so that the properties of all these functions, as well as of many others which are developable into series of the form (392), are the same as those of the right-hand member of (391).

In all these cases, when a = 1, we have from (391) and (392),

r (m) r (n—m) √ "'2"−1 (1 − x) "—m-r-1 da

=

г(n) г(n—m — r)

r(n-m) r(n-r)

(399)

and we have shewn in

146.] In these latter applications of the Gamma-function, the argument will frequently be negative; Art. 124, that the function, as defined by e-xx-1 dx, is always

infinite when n is negative. But these latter applications are legitimate deductions from the fundamental theorems of the Gamma-function; so that it is clear that they express properties of a function of wider extent than that of the definite integral. Consequently a wider definition must be given to the function.

That due to Gauss and given in (318) is sufficient for the purpose; as it holds good for negative as well as for positive values of the argument. The following theorems are easily deduced from it.

From (401) it is evident that, if n is an integer, r (—n) = ∞, since sin n = 0. So that the Gamma-function is infinite for all negative integral values of the argument; and will be finite when n is not an integer.

1

If n = from (401) we have

2

x dx

is closely allied to the

147.] The definite integral ga Gamma-function, being indeed the form taken by that function, as expressed in (248), when n = 0; except that in this case the superior limit is the general value x. This integral moreover deserves notice, because it is one of the few which have been tabulated, the necessary calculations having been made by Soldner of Munich as long ago as the year 1809. The integral is also instrumental in the determination of many other definite integrals, and occurs in the solution of certain physical problems.

Soldner devised the symbol li.x, to denote it: li being the initial letters of logarithm-integral, by which name he called it. Thus we have for the definition of the logarithm-integral

Let x in the right-hand member be replaced by e-*; then

which may also be taken as the definition of the logarithm-integral.

If is greater than 1, the range of integration in (406) includes a value, viz. x = 0, for which the element-function = ∞; we must therefore divide the integral into two parts and take the principal value of each, making the range of each approach infinitesimally near to 1; so that if i is an infinitesimal, we may express (406) in the following form,

and the value of this is to be determined, when i = 0. Now by the definition of E, Euler's Constant, given in (298),

Also, taking the second integral in the right-hand member of (407), expanding e-*, and integrating, we have

omitting terms involving i and its powers, which must be neglected. Consequently, substituting in (407), we have

If x is less than 1, although in (406) the element-function never becomes infinite within the range of integration, yet the preceding process is not applicable, because the result contains the term log log; and this is impossible when x, is less than 1, for loga, is in that case negative, and log loga, is the logarithm

of a negative quantity. In this case let us suppose (x,)-1 to be the subject-variable of li; then from (406) we have

and if we expand the element-function, and integrate as in the

n

in which every term is real, since x is greater than 1. Thus by (410) or (411) the logarithm-integral may be calculated for all values of the subject-variable; and they have been employed by Soldner for that purpose; but it is beyond the scope of this work to enter into the details of the calculation.

148.] As another instance of the logarithm-integral, let the subject-variable be (1+x); so that by (405),

where A, A1, A2,

are coefficients of the successive powers of X,

which may be calculated by the process of Derivation, as explained in Art. 95 of Vol. I.

The values thereby found are

Now as x = 0 occurs within the range of integration, and is that value for which the element-function becomes infinite, the definite integral must be divided into two parts, and we have

the sum of all the quantities in the definite integral, which are

independent of the powers of x, in the series, being equal to E when x is an infinitesimal, because in that case

li. (1+x)—log x1 =E.

The following are indefinite integrals which are expressed in terms of the logarithm-integral.

= li.. (2) log x

= li.ex.

= li.xm+1.

(4)

fe* dx = li.e*.

dx

= e ̄*li.ea+*. (6) √ a log log

(7) fli.x dx = x li.x—li.x2.