This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1917 Excerpt: ...and we have seen that the integration does not introduce any new transcendental (II, Part I, 75). Moreover, we can prove this without any integration. Let $ (x) be an analytic function except for poles such that Consider the auxiliary function (r) = ePr (x--a)/r (x), where a and p are any two constants. From the ... Read More
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1917 Excerpt: ...and we have seen that the integration does not introduce any new transcendental (II, Part I, 75). Moreover, we can prove this without any integration. Let $ (x) be an analytic function except for poles such that Consider the auxiliary function (r) = ePr (x--a)/r (x), where a and p are any two constants. From the properties of the function r (see Vol. II, Part I, 72) we have doubly periodic functions of the second kind. In the second case the integral 0, (/) alone is a doubly periodic function of the second kind. As for the integral t0(x), the quotient 02(x)/0, (x) increases by a constant C' when x increases by 2u', and it does not change when z increases by 2u. Now the function Af(x) ] Bx, where A and B are two constant coefficients, possesses the same property, provided that we have 2Ail+2Bu = 0, 2Ar, ' + 2Bu' = C'. The difference t2/tl--A (x)--Bx is therefore an elliptic function. If the coefficient C is not zero in the equations (104), we have relations between the integrals 0, (x), 02(x), 0i(z-I-2u'), 02(z-I-2 u) of the form (103), and we can again deduce from them two different expressions for 0, (x + 2- + 2u') and 02(x + 2u + 2u'). By writing that they are identical, we obtain the conditions I = 0, k = n. The integral 0i(x) is again a doubly periodic function of the second kind, while the integral 02 (x) satisfies the two relations x 20.x C t1(x+2u')_(x) t m '' is again an elliptic function. We see, therefore, that the general integral is in all cases expressible in terms of the single transcendentals e, p(x), f(x), r(x). Let us consider, for example, LamPs equation Pv (105)--n(n+l)p(x) + hy = 0, where n is an integer and h is an arbitrary constant. The integration of this equation by Hermite was the starting point for the precedin... Read Less
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