|S_(2n+1)(x) - [f](x)| < [mu] cosec [mu] [|[f](x + 2[mu]) - [f](x)|
+ |[f](x - 2[mu]) - [f](x)|] + A|m cosec [mu],
where A is some fixed number independent of m. In any interval (a, b)
in which [f](x) is continuous, a value [mu]1 of [mu] can be chosen
such that, for every value of x in (a, b), |[f](x + 2[mu]) - [f](x)|,
|[f](x - 2[mu]) - [f](x)| are less than an arbitrarily prescribed
positive number [epsilon], provided [mu] = [mu]1. Also a value [mu]2
of [mu] can be so chosen that [epsilon][mu]2 cosec [mu]2 < 1/2[eta],
where [eta] is an arbitrarily assigned positive number. Take for [mu]
the lesser of the numbers [mu]1, [mu]2, then |S_(2n+1) - [f](x)| <
[eta] + A|m cosec [mu] for every value of x in (a, b). It follows
that, since [eta] and m are independent of x, |S_(2n+1) - [f](x)| <
2[epsilon], provided n is greater than some fixed value n1 dependent
only on [epsilon]. Therefore S_(2n+1) converges to [f](x) uniformly in
the interval (a, b).
_Case of a Function with Infinities._--The limitation that [f](x) must
be numerically less than a fixed positive number throughout the
interval may, under a certain restriction, be removed. Suppose F(z) is
indefinitely great in the neighbourhood of the point z = c, and is
such that the limits of the two integrals [int][c to c[+-][epsilon]]
F(z) dz are both zero, as [epsilon] is indefinitely diminished, then
_
/[pi]/2 sin mz
| F(z) ------ dz
_/ 0 sin z
denotes the limit when [epsilon] = 0, [epsilon]^1 = 0 of
_ _
/c-[epsilon] sin mz /[pi]/2 sin mz
| F(z) ------ dx + | F(z) ------ dz,
_/ 0 sin z _/c+[epsilon]^1 sin z
both these limits existing; the first of these integrals has
1/2[pi]F(+0) for its limiting value when m is indefinitely increased,
and the second has zero for its limit. The theorem therefore holds if
F(z) has an infinity up to which it is absolutely integrable; this
will, for example, be the case if F(z) near the point C is of the form
x(z)(z - c)^-[mu] + [psi](z), where [chi](c), [psi](c) are finite, and
0 < [mu] < 1. It is thus seen that [f](x) may have a finite number of
infinities within the given interval, provided the function is
integrable through any one of these points; the function is in that
case still representable by Fourier's Series.
_The Ultimate Values of the Coefficients in Fourier's Series._--If
[f](x) is everywhere finite within the given interval -[pi] to +[pi],
it can be shown that a_n, b_n, the coefficients of cos nx, sin nx in
the series which represent the function, are such that na_n, nb_n,
however great n is, are each less than a fixed finite quantity. For
writing [f](x) = [f]1(x) - [f]2(x), we have
_ _ _
/[pi] /[xi] /[pi]
| [f]1(x) cos nxdx = [f]1(-[pi] + 0) | cos nxdx + [f]1([pi] - 0) | cos nxdx
_/-[pi] _/-[pi] _/[xi]
hence
_
/[pi] sin n[xi] sin n[xi]
| [f]1(x) cos nxdx = [f]1(-[pi] + 0) --------- + [f]1([pi] - 0) ---------
_/-[pi] n n
with a similar expression, with [f]2(x) for [f]1(x), [xi] being
between [pi] and -[pi]; the result then follows at once, and is
obtained similarly for the other coefficient.
If [f](x) is infinite at x = c, and is of the form [phi](x)/(x - c)^K
near the point c, where 0 < K < 1, the integral
_
/[pi]
| [f](x)cos nxdx contains portions of the form
_/-[pi]
_ _
/[epsilon]+[epsilon] [phi](x) / c [phi](x)
| --------- cos nxdx | --------- cos nxdx;
_/ [c] (x - c)^K _/c-[epsilon] (x - c)^K
consider the first of these, and put x = c + u, it thus becomes
_
/[epsilon] [phi](c + u)
| ------------ cos n(c + u) du, which is of the form
_/ 0 u^K
_
/[epsilon] cos n(c + u)
[phi](c + [theta][epsilon]) | ------------ du;
_/ 0 u^K
now let nu = v, the integral becomes
_ _ _ _
| cos nc /n[epsilon] cos v sin nc /n[epsilon] sin v |
[phi](c + [theta][epsilon]) | ------- | ----- dv - ------- | ----- dv |;
|_ n^(1-K) _/ 0 v^K n^(1-K) _/ 0 v^K _|
hence n^(1-K) [int]([pi] to -[pi]) [f](x) cos nxdx becomes, as n is
definitely increased, of the form
_ _ _ _
| /[oo] cos v /[oo] sin v |
[phi](c) | cos nc | ----- dv - sin nc | ----- dv |
|_ _/ 0 v^K _/ 0 v^K _|
which is finite, both the integrals being convergent and of known
value. The other integral has a similar property, and we infer that
n^(1-K) a_n, n^(1-K) b_n are less than fixed finite numbers.
_The Differentiation of Fourier's Series._--If we assume that the
differential coefficient of a function [f](x) represented by a
Fourier's Series exists, that function [f]'(x) is not necessarily
representable by the series obtained by differentiating the terms of
the Fourier's Series, such derived series being in fact not
necessarily convergent. Stokes has obtained general formulae for
finding the series which represent f'(x), [f]"(x)--the successive
differential coefficients of a limited function [f](x). As an example
of such formulae, consider the sine series (1); [f](x) is represented
by
_
2 n[pi]x /l n[pi]x
-- [Sigma] sin ------ | [f](x) sin ------ dx;
l l _/0 l
_
/l n[pi]x
on integration by parts we have | [f](x)sin ------ dx
_/0 l
_ _
l | n[pi]a |
= ---- | [f](+0) [+-] [f](l - 0) + [Sigma] cos ------ {[f]([alpha] + 0) - [f]([alpha] - 0)} |
n[pi] |_ l _|
_
l /l n[pi]x
+ ----- | [f]'(x) cos ------ dx
n[pi] _/0 l
where [alpha] represent the points where [f](x) is discontinuous.
Hence if f(x) is represented by the series [Sigma]a_n sin (n[pi]x/l),
and [f]'(x) by the series [Sigma]b_n cos (n[pi]x/l), we have the
relation
_ _
n[pi] 2 | n[pi][alpha] |
b_n = ----- [alpha]_n - -- | [f](+0) [+-] [f](l - 0) + [Sigma]cos ------------ {[f]([alpha] + 0) - [f](alpha - 0)} |
l l |_ l _|
hence only when the function is everywhere continuous, and [f](+0)
[f](l - 0) are both zero, is the series which represents [f]'(x)
obtained at once by differentiating that which represents [f](x). The
form of the coefficient [alpha]_n discloses the discontinuities of the
function and of its differential coefficients, for on continuing the
integration by parts we find
_ _
2 | n[pi][alpha] |
[alpha]_n = ----- | [f](+0) [+-] [f](l - 0) + [Sigma] cos ------------ {[f]([alpha] + 0) - [f]([alpha] - 0)} |
n[pi] |_ l _|
_ _
2l | n[pi][beta] |
+ --------- | [f]'(+0) [+-] [f]'(l - 0) + [Sigma] sin ----------- {[f]'([beta] + 0) - [f]'([beta] - 0)} | + &c.
n^2[pi]^2 |_ l _|
where [beta] are the points at which [f]'(x) is discontinuous.