In the left-hand member, let a = k cos a, x = k sin a, so that

where a is a positive quantity, m+n−1 is a positive quantity, and

m is a positive proper fraction.

If in these last formulæ a is replaced by a tan 0, then

142.] Another important use of the Gamma-function, and of its allied integrals, is the evaluation of a definite integral in terms of a series, and conversely the summation of a series in terms of a definite integral. This method, however, is only an application of the general process of development explained in the preceding section; and especially of that particular form given in Art. 121; but it is useful, because the same definite integral may be made to enter into every term of the development by reason of the properties of the Gamma-function, and consequently will be a common factor. Moreover we shall hereby arrive at certain series which have been called hypergeometrical, the series being of the type of geometrical series, but the coefficients of the several terms. being more complex.

The following is the fundamental theorem on which the application depends.

Let F(x) be a function of x, expanded into a convergent series of the form,

F(x)=1ƒ(x,0)+â1ƒ(x,1)+^ƒ(x,2) + ... + ^x ƒ (x,k) + ..... ; (371) and let both sides of this equation be multiplied by p(x) dx, and let the x-integral be taken between the limits x and xo, it being supposed that none of the element-functions is infinite for any value of its subject-variable between these limits; and let us moreover suppose that

then

[*4(x) f(x, k) dr = n. [′′ 4(x)f(x, 0) dx ;

Bk

(372)

ΤΟ

x

Thus, if [* p(x) ƒ (x, 0)da is known, the definite integral in the

[**p(x) F (x) da is known, the sum of the series

left-hand member of (373) will be given in terms of a series; and conversely the series will be given in terms of a definite integral. If the value of will be known also. The following are examples of the process. 143.] Let us in the first place employ the Gamma-function for the purpose of forming the definite integrals; and let us suppose F(x) to be a convergent series in ascending powers of x; so that F(x) = α2+α1 x + а1⁄2 x2 + α3 X3 + ... ;

(374)

then

..

= ao

0

= a。г(n) + a1r (n+1)+ a2r (n + 2) + ...

=

г(n) {α + a1n + a ̧ n (n + 1) + a3 n (n + 1) (n + 2)+...}. (376)

a+a1n + a2n (n + 1) + a, n (n + 1) (n + 2) + ...

00 1

=

r(n)

e ̄a x” ̄1 {a+a ̧ x + α12 x2 + ..... } dx. (377)

The following is an example of the process;

Again, let us suppose F(x) to be a convergent series in ascend

144.] In the next place let us employ the Beta-function, and its equivalent in terms of the Gamma-function, for the purpose of forming the definite integrals; and let us suppose F(x) to be a series in ascending powers of x; so that

F(x) = α2+α1x + A2 x2 + Az x3 + ... ;

(384)

the series being convergent for all values of a between 1 and 0; then

The value of the following integral may also be determined by means of the properties of the Beta-function.

This however is only a particular case of a much more general theorem which is of use in the higher applied mathematics.

145.] In (386) let n be replaced by nm, then

r m r(r+1) m(m+1)

1+

-a+

· a2 +

1 n

1.2

n(n + 1)

r(r+1)(r+2) m (m+1)(m+2)
1.2.3 n(n+1)(n+2)

r(n)

=

г(m) г(n—m).

xm−1 (1 − x)n—m−1 (1—ax) ̄ ̄ dx. (391)

The series in the left-hand member of this equation is that in pursuit of the properties of which Gauss has been led into his inquiries respecting the Gamma-function. His original memoir was read to the Academy of Sciences at Gottingen on Jan. 30, 1812, and is contained in the first volume of the Commentationes Recentiores of that Society. The series is evidently a function of four variables, r, m, n, a, and is the general type to which very many known series conform. vestigated its properties. (r, m, n, a), so that

This is the reason why he has inLet it be denoted by the symbol

This series is evidently convergent for all possible values of a, less than 1; and for all impossible values of a, of which the modulus is less than 1. Hence, under these circumstances, the right-hand member of (391) may be used as the equivalent value of it. The function (r, m, n, a), is called the Gaussian Function, and the equivalent series is called the Gaussian series.

PRICE, VOL. II.

Bb