them without such arguments5 .

(d) Have you given the simplest possible example in (b)? (If you feel that

this is not a proper question, let us ask instead for the smallest possible sets

A and B.)

Exercise K.84. [5.6, P, G] Show that an injective function f : [0, 1] ’

[0, 1] is continuous if and only if it is strictly monotonic (that is strictly

increasing or strictly decreasing). Is the result true if we replace ˜injective™

by ˜surjective™ ? Give a proof or a counterexample.

Exercise K.85. (Multiplication before Napier.) [5.6, T, G]

(i) The ancient Greek astronomers drew up tables of chords where

chord ± = 2 sin(±/2).

Prove that

chord θ chord(π ’ φ) = chord(θ + φ) + chord(θ ’ φ).

5

Conway refers to arguments of the form ˜Assume A is true but B is false. Since A

implies B it follows that B is true. This contradicts our original assumption. Thus A

implies B.™ as ˜absurd absurdums™.

476 A COMPANION TO ANALYSIS

If 0 < x < 2 and 0 < y < 2, show that the following procedure computes xy

using only addition, subtraction and table look up.

(a) Find θ and φ so that x = chord θ, y = chord(π ’ φ).

(b) Compute θ + φ and θ ’ φ.

(c) Find chord(θ + φ) and chord(θ ’ φ).

(d) Compute chord(θ + φ) + chord(θ ’ φ).

(ii) In fact we can multiply positive numbers using only addition, sub-

traction, division by 4 and a table of squares. Why? (Do think about this

for a little before looking at Exercise 1.2.6 (iii) for a solution.)

If we just want to multiply two numbers together, the methods of (i)

and (ii) are not much longer than using logarithms but, if we want to multi-

ply several numbers together, logarithms are much more convenient. (Check

this statement by considering how you would multiply 10 numbers together

by the various methods.) Logarithms also provide an easy method of ¬nd-

ing ±th roots. (Describe it.) We may say that logarithms are more useful

computational tools because they rely on isomorphism rather than clever

tricks.

[This question was suggested by the beautiful discussion in [39]. If the reader

wants to know more about the historical context of logarithms, Phillips™ book

is the place to start.]

Exercise K.86. [5.6, P, S] Arrange the functions

1/2 3 1/2

1 1 1 1

, log , exp log , exp

x x x x

as f1 (x), f2 (x), f3 (x), f4 (x) in such a way that fr (x)/fr+1 (x) ’ 0 as x ’ 0

through positive values of x. Justify your answer.

Exercise K.87. [5.6, P] We shall give an number of alternative treatments

of the logarithm. This is one I like lees than some of the others.

Let x0 = x > 0. The sequence of real strictly positive numbers xn is

de¬ned by the recurrence relation

x2 = x n .

n+1

Prove that the two expressions 2n (xn ’ 1) and 2n (1 ’ 1/xn ) both tend to the

same limit as n ’ ∞. Taking this limit as the de¬nition of log x, prove that,

for real, strictly positive x and y,

(i) log 1 = 0,

(ii) log xy = log x + log y,

(iii) log 1/x = ’ log x,

(iv) log x increases with x.

477

Please send corrections however trivial to twk@dpmms.cam.ac.uk

Exercise K.88. [5.7, P, S] Let f : R ’ R be a strictly positive function.

1/·

f (x + ·)

tends to a limit l, say, as · ’ 0 if and only if f is

Show that

f (x)

di¬erentiable at x. If l exists, give its value in terms of f (x) and f (x).

Exercise K.89. [5.7, S] (This is really just a remark.) I have been unable

to trace the story told in Exercise 5.7.10 to a sure source. However, one of

Pierce™s papers6 does refer to the ˜mysterious formula™

√

i’i = ( e)2π

(which he makes still more mysterious by using a notation of his own inven-

tion). By remarking that eiπ/2 = e’3iπ/2 , show that, even if we interpret this

equation formally, the left hand side of his equation is ambiguous. How did

we prevent this ambiguity in part (ii) of Exercise 5.7.10?

Exercise K.90. (Functional equations.) [5.7, T]

(i) Suppose f : R ’ R is a function such that

f (x + y) = f (x) + f (y)

for all x, y ∈ R. Let f (1) = a.

(a) Find f (n) for n a strictly positive integer.

(b) Find f (n) for n an integer.

(c) Find f (1/n) for n a non-zero integer.

(d) Find f (x) for x a rational.

Now suppose that f is continuous. Show that f (x) = ax for all x ∈ R.

(ii) Suppose ± : Rn ’ Rm is a continuous function such that

±(x + y) = ±(x) + ±(y)

for all x, y ∈ Rn . Show that ± is a linear map.

(iii) Suppose g : (0, ∞) ’ (0, ∞) is a continuous function such that

g(xy) = g(x)g(y)

for all x, y ∈ (0, ∞). Find the general form of g.

(iv) Suppose g : R \ {0} ’ R \ {0} is a continuous function such that

g(xy) = g(x)g(y)

for all x, y ∈ R \ {0}. Find the general form of g.

6

Linear Associative Algebra, Vol 4 of the American Journal of Mathematics, see

page 101.

478 A COMPANION TO ANALYSIS

Exercise K.91. [5.7, T, ‘ ] In this question we seek to characterise the

continuous homomorphisms χ from the real numbers R under addition to

the group S 1 = {» ∈ C : |»| = 1} under multiplication. In other words we

want to ¬nd all continuous functions χ : R ’ S 1 such that

χ(x + y) = χ(x)χ(y)

for all x, y ∈ R.

(i) For each t ∈ T let θ(t) be the unique solution of

χ(t) = exp iθ(t)

with ’π < θ(t) ¤ π. Explain why we cannot establish the equation

?

θ(t) = 2θ(t/2)

for all t but we can ¬nd a δ > 0 such that

θ(t) = 2θ(t/2)

for all |t| < δ.

(ii) If θ(δ/2) = γ and » = 2δ ’1 γ establish carefully that χ(t) = exp i»t

for all t. Conclude that the continuous homomorphisms χ from R to S 1 are

precisely the functions of the form χ(t) = exp i»t for some real ».

(iii) Find the continuous homomorphisms from the group S 1 to itself.

Exercise K.92. [5.7, T, ‘ ] (i) Suppose f : R ’ R is a function such that

f (2t) = 2f (t)

for all t ∈ R. Show that f (0) = 0.

(ii) Let pj be the jth prime. Show that we can ¬nd a function F : R ’ R

such that

F (2t) = 2F (t)

for all t ∈ R and F (p’1 ) = j. Show that F is not continuous at 0.

j

(iii) Suppose f : R ’ R is a function such that

f (2t) = 2f (t)

for all t ∈ R and, in addition, f is di¬erentiable at 0. Show that f (t) = At

for all t ∈ R and some A ∈ R.

479

Please send corrections however trivial to twk@dpmms.cam.ac.uk

(iv) Show that we can ¬nd a function F : R ’ R such that

F (2t) = 2F (t)

for all t ∈ R and F is continuous at 0 but not di¬erentiable there.

(v) Suppose u : R ’ R is a function such that

u(2t) = u(t)2

for all t ∈ R and, in addition, u is di¬erentiable at 0. Show that either

u(t) = 0 for all t ∈ R or there exists an ± > 0 such that u(t) = ± t for all

t ∈ R.

Exercise K.93. [6.1, P] We work in R3 with the usual inner product. Con-

sider the function f : R3 ’ R3 given by

x

f (x) = for x = 0

x

and f (0) = 0. Show that f is di¬erentiable except at 0 and

h x

’ x, h

Df (x)h = 3

x x

Verify that Df (x)h is orthogonal to x and explain geometrically why this is

the case.

Exercise K.94. (The Cauchy-Riemann equations.) [6.1, T] (This ex-

ercise is really the ¬rst theorem in a course on complex variable but it can

do the reader no harm to think about it in advance.)

We say that a function f : C ’ C is complex di¬erentiable at z0 if there

exists a f (z0 ) ∈ C such that

f (z0 + h) ’ f (z0 )

’ f (z0 ) as |h| ’ 0.

h

Observe that C can be considered as the vector space R2 and so we we can

write

f (x + iy) = u(x, y) + iv(x, y)

with x, y, u and v real, obtaining a function F : R2 ’ R2 given by

x u(x, y)

F = .

y v(x, y)

480 A COMPANION TO ANALYSIS

Show that the following statements are equivalent

(i) f is complex di¬erentiable at z0 .

x0

(ii) F is di¬erentiable at and its Jacobian matrix at this point is

y0

given by

‚u ‚u

cos θ ’ sin θ

‚x ‚y

=»

‚v ‚v

sin θ cos θ

‚x ‚y

with » real and » ≥ 0.

x0

(iii) F is di¬erentiable at and its derivative at this point is the

y0

composition of a dilation and a rotation.

x0

(iv) F is di¬erentiable at and its partial derivatives at this point

y0

satisfy the Cauchy-Riemann conditions

‚u ‚v ‚v ‚u

=’ .

= ,

‚x ‚y ‚x ‚y

Exercise K.95. [6.2, P] Let f : R2 ’ R be di¬erentiable and let g(x) =

f (x, c ’ x) where c constant. Show that g : R ’ R is di¬erentiable and ¬nd

its derivative

(i) directly from the de¬nition of di¬erentiability,

and also

(ii) by using the chain rule.

Deduce that if f,1 = f,2 throughout R then f (x, y) = h(x + y) for some

di¬erentiable function h.

Exercise K.96. [6.2, P] Consider a function f : R ’ R. State and prove

a necessary and su¬cient condition for there to be a continuous function

F : R2 ’ R with

f (x) ’ f (y)

F (x, y) =

x’y

whenever x = y.

Prove that, if f is twice di¬erentiable, then F is everywhere di¬erentiable.

Exercise K.97. [6.2, T] (The results of this question are due to Euler.) Let

± be a real number. We say that a function f : Rm \{0} ’ R is homogeneous

of degree ± if

f (»x) = »± f (x) †

481

Please send corrections however trivial to twk@dpmms.cam.ac.uk

for all » > 0 and all x = 0.

(i) By di¬erentiating both sides of † and choosing a particular value of

», show that, if f : Rm \ {0} ’ R is a di¬erentiable function which is

homogeneous of degree ±, then

m

††

xj f,j (x) = ±f (x)

j=1

for all x = 0.

(ii) Suppose conversely, that f : Rm \ {0} ’ R is a di¬erentiable function

satisfying ††. By setting up a di¬erential equation for the function v de¬ned

by v(») = f (»x), or otherwise, show that f is homogeneous of degree ±.

Exercise K.98. [6.2, T] (i) Suppose that ± : R2 ’ R2 is a linear map and

that its matrix with respect to the standard basis is

ab

.

cd

Show how ± may be calculated. (There is no particular point in carrying