Besides the power series, there is another class of infinite series which has a particularly important role both in pure and applied mathematics. These are the Fourier series, in which the individual terms are trigonometric functions and the sum is a periodic function.
9.1.1 General Remarks: Periodic functions of the time, i.e., functions which repeat their course after a definite interval of time, are encountered in many .applications. In most machines, a periodic process occurs in rhythm with the rotation of a flywheel, e.g., the alternating current developed by a dynamo. Periodic functions are also associated with all vibration phenomena.
A periodic function with period 2l is represented by the equation
true for all values of x. We call especially attention to the fact that 2l is called the period.
In representing periodic functions, it is often convenient to denote the independent variable x by a point on the circumference of a circle instead of the usual point on a straight line. If a function f(x) has the period 2p , i.e., if the equation
holds for all values of x and we denote by x the angle at the centre of a circle of unit radius, which is included between an arbitrary initial radius and the radius to a variable point on the circumference, then the periodicity of the function f(x) is expressed simply by the fact that there corresponds to each point on the circumference just one value of the function. For example, in the case of a machine, periodicity may be expressed in terms of the position of a point on the flywheel.
It is worth noting that in addition to the period 21, the function f{x) necessarily has also the period 4l, since f(x+ 4l) =f(x+ 2l) = f(x); similarly, f(x) has the periods 6l, 8l, ···; and it is also possible (though not necessarily true) that f(x) may have shorter periods such as l or l/5. Graphically speaking, the graph of the function has in any two consecutive intervals of length 21 exactly the same shape. In order to have available a second interpretation, which some readers may prefer, we may think of the variable x as the time and write, accordingly, sometimes t instead of x; the function f(x) then represents a periodic process or, as we shall also say, a vibration (or oscillation). The period 2l = T is then called the period of vibration (or period of oscillation).
If any arbitrary function f(x) is given in a definite interval, say. -l £ x £ l, it can always be extended as a periodic function; we have only to define f(x) outside the interval by the equation f(x + 2nl) = f(x), where n is an arbitrary positive or negative integer. We must point out here that, if f(x) is continuous in the interval -l £ x £ l, but f(-l) ¹ f(l), our extended periodic function will be discontinuous at the points ±l, ±3l, ···, (Figs. 7 and Fig. 8), where l = p . Moreover, in this case, the extension fails to yield a single-valued function f(x) at the points x = ±l, ±3l, ···, since, for example, we have defined f(3l) as f(l+2l), which gives f(3l) = f(l), and we also have defined it as /(-l+ 4l), which yields f(3l) = f(-l). We avoid this difficulty by not extending the function as defined for -l £ x £ l, but as defined either for -l < x £ l or -l £ x < l, i.e., we either discard the original value f(-l) or the original value f(+l).
We should point out here a general fact,, which relates to periodic functions and is expressed by the equation
or, in words: The integral of a periodic function over an interval, the length of which is one period T == 21, has always the same value, no matter where the interval lies.
In order to prove this, we need only note that,
by virtue of the equation 
, the substitution x = x - 21
yields
In particular, for a = -l -a and b = -l, it follows that
whence
which proves our statement. If we recall the geometrical meaning of the integral, the statement is made obvious by Fig. 1.
The simplest periodic functions, from which we shall later compose the most general periodic functions, are a sinw x and a cosw x, or, more generally, asinw(x - x) and acosw(x - x), where a(³ 0), w(³ 0) and x are constants. We call the processes represented by these functions* sinusoidal vibrations or simple harmonic vibrations (or oscillations). The period of vibration is T=2p/w. The number w is called the circular frequency of the vibration**; since 1/T is the number of vibrations in unit time or frequency, w is the number of vibrations in time 2p. The number a is called the amplitude of the vibration; it represents the maximum value of the function asinw(x - x) or acosw(x - x), since both sine and cosine have the maximum value 1. The number w(x - x) is called the phase and the number wx the phase displacement.
* Either of these formulae taken by itself (for all values of a and x) represents the class of all sinusoidal vibrations; the two formulae are equivalent to one another, since asinw(x - x) = acosw{x - (x + p /2w)}.
**The reader should take care to distinguish between the terms frequency and circular frequency!
We obtain these functions graphically by stretching the sine curve in the ratios 1 : w along the x-axis and a : 1 along the y-axis and then translating the curve by a distance x along the x-axis (Fig. 2) in the positive direction.
We can also repress sinusoidal vibrations by the addition formula for the trigonometric functions in the form
respectively, where a = -asinwx and b = acoswx. Conversely, every function of the form
represents a sinusoidal vibration asinw(x - x) with the
amplitude 
and the phase displacement wx given by the
equations a = -asinwx and b=acoswx. By using the expression asinwx + bcoswx, we see
that the sum of two or more such functions with the same circular
frequency w always represent another sinusoidal vibration with the
circular frequency w.
9.1.2 Superposition of Sinusoidal Vibrations. Harmonics. Beats:. Although many vibrations are found to be sinusoidal ( 5.4.4), it is nevertheless true that most periodic motions have a more complicated character, being obtained by superposition of sinusoidal vibrations. Mathematically speaking, this simply means that the motion, e.g., the distance of a point from its initial position as a function of the time, is given by a function which is the sum of a number of pure periodic functions of the above type. The sine waves of the function are then piled up on top of one another (that is, their ordinates are added), or, as we say, they are superposed. In this superposition, we assume that all the circular frequencies (and, of course, the periods) of the superposed vibrations differ, because the superposition of two sinusoidal vibrations with the same circular frequency gives us another sinusoidal vibration with the same circular frequency (but with a different amplitude and phase displacement), as shown above.
If we consider the simplest case, the superposition of two sinusoidal vibrations with the circular frequencies w1 and w2, we find that there are two fundamentally different cases, depending on whether the two circular frequencies have a rational ratio or not, or, as we say, whether they are commensurable or incommensurable. We begin with the first case and, by way of an example, take the second circular frequency to be twice the first: w2=2w1. The period of the second vibration will then be half the period of the first vibration, 2p/2w1 = T2 = T1/2, whence it will necessarily have not only the period T2, but also the doubled period T1, since the function repeats itself after this double period; and the function after their superposition will also have the period T1. The second vibration with twice the circular frequency and half the period of the first vibration is called the first harmonic of the first vibration (the fundamental).
Corresponding statements hold, if we introduce a further vibration with the circular frequency w3 = 3w1. Here again, the vibration function sin3w1x will necessarily repeat itself with the period 2p /w1 = T1. Such a vibration is called a second harmonic of the given vibration. Similarly, we can consider third, fourth, ··· , (n — l)-th harmonics with the circular frequencies w4 = 4w1, w5 = 5w1, ··· , wn = nw1, and, moreover, with any phase displacements of your choice. Every such harmonic will necessarily repeat itself after the period T1=2p /w1 and, consequently, every function obtained by superposition of a number of vibrations, each of which is a harmonic of a given fundamental circular frequency w1, will itself be a periodic function with the period 2p/w1=T1. By superposing vibrations with circular frequencies ranging from that of the fundamental to that of the (n - 1)-th harmonic, we obtain a periodic function of the form
The proportions of
Fig.3 correspond to the assumption w =1.
(The constant a, which we have introduced here in order to make the formula slightly more general, does not affect the periodicity, since it is periodic for any period.) Since this function contains 2n + 1 constants, which we can choose arbitrarily, we are thus able to generate very complicated curves which are not at all like the original curves. Figs. 3-5 display this situation.
The term harmonic originated in acoustics, where we find that there corresponds to a fundamental vibration with circular frequency a note of a certain pitch, then the first, second, third, etc., harmonics correspond to the sequence of harmonics of the fundamental, i.e., to the octave, octave + fifth, double octave, etc. (In acoustics, also the terms overtone, (upper) partial are used.
In general, in the case of superposition of vibrations in which the circular frequencies have rational ratios, these circular frequencies can all be represented as integral multiples of a common fundamental circular frequency. However, the superposition of two vibrations with incommensurable circular frequencies w1 and w2 represents an intrinsically different type of phenomenon. In that case, the process resulting from the superposition of sinusoidal vibrations will no longer be periodic. We cannot go here into the mathematical discussions that arise from this, but will merely remark that such functions always have an approximately periodic character or, as we say, are almost periodic. In 1935, such functions had just been studied in great detail.
A final remark on the superposition of sinusoidal vibrations concerns the phenomenon of beats. If we superpose two vibrations, both with unit amplitude but different circular frequencies w1 and w2, and, for the sake of simplicity, take the same value of x for both (the generalization to arbitrary phase can be left to the reader), then we are merely concerned with the behaviour of the function
A well-known trigonometrical formula yields
This equation represents a phenomenon which we may think of as follows: We have a vibration with the circular frequency ½(w1 + w2) and the period 4p/(w1+w2). However, this vibration does not have a constant amplitude; on the contrary, its amplitude is given by 2cos½(w1+w2)x, which varies with the longer period 4p/(w1-w2)). This point of view is particularly useful and easy to interpret when the two circular frequencies w1and w2 are relatively large, while their difference w1-w2 is small in comparison with them. Then the amplitude
of the vibration 
with
period 
will vary only slowly compared with the period of
vibration and this change of amplitude will repeat itself
periodically with the long period 
These rhythmic changes
of amplitude are called beats. Everyone is acquainted with this phenomenon in
acoustics, and perhaps also in wireless telegraphy, in which the
circular frequencies w1and w2 are, as a rule, far above those which the
ear can detect, while the difference w1- w2
falls in the range of audible notes. The beats can then be heard,
while the original vibrations remain imperceptible to the ear.
Fig. 6 illustrates the phenomenon.
9.2.1 General Remarks: An investigation of vibration phenomena and periodic functions gains in formal simplicity if we employ complex numbers, combining each pair of trigonometric functions cosw x and sinw x to form an expression of the type cosw x + isinw x = eiw x(8.7.1). We must here bear in mind that one equation between complex quantities is equivalent to two equations between real quantities and that our results must always be interpreted and made intelligible in the real domain.
If we replace everywhere the trigonometric functions by exponential functions in accordance with the formulae
we express sinusoidal vibrations in terms of the complex quantities eiw x, e -iwx or
respectively, where a, w and wx are the real quantities: Amplitude, circular frequency and phase displacement. The real vibrations are obtained from this complex expression simply by taking real and imaginary parts.
The convenience of this mode of representation for many purposes is due to the fact that the derivatives of the real vibrations with respect to the time are obtained by differentiating the complex exponential function just as if it were a real constant, as is expressed by the formula
or
9.2.2 Application to the Study of Alternating Currents: We shall now illustrate these matters by means of an important example and denote here the independent variable, the time, by t instead of x.
We consider an electric circuit with resistance R and inductance L, on which an external electromotive force (voltage) E is impressed. In the case of a direct current, E is constant and the current I is given by Ohm's law E = RI.
However, if we are dealing with an alternating current, E is a function of the time t, and consequently so is I; Ohm's law then becomes (cf. 3.7.6)
In the simplest case, to which we will restrict ourselves here, the external electromotive force E is sinusoidal with circular frequency w. Now, instead of taking this oscillation in the form accost t or asinw t, we combine both possibilities formally in the complex form
where e ( >0) represents the amplitude. We shall operate
with this complex voltage as
if it were a real parameter and thus obtain a complex current I.
Then, the significance of the relation thus found between the
complex qnantities E and I is that the current
corresponding to an electromotive force ecosw t
is the real part of I, while the current corresponding
to an electromotive force esinwt is the imaginary part of I. The
complex current can be calculated immediately, if we write down
for I an expression of the form 
i.e., if we assume
that I is also sinusoidal with circular frequency w. The
derivative of I is then given formally by the expression
On substitution of these quantities into the generalized form of Ohm's law and dividing by the factor eiw t, we obtain the equation e - aLicw = Ra, or
so that 
We may regard this last equation as Ohm's law for alternating currents in complex form, if we call the quantity 
the complex resistance of the circuit. Ohm's law is then the same as for direct current: The current is equal to the voltage
divided by the resistance.
If we write the complex resistance in the form
where
we obtain 
According to this formula, the current has the same period (and circular frequency) as the voltage; the amplitude a of the current is connected with the amplitude e of the electromotive force by the equation
and, in addition, there is a phase difference
between the current and the voltage. The current does not reach
its maximum at the same time as the voltage, but at a time d /w later and
the same is of course true for the minimum. In electrical
engineering, the quantity 
is frequently called the impedance or the alternating
current resistance of the circuit
for the circular frequency w; the phase displacement, usually stated in degrees, is
called the lag.
9.2.3 Complex Representation of the Superposition of Sinusoidal Vibrations:. So far, the complex notation has been used to denote the combination of two sinusoidal vibrations. But a single vibration or a compound vibration of the type
(for the sake of simplicity, we have taken w = 1) can also be reduced to complex form by the substitution
The above expression then assumes the form
where the complex numbers an are linked to the real numbers a, an and bn by the equations
In order that the equation an = an + a-n shall formally include the case n = 0, we often put a = a0 = a0/2.
Conversely, we may regard any arbitrary expression of the form
as a function which represents the superposition of vibrations written in complex form. In order that the result of this superposition may be real, it is only necessary that an + a-n should be real and an - a-n pure imaginary, i.e., that an and a-n are conjugate complex numbers.
9.2.4 Deduction of a trigonometric Formula: By using complex notation, we obtain a very simple proof of a formula which we shall require below. This is the trigonometric summation formula
which holds for all values of a except the values 0, ±2p, ±4p, ···. In order to prove this, we replace the cosine function by its exponential expression and thus write the sum sn(a) in the form
On the right hand side, we have a geometric progression with the common ratio q = eia ¹ 1. Using the ordinary formula for the sum, we have
On multiplying the numerator and denominator by e-ia/2, we obtain
as has been stated above.
1. Sketch the curve 
for N = 3, 5
6.
2. Sketch the curve 
for N = 3, 6,
8.
3. Evaluate the sum 
4. If
prove that
(The expression sm is called the Fejér kernel and is of great importance in the more advanced study of Fonrier series.)
5. Show that 
where sm(a) is the Fejér kernel of
Exercise 4.
The function
resulting from the superposition of sinusoidal vibrations, contains 2n + 1 arbitrary constants a, an, bn. There arises now the question whether these constants can be chosen such that in the interval -p £ x £ +p the sum S(x) shall approximate to a given function f(x), and if so, how they are to be found. More precisely, we ask whether the given function f(x} can be expanded in an infinite series
If we assume for the moment that this expansion of the function f(x) is actually possible and that the series converges uniformly in the interval -p £ x £ +p, we readily obtain a simple relation between the function f(x) and the coefficients a = ½a0, an and bn. (We shall soon see that the notation a = ½a0 is justified by its convenience.) We multiply the above hypothetical expansion by cosnx and integrate term by term, which is permissible on account of its uniform convergence. By virtue of the orthogonality relations
proved at the end of 4.3, we obtain at once for the coefficients the formula
Similarly, multiplying the series by sin nx and integrating, we find
These formulae assign a definite sequence of coefficients an and bn, usually called Fourier coefficients, to every function f(x) which is defined and continuous in the interval -p £ x £ +p, or has only a finite number of jump discontinuities there. If the function f(x) is given, we can employ these quantities an, bn to form the Fourier partial sum
and we may also write down formally the corresponding infinite Fourier series. Our problem is now to distinguish simple classes of functions f(x) for which these Fourier series actually converge and represent them.
In order to formulate the result which we wish to prove, we introduce the definition:
A function f(x) is said to be sectionally continuous in an interval, if it is itself sectionally continuous (i.e., is continuous in the interval except for a finite number of jump discontinuities) and if, in addition, its first derivative f(x) is sectionally continuous.
We shall imagine that the function f(x), originally defined in the interval -p£x£p, is periodically extended. At each point at which the function f(x) has a jump discontinuity, we shall, if necessary, alter the function and assign to it the value which is the arithmetic mean of the left hand and right-hand limits of f(x), i.e., we shall write
where f(x - 0) and f(x + 0) are simply the limits of f(x) as x approaches from the left and from the right hand side, respectively. This equation is obviously true for every point x at which f(x) is continuous.
Our goal is now the theorem: If the function f(x) if sectionally smooth and satisfies the above equation, then its Fourier series converges at any point x and represents the function.
Note that this theorem can be proved for more general classes of functions. However, the result formulated here is sufficient for all applications.
Moreover, we shall prove the theorem: In every closed interval in which the function f(x) (imagined to be periodically extended) is continuous as well as sectionally smooth, the Fourier series converges uniformly.
Finally: If the function f(x) is sectionally smooth and has no discontinuities, the Fourier series converges absolutely.
The proofs of these theorems will be postponed (9.5). We merely wish to emphasize here that the functions, which can be expanded according to these theorems, have a very high degree of arbitrariness; it is by no means necessary that the function should be given by a single analytical expression.
In the next section, we shall display the extraordinary fertility of Fourier expansions by discussing a number of examples.
9.4. Examples of Fourier Series
9.4.1 Preliminary Remarks: We shall assume that our functions f(x) have the period 2p and are defined in the interval -p £ x £ +p, . Beyond this interval, to the left and right hand sides, they are to be extended periodically as in 8.1.1.
Before going into details, we note that if f(x) is an even function, then clearly f(a) sin nx is odd and f(x) cos nx is even, so that
We thus obtain a cosine series. On the other hand, if f(x) is an odd function, then
and we obtain a sine series.
Consequently, if the function f(x) is initially given only in the interval 0 < x <p, we can extend it in the interval -p < x < 0 either as an odd function or as an even function, and, correspondingly, expand it in the interval 0 < x < p in a sine series or in a cosine series.
9.4.2 Expansion of the Functions y (x) = x and j (x) = x²: For the odd function x, we have
and, on integration by parts,
Hence, we obtain for the periodic function y(x), which is equal to x in the interval -p < x < p (Fig. 7 below)
If we set x = p /2 , we obtain Gregory's series
with which we are already familiar. The function y(x), represented by this series, is not a continuous function; on the contrary, it jumps by 2p at the points x = kp, k= ±l, ±3. ±5, ···. At these points of discontinuity, that is, at the points x = kp, k = ±l, ±3, ±5, ···, each term of the series is zero, whence the function itself is zero. Hence, at the points of discontinuity, the series represents the arithmetic mean of the left hand and right hand limits.
If x is any fixed number between -p and p and we replace x in the above series by (x - x), we obtain the series
This series may also be written in the form of a Fourier series with the coefficients
which tend to zero as n increases; this series represents a function with the discontinuities, described above, at the points x = x ±p, x = x ± 3p, ···.
For the even function j (x) = x², on integrating twice by parts, we find
Differentiating this series term by term and dividing by 2, we recover formally the series for y(x) = x.
9.4.3 Expansion of the Function xcos x:. For this odd function, we find
Using the formula
above, we find
Hence we obtain the series
and, if we add the series in 9.4.2, the series
When the function, which is equal to xcos x in the interval -p < x < p, is extended periodically beyond this interval, the same discontinuities (Fig. 8, below) appear as are exhibited by the function y(x) considered in 9.4.2. On the other head, if the function x(l + cos x) is periodically extended, it remains continuous at the end-points of the interval and, in fact, its derivative also remains continuous, since the discontinuities are eliminated by the factor 1+cosx which vanishes together with its derivative at the end-points.
9.4.4 The Function f(x) = |x|: This function is even, whence bn = 0 and
and, on integration by parts, we obtain
whence
Setting x = 0, we obtain the remarkable formula
9.4.5 The Step Function: The function defined by the equations
as, indicated by Fig. 9, is an odd function, whence an = 0 and
so that its Fourier series is
For x = p/2, in particular, this again yields Gregory's series.
Note that this series can be derived formally from that for |x| by term by term differentiation.
9.4.6. The Function f(x) = |sin x|: The even function f(x) = |sin x| can be expanded in a sine series with the coefficients an given by
We thus obtain
9.4.7 Expansion of the Function cos mx. Resolution of cotan into Partial Fractions. The Infinite Product for sin x:. Let f(x) = cos mx for -p < x < p, where m is not an integer. Since f(x) is even, we obtain bn = 0, while
We thus find
This function remains continuous at the points x = ± p. If we set x = p, divide both sides of the equation by sin m x and then write x instead of m, we obtain
This is the so-called resolution of cotan x into partial fractions, a very important formula which is frequently discussed in analysis. We now rewrite this series in the form
If x lies in an interval 0 £ x £ q <
1, the absolute value of the n-th term on the right hand
side is less than 
whence the series converges uniformly in this
interval and can be integrated term by term. We thus obtain
on the left hand side and
on the right hand side after multiplying both sides by p. If we step over from the logarithm to the exponential function, we find
Hence
Thus, we have obtained the famous expression for sin x as an infinite product. Setting x = ½, we obtain Wallis' product
The formula for sin p x is particularly interesting, because it shows directly that the function sin p x vanishes at the points x = 0, ± 1, ±2, ··· . In this respect, it corresponds to the factorization of a polynomial when its zeroes are known.
9.4.8 More Examples: Brief calculations, similar to the preceding ones, yield the following examples of expansions in series.
The function, defined by the equation f(x) = sin m x for -p < x < p can be expanded in the series
Setting x p/2 and using the
relation 
, we obtain the resolution of the secant into partial fractions, i.e., of the function 
this
expansion is
where we have replaced m/2 by x.
The series for the hyperbolic functions cosh m x and sinh m x (-p < x < p) are
1. Find the Fourier expansions for the functions which are periodic with period 2p and which are defined in -p < x < p by:
2. The function f(t) is
periodic with period 1 and is given in 0 < x < 1
by f(t)=t. Prove that
3. The Bernoulli polynomials Bn(t) are defined by the relations:
Find B2(t), B3(t), B4t). (Note.—The numbers Bn(0) are rational and , in fact, are the same as Bemoulli's numhers Bn (cf. A8.4)
4. Verify the Fourier expansions for the Bernoulli polynomials:
5. Prove that
6. Prove that
7. Prove that
8. Obtain the infinite product for the cosine from the relation
9.5 The Convergence of Fourier Series
We now proceed to establish rigorously the theorems which were stated in 9.3 and illustrated in 9.4.
9.5.1 The Convergence of the Fourier Series of a Sectionally smooth Function:. We first recall that, if f(x) is any function which is defined and sectionally continuous (i.e., continuous, except at most at a finite number of jump discontinuities) in the interval -p £ x £ p, we can form its Fourier coefficients according to the formulae
and write down formally the series
This series is called the Fourier series corresponding to f(x), irrespectively of whether it converges or not. We will now find the conditions, which must be imposed on f(x), in order to ensure that the Fourier series corresponding to f(x) converges and represents f(x). We will assume that f(x) is extended periodically beyond the interval -p < x £ p.
We shall now prove the theorem: If the function f(x) is sectionally smooth, i.e. f(x) and its derivative f '(x) are sectionally continuous, and satisfies at each point of discontinuity (s) the condition f(x)=½{f(x-0)+f(x+0)}, then the Fourier series corresponding to f(x) converges at every point and represents the function f(x).
In order to prove this theorem, we consider the partial sums
If we substitute for the coefficients the above integral expressions and then interchange the order of integration and summation, we obtain
or, by the addition theorem for the cosine,
If we now apply the summation formula obtained above, this becomes
Finally, applying the transformation t = (t - x) and noting the periodicity of the integrand, we obtain
Starting with this form of the partial sum Sn(x), we can prove that it tends to f(x), by means of the lemma:
If the function s(x) is sectionally continuous in the interval a £ x £ b, then the integral
tends to 0 as l increases.
In the proof, we may assume that s(x) is continuous in the entire interval, since otherwise we merely need carry out the argument for each sub-interval in which s(x) is continuous
As in the similar argument in A8.2, we note that, if l is positive, the function sinlt is alternately positive and negative in successive intervals of length p /l. For large values of l, the contributions to the integral from adjacent intervals almost cancel one another, since, on account of the continuity, the values of s(x) in two such adjacent intervals differ only slightly from each another. We make use of this circumstance by transforming the integral I by the substitution t = t + h, where h = p /l; then, since sin lt = - sin lt, we obtain
If we again replace the letter t by t and then add the two expressions for I, we find
If M is an upper bound for the absolute value of s(x), i.e., if for all values of x in the interval under consideration |s(x)| £ M, then there follows at once from this expression for I the inequality
Now, let e be any positive number; if we choose l so large that in the entire interval a £ t £ b - h the expression |s(t) - s(t + h)| remains less than e /(b - a) and also Mh = Mp /l < e /2, then |I|] < e ; consequently, since e can be chosen as small as we please,
If we assume that s(x), besides being continuous, has a sectionally continuous derivative s'(x), the proof of this lemma follows simply on integration by parts. In fact,
We see here at once that, as l increases, the right hand side tends to zero.
Besides this lemma, we need the integration formula
which is true for every positive integer n. We establish this readily by using our summation formula for the cosine, since
Proof of the Main Theorem: By means of the lemma, the main theorem is readily proved, i.e., the formula
We begin by subdividing the interval of integration at the origin. For fixed values of x, the function
is sectionally continuous in the interval 0 £ t £ p. In fact, this is obvious when 0£ t £ p, while the continuity at t = 0 follows from the assumed existence of the right hand derivative
Hence, as l = n + ½ increases, the integral
tends to zero.
However, since the factor f(x
+ 0) can be taken out of the second integral on the right hand
side and, for l = n + ½, the integral 
is equal to p/2, we
obtain immediately
By setting in this equation x
= 0, f(t) = (sin ½t)/t and
then replacing t by u/l, we obtain the
important relation 
(cf. 4.8.5).
In the same way, we obtain for the interval -p £ t £ 0
and by addition
9.5.2. Further Investigation of Convergence: In the neighbourhood of those points, where the function f(x) is discontinuous, the Fourier series does not converge uniformly; in fact, by 8.4.3, a uniformly convergent series of continuous functions possesses a continuous sum. Nevertheless, we have the important theorem:
If a sectionally smooth periodic function has no discontinuities, its Fourier series converges absolutely and uniformly. The convergence of the Fourier series for any sectionally smooth function whatsoever is uniform in every closed interval which contains no point of discontinuity of the function.
In order to prove this theorem, we start from a fundamental inequality satisfied by the Fourier coefficients of any function f(x) which is sectionally continuous (note that f(x) is not assumed to be sectionally smooth). This so-called Bessel inequality states that for all values of n
The proof follows from the fact that the expression
is always positive or zero. If we evaluate the integral by expanding the bracket under the integral sign and recall the orthogonality relations and definitions of the Fourier coefficients, we obtain at once Bessel's inequality in the form
In addition to Bessel's inequality, we employ Schwarz's inequality : If u1,u2,···,un and v1,v2,···,vn are arbitrary real numbers, it is always true that
the equality sign occurring only when the sequence u is proportional to the sequence v.
We now assume that the periodic function f(x) is sectionally smooth as well as continuous. The derivative g(x) = f '(x) is sectionally continuous and we easily show that cn and dn , the Fourier coefficients of g(x), satisfy the relations
in fact, on integration by parts, we have
similar proofs holding for the other statements.
Hence, Bessel's inequality applied to the function g(x) yields
If, for the sake of brevity, we denote the right hand side of this inequality by M² and apply Schwarz's inequality, we find that for m > n
since 
is the amplitude of the periodic
function 
However, owing to the convergence of 
, the
right hand side, which is independent of x, can be made
as small as we please by choosing n and m large
enough, which proves the absolute
and uniform convergence of the series.
Incidentally, the same considerations show that
for periodic functions with continuous derivatives of the (h-
l)-th order and derivatives of the (h-1)-th order, which
are at least sectionally continuous, the sum 
remains below a fixed
bound. This gives us a definite statement about the order to
which the Fourier coefficients vanish. For such a function, the
Fourier series of the derivatives up to the order (h - 1) converge
absolutely and uniformly.
In order to prove the above theorem for sectionally smooth functions which are discontinuous, we first consider a special function y(x) of this type.
In the interval -p < x < p , we define y(x) as equal to x, outside this interval, y(x) is extended periodically. According to 9.4.1, its Fourier series is
This series cannot be uniformly convergent, because its sum is the discontinuous function y(x). However, we shall show that the convergence is uniform in every interval -l £ x £ l for which 0 < l < p.
The proof is based on a special artifice*. We observe that in the interval -l£x£l the function cos x/2 is never less than the positive quantity cos l/2 = k. If we multiply the absolute value of the difference between the m-th and n-th partial sums of the above series (m > n), i.e., the expression
by the function cos x/2, then, in accordance with the well-known trigonometric formula
we obtain the absolute value of the expression
* We are led to this artifice naturally by observing that the function 2y cos y, when extended periodically beyond the interval -p/2 £ y £ p/2 remains continuous, so that, according to the first part of the theorem, its Fourier series must converge uniformly and must represent the function. However, this series is obtained, if we multiply the Fourier series for 2y by cos y. If we now put y = x/2, this multiplication leads to the steps in the text.
If we combine the terms on the right hand side with the same numerators, we obtain
and, since cos x/2 ³ k and |sin u| £ 1, the estimate
But the expression on the right hand side does
not depend on x and, by virtue of the convergence of the
series
, it can be made as small as we please by choosing n
and m large enough. This implies the uniform convergence
of the Fourier series, as we have asserted.
Now that we have obtained the expression for a particular discontinuous function, we can (cf. 9.4.2) transfer the discontinuity to any arbitrary point in the interval by translation of the curve or of the co-ordinate system. In fact, the function
is continuous except at the points (2k+1)p + x, where k is an integer. However, on passing these points, the function jumps by an amount -2p from the value p to the value -p, while at these points themselves the value of the function is zero.
If now f(x) is any sectionally smooth function, which in the interval -p £ x £ p is discontinuous only at the points x1, x2, ··· ,xm and if on passing these points from the left to the right hand side the function jumps by the amounts d1,d2,···,dm, respectively, then the function
will be continuous and sectionally smooth, whence, by the previous proof, it can be expanded in a uniformly convergent Fourier series. We now obtain the Fourier series of the function f(x) by adding term by term the finite number of Fourier series corresponding to the functions
Hence the theorem is proved.
This result is quite adequate for most mathematical investigations and applications. However, we point out that the investigation of Fourier series has been advanced much further. The conditions for an expansion in Fourier series, which we have found here to be sufficient, are by no means necessary. Functions with far fewer continuity properties than those discussed here can be represented by Fourier series. There is an extensive literature devoted to these questions and to the general problem of the expandability of a function in a Fourier series. As a remarkable result of such investigations, we mention the fact that there are continuous functions, the Fourier series of which do not converge in any interval, no matter how small. Such a result does not in any impugnway reduce the usefulness of Fourier series; on the contrary, it must be regarded as evidence that the concept of a continuous function involves fairly complicated possibilities, as has already been shown by the example of continuous functions which nowhere have a derivative.
Appendix to Chapter IX
A9 Integration of Fourier Series
One of the remarkable properties of Fourier series is their term by term integrability. In general, a series can be integrated term by term, if it is uniformly convergent; otherwise, term by term integration may lead to false results. In contrast to this, for Fourier eeries, we have the theorem:
If f(x) is sectionally continuous in -p £ x £ p and if the Fourier series
corresponds to f(x), then this series can be integrated term by term between any two limits x and x lying in the interval -p £ x £ p , i.e., in symbols,
Moreover, for every fixed value of x, the series on the right hand side converges uniformly in x.
The remarkable feature of this theorem is that not only do we not require that the Fourier series for f(x) shall be uniformly convergent, but we need not even assume that it converges at all.
In order to prove this, let the function F(x) be defined by the equation
This function is sectionally smooth and, by the definition of a0, we have F(p ) = F(-p) = 0, so that f(x) can be extended periodically and continuously. Hence the Fourier series
of the function F(x) converges uniformly to F(x). We will now investigate the coefficients An and Bn. By integration by parts, as in 9.5.1, we find that, for n>0, An = bn /n and Bn = an /n. Hence for any values x and x in the interval -p£ x£p, we have
converging uniformly in x. If we replace F(x) by its definition, this becomes
as was to be proved.
It is easy to see that, if f(x) is periodic and sectionally continuous, the term by term integration can be performed over any interval whatever.