r=0

555

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

Deduce the existence of an in¬nitely di¬erentiable function f : R ’ R

whose Taylor series

∞

f (n) (q)

(q + h)n

n!

n=0

diverges for all h = 0 and all rational numbers q.

(iii) The result of (ii) is even stronger than at ¬rst appears. Use the ideas

of Exercise K.230 to show that, if a function F : R ’ R has derivatives of

all orders at x and

∞

F (n) (x) n

F (x + h) = h

n!

n=0

for all |h| < R, then F is in¬nitely di¬erentiable at t for all |t ’ x| < R and

∞

F (n) (t) n

F (t + k) = k

n!

n=0

whenever |k| < R ’ |t ’ x|.

Conclude that the function f obtained in (ii) can not have a valid Taylor

expansion about any point.

Exercise K.227. [11.4, T!] The object of this question and the next two is

to de¬ne x± and to obtain its properties directly rather than by the indirect

de¬nition used in Section 5.7. In this ¬rst question we deal with the case ±

rational. We assume the properties of xn when n is an integer.

(i) Carefully quoting the results that you use, show that, if n is a strictly

positive integer, then, given any x > 0, there exists a unique r1/n (x) > 0 with

r1/n (x)n = x. Show further (again carefully quoting the results that you use)

that the function r1/n : (0, ∞) ’ (0, ∞) is di¬erentiable with

r1/n (x)

r1/n (x) = .

nx

(ii) Show that if m, m , n and n are strictly positive integers with m/n =

m /n then

(r1/n (x))m = (r1/n (x))m

for all x > 0. Explain why this means that we can de¬ne rm/n (x) = r1/n (x)m

for all m, n positive integers.

(iii) If ±, β ∈ Q and ±, β > 0 prove the following results.

556 A COMPANION TO ANALYSIS

(a) r±+β (x) = r± (x)rβ (x) for all x > 0.

(b) r±β (x) = r± (rβ (x)) for all x > 0.

(c) r± (xy) = r± (x)r± (y) for all x, y > 0.

(d) r± (1) = 1.

(e) The function r± : (0, ∞) ’ (0, ∞) is di¬erentiable with

±r± (x)

r± (x) = .

x

Show further that, if n is a strictly positive integer,

(f) rn (x) = xn for all x > 0.

(iv) Show carefully how to de¬ne r± for all ± ∈ Q and obtain results

corresponding to those in part (iii).

Exercise K.228. [11.4, T!, ‘ ] We continue the arguments of Question K.227.

(i) Suppose that x > 0, ± ∈ R and ±n ∈ Q with ±n ’ ± as n ’ ∞.

Show that the sequence r±n (x) is Cauchy and so tends to a limit y, say.

Suppose that βn ∈ Q with βn ’ ± as n ’ ∞. Show that rβn (x) ’ y as

n ’ ∞. We can thus write r± (x) = y.

Show also that r± (x) > 0.

(ii) Suppose that ± ∈ R, ± = 0 and ±n ∈ Q with ±n ’ ± as n ’ ∞.

Show that, if 0 < a < b, we have r±n ’ r± uniformly on [a, b]. Deduce,

quoting any results that you use, that the function r± : (0, ∞) ’ (0, ∞) is

di¬erentiable with

±r± (x)

r± (x) = .

x

(iii) Prove the remaining results corresponding to to those in part (iii) of

Question K.227.

Exercise K.229. [11.4, T!, ‘ ] Question K.228 does not give all the prop-

erties of x± that we are interested in. In particular, it tells us little about

the map ± ’ x± . If a > 1, let us write P (t) = Pa (t) = at .

(i) Show that P : R ’ (0, ∞) is an increasing function.

(ii) Show that P (t) ’ 1 as t ’ 0.

(iii) By using the relation P (s+t) = P (s)P (t), show that P is everywhere

continuous.

(iv) Show that, if P is di¬erentiable at 1, then P is everywhere di¬eren-

tiable.

(v) Let f (y) = n(y 1/n ’ 1) ’ (n + 1)(y 1/(n+1) ’ 1). By considering f , or

otherwise, show that f (y) ≥ 0 for all y ≥ 1. Hence, or otherwise, show that

P (1/n) ’ P (0)

tends to a limit, L say,

1/n

557

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

as n ’ ∞.

(vi) Show that, if 1/n ≥ x ≥ 1/(n + 1) then

P (x) ’ P (0) P (1/n) ’ P (0) n + 1

¤ — .

x 1/n n

By using this and similar estimates, or otherwise, show that

P (x) ’ P (0)

’L

x

as x ’ 0 through values of x > 0.

(vii) Show that, if x = 0

P (x) ’ P (0) P (’x) ’ P (0)

= P (x)

’x

x

and deduce that

P (x) ’ P (0)

’L

x

as x ’ 0. Conclude that P is di¬erentiable at 0 and so everywhere with

P (t) = LP (t).

(viii) If a > 0 and we set Pa (t) = at , show that Pa is di¬erentiable with

Pa (t) = La P (t) for some constant La . Show that La > 0 if a > 1. What can

you say about La for other values of a?

(ix) If a, » > 0 show that La» = »La . Deduce that there is a real number

e > 1 such that, if we write

e(t) = et ,

then e is a di¬erentiable function with e (t) = e(t).

(x) Tear o¬ the disguise of La .

We have now completed a direct attack on the problem of de¬ning x± and

obtaining its main properties. It should be clear why the indirect de¬nition

x± = exp(± log x) is preferred.

Exercise K.230. [11.5, P] Suppose ∞ an z n has radius of convergence

n=0

greater than or equal to R > 0. Write g(z) = ∞ an z n for |z| < R. Show

n=0

∞

that, if |z0 | < R then we can ¬nd b0 , b1 , · · · ∈ C such that n

n=0 bn z

has radius of convergence greater than or equal to R ’ |z0 | and g(z) =

∞ n

n=0 bn (z ’ z0 ) for |z0 ’ z| < R ’ |z0 |. [This is quite hard work. If you

have no idea how to attack it, ¬rst work out formally what the values of the

bn must be. Now try and use Lemma 5.3.4.]

558 A COMPANION TO ANALYSIS

Show that g is complex di¬erentiable at z0 with g (z0 ) = b1 = ∞ nan z0 .n’1

n=1

Deduce that a power series is di¬erentiable and that its derivative is that ob-

tained by term by term di¬erentiation within its radius of convergence. (We

thus have an alternative proof of Theorem 11.5.11.)

Exercise K.231. [11.5, P] Here is another proof of Theorem 11.5.11. Sup-

pose ∞ an z n has radius of convergence greater than or equal to R > 0.

n=0

(i) Show that ∞ nan z n’1 and ∞ n(n ’ 1)an z n’2 also have radius

n=1 n=2

of convergence R.

(ii) Show that

n’2

n

¤ n(n ’ 1)

j j

for all n ’ 2 ≥ j ≥ 0 and deduce that

|(z + h)n ’ z n ’ nz n’1 h| ¤ n(n ’ 1)(|z| + |h|)n’2

for all z, h ∈ C.

(iii) Use (i) and (ii) to show that, if 0 < δ and |z| + |h| < R ’ δ, then

∞ ∞ ∞

n n

nan z n’1 ¤ A(δ)|h|2

an (z + h) ’ an z ’ h

n=0 n=0 n=1

where A(δ) depends only on δ (and not on h or z).

(iv) Deduce that a power series is di¬erentiable and that its derivative is

that obtained by term by term di¬erentiation within its radius of convergence.

Exercise K.232. [11.5, P] Consider the Legendre di¬erential equation

(1 ’ x2 )y ’ 2xy + l(l + 1)y = 0,

where l is a real number. Find the general power series solution. Show that,

unless l is non-negative integer, every non-trivial power series solution has

radius of convergence 1.

Show, however, that, if l = n a non-negative integer the general series

solution can be written

y(x) = APn (x) + BQn (x)

where Qn is a power series of radius of convergence 1, Pn is a polynomial of

degree n, and A and B are arbitrary constants.

559

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

Verify that, when n is a non-negative integer, the function given by v(x) =

(x2 ’ 1)n satis¬es the equation

(1 ’ x2 )v (x) + 2nxv(x) = 0.

Deduce that v (n) (x) satis¬es the Legendre di¬erential equation with l = n.

[Consult Exercise K.270 if you need a hint.] Conclude that Pn is a constant

multiple of the function pn de¬ned in Exercise K.212.

Exercise K.233. [11.5, P] Obtain the general power series solution of

2

2d w dw

+ (z 2 ’ 1)w = 0.

z +z

dz 2 dz

For what values of z is your solution valid and why?

Answer the same questions for

2

2d w dw

+ (z 2 ’ 1)w = z 2 .

z +z

dz 2 dz

Exercise K.234. [11.5, P] Using Lemma 11.5.19 or otherwise, ¬nd the

power series expansion ∞ aj xj of (1 + x)1/2 for x real and |x| < 1.

j=0

Show that the complex power series ∞ aj z j has radius of convergence

j=0

1. What is the power series for the product ∞ aj z j ∞ aj z j and why?

j=0 j=0

What is its radius of convergence?

By considering

∞ ∞ ∞

aj K ’j z j

aj z j aj z j ,

j=0 j=0 j=0

or otherwise, show that, if two power series of the same radius of conver-

gence R are multiplied, the resulting power series may have any radius of

convergence with value R or greater.

By considering expressions of the form

∞ ∞ ∞

Aj z j Bj z j + Cj z j ,

j=0 j=0 j=0

or otherwise, show that, if two power series of radius of convergence R and S

are multiplied, the resulting power series may have any radius of convergence

with value min(R, S) or greater.

560 A COMPANION TO ANALYSIS

Exercise K.235. [11.5, P] By modifying the proof of Abel™s test (Lemma 5.2.4),

or otherwise, prove the following result. Let E be a set and aj : E ’ Rm a

sequence of functions. Suppose that there exists a K such that

n

aj (t) ¤ K

j=1

for all n ≥ 1 and all t ∈ E. If »j is a decreasing sequence of real positive

numbers with »j ’ 0 as j ’ ∞ then ∞ »j aj converges uniformly on E.

j=1

Deduce, in particular, that if bn ∈ R and ∞ bn converges, then ∞ bn xn

n=0 n=0

converges uniformly on [0, 1]. Explain why this implies that

∞ ∞

1

bn

n

bn x dx = .

n+1

0 n=0 n=0

Show that, provided we interpret the left hand integral as an improper

1’

integral, lim ’0, >0 0 , equation remains true under the assumption

∞

that n=0 bn /(n + 1) converges.

Show that

111

1 ’ + ’ + · · · = log 2

234

and

111 π

1 ’ + ’ + ··· = .

357 4

Do these provide good methods for computing log 2 and π? Why?

Recall that, if a, b > 0, then aloga b = b. Show that

log2 e ’ log4 e + log8 e ’ log16 e + . . .

converges to a value to be found.

Exercise K.236. [11.5, G, S] (Only if you have done su¬cient probability,

but easy if you have.) Let X be a random variable taking positive integral

values. Show that

∞

φX (t) = Et = X

Pr(X = n)tn

n=0

is well de¬ned for all t ∈ [’1, 1].

If EX is bounded, show that

φX (1) ’ φX (t)

’ EX

1’t

as t ’ 1 through values t < 1. Give an example of an X with EX in¬nite.

561

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

Exercise K.237. [11.5, T, S] Use the ideas of Exercise K.78 to prove the

following extension. Suppose that we have a sequence of functions an : „¦ ’ C

and a sequence of positive real numbers Mn such that ∞ Mj converges

j=1

absolutely and |an (z)| ¤ Mn for all z ∈ „¦ and all n. Then

∞

N

(1 + aj (z)) ’ (1 + aj (z))

j=1 j=1

uniformly on „¦.

Exercise K.238. [11.5, P, S, ‘ ] We do not have the apparatus to exploit

Exercise K.237 fully but the following exercise is suggestive.

(i) Show that SN (z) = z N (1 ’ z 2 n’2 ) converges uniformly to S(z) =

n=1

z ∞ (1 ’ z 2 n’2 ) on any disc {z : |z| < R}. Deduce that S is a well de¬ned

n=1

continuous function on C.

(ii) By writing SN (z) = ’(N !)’2 N n=’N (n ’ z) and considering SN (z +

1)/SN (z), or otherwise, show that S(z + 1) = S(z) (that is, S is periodic

with period 1). Show also that S(z) = 0 if and only if z is an integer.

[The product z ∞ (1 ’ z 2 n’2 ) goes back to Euler who identi¬ed it with

n=1

a well known function which the reader should also be able to guess. An

advantage of in¬nite products over in¬nite sums is that we can ¬nd zeros of

the function much more easily.]

Exercise K.239. [11.5, T] (i) Suppose that an ∈ C and ∞ an z n has

n=0

radius of convergence R > 0. If aN = 0 and an = 0 for n < N , use the

inequality

q q

n N

|an z n’N |

¤ |z| |aN | ’

an z

n=0 n=N +1

(for q > n), together with standard bounds on |an z n |, to show that there

exists a δ > 0 such that

q

|aN ||z|N

n

≥

an z

2

n=0

for all |z| < δ.

(ii) Suppose that an ∈ C and ∞ an z n has radius of convergence R > 0.

n=0

∞ n

Set f (z) = n=0 an z for |z| < R. Show, using (i), or otherwise, that, if

there exist wm ’ 0 with f (wm ) = 0 and wm = 0, then an = 0 for all n and

f = 0.

562 A COMPANION TO ANALYSIS

Exercise K.240. [11.5, P, ‘ ] This is a commentary on the previous exer-

cise. We shall consider functions from R to R.

(i) Suppose that an ∈ R and ∞ an xn has radius of convergence R > 0.

n=0

Set f (x) = ∞ an xn for |x| < R. Show that, if there exist xm ’ 0 with

n=0

f (xm ) = 0 and xm = 0, then an = 0 for all n and f = 0.

(ii) Explain why a polynomial of degree at most n with n + 1 zeros must

be zero.

(iii) Give an example of a power series of radius of convergence ∞ which

has in¬nitely many zeros but is not identically zero.

(iv) Let F be as in Example 7.1.5. Set G(x) = F (x) sin x’1 for x = 0

and G(0) = 0. Show that G is in¬nitely di¬erentiable everywhere (this will

probably require you to go through much the same argument as we used for

F ). Show that there exist xm ’ 0 with G(xm ) = 0 and xm = 0 but G is not

identically zero.

Exercise K.241. [11.5, P] This question prepares the way for Exercise K.242.

The ¬rst two parts will probably be familiar but are intended to provide a

hint for the third part.

n n n+1

(i) If n ≥ r ≥ 1, show that + = .

r’1 r r

(ii) Use induction to show that

n

n r n’r

(x + y)n = xy

r

r=0

whenever n is a positive integer and x and y are real.

(iii) Suppose that n is a positive integer and x and y are real. Verify the

next formula directly for n = 0, 1, 2, 3 and then prove it for all n.

n r’1 n’r’1 n’1

n

(x ’ j) (y ’ k) = (x + y ’ q).

r

r=0 q=0

j=0 k=0

Exercise K.242. [11.5, T, ‘ ] The following is a modi¬cation of Cauchy™s

proof of the binomial theorem. We shall work in R. We take w as a ¬xed

real number with |w| < 1.