Page images
PDF
EPUB

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 + 1)

0

[blocks in formation]
[blocks in formation]

Let be replaced by x-z; and this being done, let every term be multiplied by ƒ(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

1

} ['fƒ (z) dz + ['ƒ (z) cos(x−2) dz + [ƒ (z) cos 2 (x −z) dz + ...

a

| |

[blocks in formation]

It is evident that every term in the left-hand member is unaltered when a 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+2kπ and а+2kл 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 z, and we shall have three cases; (1), when x falls within the range of integration; (2), when a coincides with either limit of the integration; (3), when a falls beyond the range.

[ocr errors]

(1). Let x fall within the range of integration; that is, let a 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

of z; for since (n+1)(x−≈) varies by 27, when z varies by

[blocks in formation]

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

[ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

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

હું

so that

ƒ (x)+Eƒ'(x), and sin§ by §;

[ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small]

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

[ocr errors]

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

[merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors]

2= [f(z)dz + = ['f(z) con (x — z) dz

1

1

=

[ocr errors]
[ocr errors]
[ocr errors]

П

[blocks in formation]

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

As the general term of this series is

and this is equal to

[merged small][merged small][ocr errors][merged small][ocr errors][merged small]
[ocr errors]

f(z)

f(z) cos n (x-z) dz,

[ocr errors]
[blocks in formation]

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+27 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 § 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

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

π

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

π

1

[ƒ (b) = } ["ƒ (z)dz + Σm=ï [["ƒ (z) cos n (b−z) dz ;

π

1

a

1

= ƒ (a) = } ["ƒ (=)dz + Σh=i['ƒ (2) cos n (a—z) dz.

[ocr errors]

(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

[merged small][merged small][ocr errors][subsumed][merged small][merged small][merged small]

Thus in recapitulation, b—a being not greater than 27,

[merged small][merged small][ocr errors][merged small][merged small][merged small][merged small]

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

[blocks in formation]

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 a; 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 repre

sented by the curve y =ƒ (»); when æ = a, y =ƒ(a); and when

f(b)

2

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

x=

2

are evidently discontinuous points. Also for values of a greater

than b, and for values less than a, y = 0, so that the locus becomes the x-axis. This last part of the locus however is subject to a

[merged small][ocr errors][merged small]

x+47, =

[blocks in formation]

but also when z = x+2π, = =x+2kπ; therefore the right-hand member of (85) = f(x), not only when x has values between b and a, but also when a has values between b+2π and a+2, between b+4 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 when x=a+2π,

=2π. Also as y =

f(a), when x = a, so y=

2

f(b)
2

f(a)

2

=α+4π, = ; 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 a increases, y = f(x); and this locus continues until a=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 + 47, 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-a2, 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

y =

2

2 {f(a)+f(x+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 PRICE, VOL. II.

M m

« PreviousContinue »