The History of Mathematics: A Brief Course

QUESTIONS AND PROBLEMS 507

A set is of first category if it is the union of a sequence of sets Ak such that
every interval (a,b) contains an interval (c, d) disjoint from A^. All other sets
are of second category.^15 Although interest in the specific problems that inspired
Baire has waned, the importance of his work has not. The whole edifice of what
is now functional analysis rests on three main theorems, two of which are direct
consequences of what is called the Baire category theorem (that a complete metric
space is of second category as a subset of itself) and cannot be proved without it.
Here we have an example of an unintended and fortuitous consequence of one bit
of research turning out to be useful in an area not considered by its originator.

17.1. The familiar formula cos# = 4cos^3 (0/3) - 3cos(0/3), can be rewritten as
p(cos#/3, cos0) = 0, where p(x, y) = 4x^3 -3x-y. Observe that cos(6+2mn) = cosf?
for all integers m, so that

for all integers m. That makes it very easy to construct the roots of the equation
p(x,cosO) = 0. They must be cos((f? + 2ôçð)/3) for m = 0,1,2. What is the
analogous equation for dividing a circular arc into five equal pieces?
Suppose (as is the case for elliptic integrals) that the inverse function of an
integral is doubly periodic, so that f(x+mu>i +çù 2 ) = f(x) for all m and n. Suppose
also that there is a polynomial p(x) of degree n^2 such that p(/(0/n)) = /(#). Show
that the roots of the equation p(x) — /(È) must be /(è/ç + (k/n)u)\ + (1/ç)ù 2 ),
where fc and / range independently from 0 to ç - 1.
17.2. Show that if y(x,t) = (f(x + ct) + f(x - ct))/2 is a solution of the one-
dimensional wave equation that is valid for all ÷ and t, and y(0, t) = 0 — y(L, t) for
all t, then f(x) must be an odd function of period 2L.

17.3. Show that the problem X"(x) - \X(x) = 0, Y"(y) + XY(y) = 0, with

boundary conditions Y(0) = Y(2n), Y'(0) = Y'(27r), implies that ë = ç^2 , where

ç is an integer, and that the function X(x)Y(y) must be of the form (cnenx +

d„e~nx) (an cos(ny) + bn sin(ra/)) if ç ö 0.

17.4. Show that the differential equation

has the solution y = [(1 - x^2 )/(l +x^2 )]1/2. Find another obvious solution of this

equation.

17.5. Show that Fourier series can be obtained as the solutions to a Sturm-Liouville

problem on [0,2ð] with p(x) = r(x) Î 1, q(x) = 0, with the boundary conditions

y(0) = y(2ix), y'(0) = 2/(2ð). What are the possible values of A?

Questions and problems

(^15) In his work on set theory, discussed in Section 4 of Chapter 12, Hausdorff criticized this termi-
nology as "colorless."