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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. 1916 Excerpt: ... 2/J/an, and the th root of the general term of this last series, or R/an, approaches zero as n increases indefinitely.t Therefore we can always choose the integer v so that the infinite product Ft(z) will be absolutely and uniformly convergent in the circle of radius R. Such a product can be replaced by the sum of a uniformly convergent series ( 176, 2d ed.) whose terms are all analytic. Hence the product Ft(z) is itself an analytic function within this circle ( 39). Multiplying Ft(z) by the product Fz), which contains only a finite number of analytic factors, we see that the infinite product is itself absolutely and uniformly convergent in the interior of the circle C with the radius R, and represents an analytic function within this circle. Since the radius R can be chosen arbitrarily, and since For example, let a = logn(/i 2). The series whose general term is (logn)-p Is divergent, whatever may he the positive numherp, for the sum of the first (i-1) terms is greater than (n-l)/(logn)p, an expression which becomes infinite with n. t Borel has pointed out that it is sufficient to take for v a number such that r +1 shall be greater than logn. In fact, the series 2 /'/?nl'--is convergent, for the general term can be written e'K 'ok I I = n'o I s/an I. After a sufficiently large value of n, an/R will be greater than e2, and the general term less than 1/n2. v does not depend on R, this product is an integral function G, () which has as its roots precisely all the various numbers of the sequence (1) and no others. If the integral function G () has also the point z = 0 as a root of the pth. order, the quotient g( ) z Gz) is an analytic function which has neither poles nor zeros in the whole plane. Hence t...

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