Page:Encyclopædia Britannica, Ninth Edition, v. 13.djvu/57

INFINITESIMAL CALCULUS 47 The corresponding result when m is an integer is obtained by interchanging the letters m and n. We now proceed to show that B(m, n) can in all cases be expressed gamma-functions. For if we substitute zx for x in the equation in gamma-functions o we get TO) =fc-**? x l - l dx . Jo Hence 1 n)c - l ~ m ~ ^/""V^ 1 +%>+ -V" 1 o o o Let z(l +.*) = #, and we get Hence Next let x n ~ l dx /-* ;V n-l ^1 / , m+, t dx = / u n /o U+^J T yo This fundamental relation is due to Euler. Again, if i = l -?i, we get from the preceding - 1 + a; , (144). If = , this becomes F() = Vir. This" result agrees with 148, for, if we write 2 2 for x, 154. Many definite integrals are reducible to gamma-functions, of which a few elementary cases are here given. (1) To express the definite integral in gamma-functions. Let ,/j=sin J 0, and the integral transforms into JO Y ( (2) To find the value of /-I / x m (l-x) r dx. Jo Let A" = ~, and the transformed integral is tl I r l> n jo (3) If in the last -/ = - i, we get Ml irj (4) = / (a -)"( -j8)rfa;. yp Let a; = /8 + (a- j3)s, and we readily find (5) To prove the equations /-x / e -<%-] /o yo where taft- 1 ( ) . In the equation cos n6 Ix dx= let re - /& be substituted for a, and we get /-co / c - "*(cos Ix + i sin l)x}x n - lx = yo If b = a tan 9, we have F(n) (a -ib) n (a? Hence the proposed equations are obtained by equating the real and imaginary parts respectively. /"* - r(rt) yo (6) cos Ix x" - hlx /"S yo These follow from the preceding by making n = 0. A more rigorous demonstration of this and of the preceding example will be found in Serret s Calcul integral, pp. 194-198. (7) Find the value of Here, since it is easily seen that But it can be shown, by trigonometry, that TT . 2ir . (n, -1W 7i . sin sin ... sin - -r , (8) Prove that cos m0 dO f ,, 155. We next proceed to show that Y(n) admits of being exliiliited as the limit of the product of an infinite number of factors, a form which was adopted by Gauss as the definition of the function. If in the equation - we make e x --=z, we get But ( 63, Ex. 9) log is the limit of yu(l -j^ when /x increases beyond limit. o -y)"-^!/, making 2 = 7^. 1 . 2. 3. . . , ( 123), .( + !)...(+/*) when n is increased indefinitely. AH an application of this definition of T(n) suppose n + l and n - I respectively substituted for n, and we readily obtain 1--^ n* 71 . ?7T -T sin <7T 71 _ (7i+l) a by a well-known trigonometrical relation. If we make n l, this gives HI - Z)r(l + ?) = -. 7 V ; . . T(/)r(l - ?) =, . , - , as before. Sin ITT Sin lir 156. Again, if we make x = az, we get f 1 m -l _ n-l , am +n-l f m -l (l --)- Irf- yo yo + 71)