Download Fundamentals of Mathematics, Volume II: Geometry by H. Behnke, F. Bachmann, K. Fladt, W. Süss, H. Kunle, S. H. PDF

By H. Behnke, F. Bachmann, K. Fladt, W. Süss, H. Kunle, S. H. Gould

Quantity II of a special survey of the full box of natural mathematic

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2, p. 169, Entry 7]. A proof via the Poisson summation formula is sketched in [21]. ENTRY 21. 1) Log(-a;q) ro - V_ f ^ — f ? 2) Log f(a,b) = Log(ab;ab) PROOF. Log(-a;q) + I -^ [a. + k=l k(l - a V ) For |q|, |a| < 1, = I Logd + aqn ) = I I {~]) n=0 n=0 k=l r (-1)k"1ak ^ K k=l n=0 f k,n ,{^ ] K ^ (-l)k-]ak k=l k(l - q K ) b } . 32 C. ADIGA, B. C. BERNDT, S. BHARGAVA, AND G. N. 1). ENTRY 22. (i) If | q | < 1, *(q) = f(q,q) = 1 + 2 then ~ (WlJqV^ k2 X q = * j—^— , k=l (q;q^)J-q W L (q2;q2) . -. 3.

186, 187], [23]. Apply the Euler-Maclaurin summation formula [77, p. 128] to the function n(v\ = (m-n) y v x ; - nq = 2 /8(m+n)+mx(x+l)/2+nx(x-l)/2 (2(m+n)x+(m-n)}2/8(m+n) on the interval 1 , x. " 2 9(-'h- -°° < x < °°. The series v B2k (2k-l),( Nu)h2k -°° • k l 1 T2kTT9 where B. denotes the jth Bernoulli number, terminates and, in fact, is J identically equal to zero. Thus, according to Ramanujan, G(q) = op i g(k) k=-°° CHAPTER 16 OF RAMANUJAN'S SECOND NOTEBOOK 39 is perfect and complete, respectively.

WATSON f h1 + /• OO I ^ k=l k even 1 + I a k=l = 2! 1 + + CO I > ak(k+l)/2 k=l k odd k(2k+l) + r ak(2k-l. k=l K K u/ J K K+l y aa k(k-l)/2 ^ " ^ ((aa 3jk(k+l)/2 ) ^ ^^ k=l + y ak(k+l)/2(a3jk(k-l)/2" k=l 2 f(a,a 3x Thirdly, f(-l,a) = I (-i)k(k + l)/2 a k(k-l)/2 + J k=2 k=l upon the replacement of Fourthly, replacing k by M ) k(k-l)/2 a k(k + l)/2 = 0f. k + 1 in the first sum on the right side. (k+n)(k+n-l)/2 ^(n+1)/2 b n(n-l)/2 = a n(n + l)/2 b n(n-l)/2 J a k(k+2n+l)/2 b k(k+2n-l)/2 k=-°° £ {a(ab)n}k(k+1)/2{b(ab)-n}k(k-1>/2 , k=-°° which completes the proof of (iv).