Special Functions

This is an outstanding book on special functions though it doesn't seem well known in the West. At first sight it appears to follow the path of Whittaker and Watson, but on closer examination it actually treats the subject(s) quite differently, sometimes better, dare I say. Among its strengths are (a) all derivations are carried out in detail; (b) the author takes great care to motivate various techniques so that they seem perfectly natural; (c) the contour-integral method is used extensively to solve the differential equations associated with the special functions; and (d) the infinite-series approach to solving the differential equations, which Whittaker and Watson develops theoretically but does not apply, is carried out more thoroughly here than anywhere else.Point (b) should greatly appeal to the physics type, and it came somewhat as a surprise to me, since I had the impression that most Chinese professors had a very condensed writing style, in which motivation isn't the top priority. On the other hand, the contour-integral-solution approach to ODEs is basically absent (at least not systematically employed) in Whittaker and Watson. When you look at the integral representations of the special functions in the book, there is less of the feeling that they just dropped out of the sky. Point (d) should appeal tremendously to most of the readers, since a typical physics/mathematics student learns the series technique in his/her second course on ODE. The coverage here is outstanding because the author does not summarily dispatch, as most others do, treatment of the irregular solutions, ie, the "bad-boy" solutions which arise when the difference of the roots of the indicial equation equals zero or an integer. Whittaker and Watson, for example, relegates the subject to a footnote in their treatment of the hypergeometric function.The original author (Wang) wrote the book in Chinese, which was translated into English by two of his students. You can easily tell even without seeing the author list that two translators were involved. One has a better command of English and his prose is more fluid.Of course, for a subject as classical as special functions, there is bound to be a great deal of overlap between any two books in terms of the topics covered. Without a doubt Whittaker and Watson is still King in this area, but at least for me this book is Queen. Highly recommended. The Expansion Of Functions In Infinite Series And Infinite Products -- Linear Ordinary Differential Equations Of The Second Order -- The Gamma Function -- Hypergeo,etric Function -- Legendre Functions -- Confluent Hypergeometric Functions -- Bessel Functions -- Weierstrass Elliptic Functions -- Jacobian Elliptic Functions -- Lame Functions -- Mathieu Functions. Z.x. Wang And D.r. Guo ; Translated By D.r. Guo, X.j. Xia. Translated From Chinese. Includes Bibliographical References. In this book, expansions of functions in infinite series and infinite product and the asymptotic expansion of functions are discussed. This may be the best reference book on Special Functions.