velopment; and acting on this presumption, investigate the series by an exact and precise process.

Now whatever is the angle denoted by 0, we have the following equivalence,

sin (n +

(79)

Let be replaced by x-z; and this being done, let every term be multiplied by f(z)dz; and let the definite integral of each term thus found be taken for the limits b and a, f(z) being finite and continuous throughout this range of integration; then

It is evident that every term in the left-hand member is unaltered when x is increased by 2. Consequently both members denote periodic functions of x, whose period is 2π, whatever is the value of n; and if we trace the value of either member of (80) through this range, that value will be repeated when b and a are replaced by b+2kя and а+2kπ respectively; we may therefore without loss of generality confine our attention within limits of integration, such that b-a is not greater than 2π.

When n∞, the number of terms in the first member of (80) is infinite, and the sum of them is denoted by the value which the second member takes when n = ∞∞. As this is the case of the series which we have to investigate, this latter value must be determined.

Its value evidently depends on the relative values of x and the several values of 2, and we shall have three cases; (1), when a falls within the range of integration; (2), when a coincides with either limit of the integration; (3), when a falls beyond the range.

Ꮖ

(1). Let x fall within the range of integration; that is, let x be greater than a and less than b; then, when z = x, the righthand member of (80) takes an indeterminate form, and must be evaluated. Now the element-function vanishes for all other values (x-z) varies by 27, when z varies by x-z) varies

of z; for since (n+1)

0, when n∞; it is evident that over that

2n+1'

range of z, which is infinitesimal,

f(z)

may be considered

sin

2

constant, and (sin (n + 1)(~—z)dz = 0.

Hence the right-hand member of (80) = 0, except when z = x. To determine the value when z = x, we may consider it only for values of z infinitesimally near to x; that is, if i is an infinitesimal, we may take the limits of the z-integration to be x-i and x+i; and if we replace z-x by §, § is the variable of integration, and is always infinitesimal, its limits of integration being -i and i; so that

and as έ is always infinitesimal f(x+8) may be replaced by

હું

so that

ƒ (x)+Eƒ'(x), and sin§ by §;

Of the integrals in the right-hand member of this equation, the

latter vanishes, because for all values of έ, sin (n + 1) = 0, when

2

n = ∞. And to determine the value of the former, we may enlarge the range to ∞ and +∞, because all the elements of that definite integral vanish when n∞, except those corresponding to small values of §. Hence

1

"b

=== [f(z)dz += f(z) cosn (x−2) dz

1

2π

π

a

which gives the periodic series into which f(x) may be developed, when a falls within the limits of the z-integration.

and this is equal to

As the general term of this series is

x

П

a

cos nx

π

[f(z) cos nz dz + sinn [f(z) sin nz dz,

it is of the form A, cos nx+B, sin na; and consequently by means of (85) f(x) is developed into a series of the form given in (70), Art. 196. This series is however more general, because the limits of integration in it are b and a, whereas those in (70) Art. 196 are a+2π and a.

=

b, say,

(2). Let x = one of the limits of the z-integration; the superior limit; then the limits of integration in the righthand member of (81) must be b and b-i, otherwise values of z would be included which are not within the range of integration; and consequently the limits of έ in the -integration are 0 and -i; and the value of the definite integral in the right-hand member of (82) is, the limits being 0 and — ∞ ; and the series is consequently equal to f(b). Similarly if x = a, the inferior limit, the limits in the right-hand member of (81) are a+i and a; so that the limits of έ in the -integration are i and 0; and the value of the definite integral in the right-hand member of

π

Π

π

(82) is, the limits being ∞ and 0; and the series is consequently

π

equal to ƒ (a). Hence if x is equal to the limit b or a, we have respectively

π

Tƒ (b) = 1} [°ƒ (z)dz + Σm=ï ['f (z) cos n(b−z) dz ;

a

1 b

1

}ƒ (a) = } ["ƒ (z)dz + Σm=ï [ƒ(z) cos n (a—z) dz.

f(

(86)

(87)

(3). Let x fall beyond the range of integration; that is, let x be less than a, or be greater than b; then, as we have shewn in case (1), the element-function vanishes for all values of z, and the right-hand member of (80) always 0. Thus

1

=

} ['f(z)dz + Σ== ['ƒ (2) cos n (x−2) dz = 0.

1

a

Thus in recapitulation, b—a being not greater than 2,

(88)

(89)

a

represents a function of x which is periodic, and of which the period is 2π; and which

and these are the values of (89) through a complete period, and are repeated in both directions. This theorem contains the doctrine of periodic series in its most general form.

198.] The preceding results may be geometrically interpreted, and the interpretation is interesting and curious, inasmuch as it exhibits the discontinuity of the function in a striking manner. Take the right-hand member of (85) to be the ordinate of a plane curve, whose abscissa is ; so that if the left-hand member is denoted by y, y = f(x) is the equation to the curve. Then for all values of a between b and a exclusively, the locus is repref(a) sented by the curve y = f(x); when x = a, y = ; and when 2

x

x = b, y =); so that the locus then becomes two points which

x=

(b)
2

x

are evidently discontinuous points. Also for values of a greater

than b, and for values less than a, y = O, so that the locus becomes the x-axis. This last part of the locus however is subject to a certain restriction; for since sin 272

O, not only when z = x,

but also when z = x+2π, = x+4π, = ... =x+2k; therefore the right-hand member of (85) = f(x), not only when a has values between b and a, but also when a has values between b+2π and a+2, between b+47 and a+4, ...; accordingly the curve, which is the locus between x = b and x = a, is repeated at intervals, such that the distance between two similar points ƒ (a), when x = a, so y=' when x=a+2π,

2. Also as y =

= a+4π, = .;

=

2

f(b)
2

f(a)

2

and y = when xb, b+2π, = b+4π,

=

; and consequently we have a repetition of the points of discontinuity. Also the portions of the x-axis which the series expresses are of a finite length, and these finite portions are repeated for as y = 0 for values of a greater than b, so this zero-value continues until x=a+2π, when y abruptly takes the

2

value (a); and then as aæ increases, y = f(x); and this locus continues until x=b+2, when y abruptly takes the value (b);

2

and again as a increases, y = 0; and this zero-value continues until x = a+4, when another point of discontinuity occurs. And so on to infinity, in both the positive and negative directions. Thus 2-(b-a) is the length of each portion of the x-axis which the series represents, and these are repeated at intervals, each of which b-a.

If b-a = 2π, so that the limits of the z-integration are a and a+2, then these portions of the x-axis disappear, and the locus consists of discontinuous branches of the curve y = f(x); and at those values of x at which the discontinuity takes place,

1 2

y =

} {ƒ (a)+f (z+a)}.

(91)

199.] This last remark leads to a very important result. It shews that f(x) need not be a continuous function, and that it may have points of discontinuity between the limits of the z-integration. When however this is the case, the integrals must be taken separately over the ranges through which the continuity of