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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: ...+ od--6c = 0. If this equation has two distinct roots sl, s2, there exist two independent integrals 0!(x), 02(x) such that we have (102) 0z + 2w) = slx), 02(x + 2 w) = s202(x), and the relations (101) can be replaced by the two relations of the same form (103) 0 + 2 /) = By means of the relations (102) and (103), we can now obtain two different expressions for 0l(x + 2 a + 2 w') and 02(x + 2 a + 2 w'). We have, on the one hand, 0!(x + 2 w + 2 /) = Simz + 2 of) = s0! (x) + sx). On the other hand, proceeding in the inverse order, we may also write 0!(z + 2 w + 2 w') = t0i(x + 2 w) + j02(x + 2 -) = fcsx) + fei02(x). Since these two expressions must be identical, we have I = 0, for sl--s2 is not zero. Similarly, by considering the two expressions for 02(x + 2w + 2w'), we find m = 0. The integrals 0l(x), 02(x) are therefore analytic functions except for poles, which reproduce themselves multiplied by a constant factor when the variable z increases by a period; these are called d0ubly periodic functions of the second kind. Every function 0 (x) analytic except for poles which possesses this property can be expressed in terms of the transcendental functions p, f, a, since the logarithmic derivative 0'(x)/0 (x) is an elliptic function, 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 0 (x) be an analytic function except for poles such that Consider the auxiliary function (x) = efxa(x--a)/r(x), where a and p are any two constants. From the properties of the function a (see Vol. II, Part I, 72) we have where C' is not zero. In the first case the integrals 0i(x), 02(x) are again doubly periodic functions of the second kind. In the second case the...

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