SPIRIT OF RAMANUJA N xix are not familiar with infinite products. A reader desiring to learn a few basic facts about the convergence of infinite products may con- sult a good text on complex analysis, such as that of N. Levinson and R. Redheffer [142, pp. 382-385], for basic properties of infinite products. In particular, all the infinite products in the present text converge absolutely and uniformly on compact subsets of \q\ 1. In particular, taking logarithms of infinite products and differentiating the resulting series termwise is permitted. At first, you may find that working with the products (a q)n and (a q)oo is somewhat tedious. In order to verify g-product identities or to manipulate g-products, it may be helpful to write out the first three or four terms of each g-product. This should provide the needed insight in order to justify a given step. After working with ^-products for awhile, you will be- gin to handle them more quickly and adroitly, and no longer need to write out any of their terms longhand. When you reach this stage, you should feel quite comfortable in manipulating ^-series. It is as- sumed throughout the entire book that \q\ 1. Over 50 exercises are interspersed within the exposition. The author is grateful for the comments of graduate students at the Universities of Illinois and Lecce where this material was taught. Y.-S. Choi, S. Cooper, D. Eichhorn, AMS copy editor M. Letourneau, J. Sohn, and K. S. Williams provided the author with many helpful suggestions and corrections for which he is especially thankful. The author also thanks N. D. Baruah, S. Bhargava, Z. Cao, H. H. Chan, W. Chu, AMS Acquisitions Editor E. Dunne, M. D. Hirschhorn, M. Somos, A. J. Yee, and the referees for their suggestions and cor- rections.
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