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a particular case of this equation, viz. when n is positive and integral, has been proved in Ex. 4, Art. 98.
123.] In the first place I propose to shew that r(n), as defined by (247), is finite and determinate for all positive values of n. For this purpose let the definite integral be divided into the three following integrals, the sum of which is equal to it; ["e-*zu-1dx = (e-gn-1dæ+"e-***-1 dx
+ 16-22n-1dx, (251) where i is an infinitesimal, and x, is a finite quantity, to which a convenient value will be given. As to the first of these three integrals, by (228) we have
rrni eli in e-xxn-1 dx = e-oi / 20"-1dx = e-i
Indon which is an infinitesimal. As to the second integral, the limits of integration are finite, and the element-function does not become infinite within the range; consequently the definite integral is finite and determinate. As to the third integral, let x, be that value of x for which, and for all quantities greater than which, e-- is less than x-(n+1); so that
which is a finite quantity; and consequently the whole definite integral denoted by r(n) is finite and determinate.
As to Xy; we must shew that our hypothesis of its value is possible; we have assumed e-*, to be less than X, -(n+1); and therefore, taking logarithms, i 1 is greater than n+1; and
parithms, log in
this is possible because = oo, when x = 1; = e, when x = e, and this is the minimum value; = oo, when x = 0o.
r(n) is also a positive quantity because all the values of the element-function within the range of integration are positive.
It is evidently a continuous function of n; because as n continuously varies, the values of the definite integral will also continuously vary. The continuity of the function may also be demonstrated by means of the n-differential of it.
Thus, differentiating (247) with respect to n,
d.f(n) = + log x 2"-1 dx ; (252)
dn which is evidently determinate for all values of x within the range of integration; and consequently r(n) varies continuously as n varies continuously.
124.] The following are some values of r(n) corresponding to particular values of the argument. (1) Let n be negative; then, if xy is a positive finite quantity,
fre* dx po e-* dx o r(-n) = | e-fx-n-1 dx =
bo n+1 tl. mit T. (253) Now, giving approximate values to these two latter integrals by means of the theorem contained in (228), if 0 is a positive proper fraction, and X, is a value of x intermediate to x, and co, we have,
1 78 e-X
(254) (2) Let n=0; then, employing the same symbols and theorems,
(255) Thus it appears that the definition of r(n) given in (247) is applicable only when n is a positive quantity. It will appear hereafter that another definition may be given of the function which will place it on a wider basis, and will not exclude all except positive values of n. (3) Let n = 1; then f(1) = |-* dx = 1.
(256) (4) Let n = 2; then
r (2) = | e-*xdx
(257) Taking these values in connection with (252), we can determine
the general course of the value of the Gamma-function; or in other words we can trace the curve y = f(x).
Expressing the equation (252) in the following manner,
it appears that “..") is equal to the difference of two definite integrals, which are necessarily positive; the former of which increases, and the latter decreases, as n increases ; therefore "I") is positive, and r(n) has a minimum value corresponding to
d.r(n)_o that value of n for which != 0. It is clear then that r(n)
dn has one minimum value ; and since r(0) = 00, r(1)= 1, r(2)=1, that minimum must correspond to a value of n greater than 1 and less than 2 ; and the minimum value of r(n) is less than 1. Also, beyond that value, r (n) increases as n increases; and r(n) = 00, when n = 0.
125.] In (250) let n be replaced by m+n, and let a be replaced by 1+2, where z is a new variable independent of x; then
Now, as x and z are independent variables, the order of the in. tegrations may be changed; and consequently we have from the left-hand member of (260),
dz e-(1+4)*7"-1pm+-1 dx
= f(m) r(n); substituting which in (260), and replacing z by x, we have,
pos ma-1 dx r(m) r(n) Jo (1 + x)*** = (m+n)
(261) In this process no restriction has been put on the values of m and n, except that they are positive quantities.
The integral in the left-hand member of this equation has been called by Legendre the first Eulerian Integral and is of considerable importance in its relation to the Gamma-function. It is evidently a function of two parameters m and n, and has been denoted by the symbol B(m, n), being called the Beta-function. So that for the definition of this function we have
poo 29-1dx s(m)r(n) B(m, n) =
(262) Jo (1 + x)"+" r(m + n)
If in the definite integral z is replaced by 3., then
po xn-ı dx po am-1 dx
o ano1 dx o xm- dx :: B(m,n) = B(n,m) =
Td (1 + x)m+
n do (1 + x)*+ * ' so that the value of the Beta-function is unaltered by the interchange of m and n. This theorem might also have been inferred from the symmetry with respect to m and n of the last member of (262).
As the Beta-function is a function of two variables m and n, it evidently represents a surface; and if x, y, z are the coordinates to any point on it,
r(w)r(y). 2 = B(x, y) =
r(x + y) and the general course of the surface may be traced from the previously known values of the Gamma-function.
126]. The following are other and equivalent forms of the Beta-function, being derived from (262) by transformation.
= [ (€* – 1)m-1e+(6+n–1)= dx. (266) (3) In (263) let x be replaced by 1 mg; then B(m, n) = | 22–(1 — x)m-1 dx = | Xm-(1-x)"-1 dx ; (267) which is a definite integral of great importance in the theory of Probabilities. (4) In (267) let x be replaced by ; then
i pamem-(a — )n-1dx B(m, n) = |
(268) lo qm+-1 20m-(a — X)*–1dx = @m+n-1 B(m,n)
B(m,n) = 2 / (sin 0)21–1(cos 6)2m-1 de. (271) All these values of B(m,n) are of course equivalents of
sor(m + n) since this is the relation which exists between the Beta- and the Gamma-functions; and since by it the Beta-function may be expressed in terms of the Gamma-function, it is unnecessary to consider separately the properties of both, so that we shall henceforth investigate the properties of only the Gamma-function.
127.] The first fundamental theorem of the Gamma-function. By (261) and (263),
r(m)r(n) po wm-1 dx
r(m + n) To (1 + x)"+n
- (n) o dx
(272) ... r(n+1) = nr(n).
(273) This is the first fundamental theorem of the Gamma-function.