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2nd. When n is odd, there is no middle factor, and the number of double factors is 3 (n-1). Also in this case we have
but for the demonstration the reader is referred to Legendre's work.
To this definite integral r(n) others may be easily reduced.
Thus if we have the integral
on differentiating en times with respect to a and then making a = 1, we obtain
2 1.3.5 ... (21 – 1) 71 (k) 6* d x 221 € -az? = =
So'dx — n-y = (
(1 – 1x") " according to the notation adopted by Euler, and, after him by Legendre; the value of the integral being supposed to change in consequence of the variation of p and q, n remaining constant. This form of the integral however is not the most convenient in practice, and we shall use another, formed from the present by putting r" = y, when it becomes
Putting = 1, ? = m, we shall designate the definite integral by a symbol of functionality applied to these letters as the symbol r is used for the second Eulerian integral : the letter we shall use is the digamma F, so that we write
(a) So dx a?-1 (1 – x)m-1 = F (1, m).
The most important properties of this integral are those by which it is connected with the second Eulerian integral founded on the theorem
r () T (m)
(3) The property (b) of the first Eulerian integral may be extended to a large class of multiple integrals by the fol. lowing theorem due to M. Lejeune Dirichlet*.
(a) Let V = sd x sdy sdz..."-yo-2Z0-1... in which the limiting values of x, y, z... are given by the condition
(0) + ()+ (%)*+ &c. 21, a, b, c...a, b, y... p, q, r...being positive quantities ;
Hence if we know the integral
U = sdx sdy sdz...x'-1 ym=1&v=.., with the previous condition for determining the limits, we can find V.
When the variables are two in number, it is easy to see that the integral is identical with that called the first Eulerian, Let us suppose therefore that there are three variables. Then
U = lo dx a'-7 foi dy ym- f*ids z"-?, where y = 1 – X, 1 = 1 – X – y.
• Liouville's Journal, Vol. iv. p. 168.