By I. R. Shafarevich

This quantity of the Encyclopaedia involves elements. the 1st is dedicated to the speculation of curves, that are handled from either the analytic and algebraic issues of view. beginning with the elemental notions of the speculation of Riemann surfaces the reader is lead into an exposition protecting the Riemann-Roch theorem, Riemann's basic lifestyles theorem, uniformization and automorphic features. The algebraic fabric additionally treats algebraic curves over an arbitrary box and the relationship among algebraic curves and Abelian kinds. the second one half is an advent to higher-dimensional algebraic geometry. the writer bargains with algebraic forms, the corresponding morphisms, the speculation of coherent sheaves and, eventually, the idea of schemes. This booklet is a really readable advent to algebraic geometry and may be immensely invaluable to mathematicians operating in algebraic geometry and complicated research and particularly to graduate scholars in those fields.

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E. C. 1 (p. 207). Provided all arguments below are of modulus less than 1, we have f (−λq, −q 2 ) f (−λq 3 ) = f (−λ2 q 3 , −λq 6 ) + qf (−λ, −λ2 q 9 ). f (−q, −λq 2 ) An excellent survey describing all known proofs of the quintuple product identity has been prepared by S. Cooper [140]. A ﬁnite form of the quintuple product identity was established by P. C. Chen, W. S. Gu [122], and by Chu [128]. A proof of the quintuple product identity by S. Bhargava, C. S. Mahadeva Naika [82] was written after the appearance of Cooper’s survey.

2 of Part I [31, p. 229]. 3 (p. 28). For any complex number a, ∞ ∞ (−1)n an q n(n+1)/2 = n=0 (−1)n (q; q 2 )n a2n q n(n+1) . (−aq; q)2n+1 n=0 Proof. 1), let h = 2, replace a by a2 q 2 /t, then set b = q and c = −aq 2 , and let t → 0. 1) with h = 1 and replaced a, b, c, and t, respectively, by 0, 0, aq, and q. 4 (p. 38). For |aq| < 1, ∞ ∞ (−aq)n (−1)n an q n(n+1)/2 = . 2) Proof. 8), set a = 0 and b = q 2 , and then replace t by −a. 9), wherein we replaced a by −q and b by q, then set t = −a, and let c → 0.

Corteel and J. J. Yee [285], W. W. Guo and Schlosser [169]. The proof given in [3] and reproduced in [54, Entry 17, p. 32] was, in fact, ﬁrst given in lectures at the University of Mysore by K. Venkatachaliengar in the 1960s and appeared later in his monograph [272, p. 30]. The proof of Chan [116] employs partial fractions, which appear to have been central in much of Ramanujan’s work. 4) was given by Ismail [184]. 2). 4), because the identity holds on a convergent sequence within the domain of analyticity |b| < 1.