Titlebook: Ramanujan’s Notebooks; Part IV Bruce C. Berndt Book 1994 Springer Science+Business Media, LLC, part of Springer Nature 1994 Ramanujan.calcu

东西

客观

-Series,rmulas given in Entries 6–8. Thus, in this chapter, only eight results are proved. Additional .-analysis can be found in Chapter 32, entitled Continued Fractions (Berndt [9]), or in the memoir of Andrews, Berndt, Jacobsen, and Lamphere [1]. Further related material can be found in Chapter 25 on theta-functions.

Anticlimax

向外供接触

Number Theory,lts on number theory found in the notebooks lie in the 100 unorganized pages at the end of the second notebook and in the short third notebook consisting of 33 pages. Most of the material on these 133 pages was probably recorded in approximately the years 1912-1914, before Ramanujan departed for Cam

Induction

craving

Theta-Functions and Modular Equations,s. However, the unorganized pages in the second and third notebooks also embrace a large amount of Ramanujan’s findings on these topics. In this chapter, we shall discuss most of this material. In Chapter 33 (Part V [9]), we relate Ramanujan’s fascinating theories of elliptic functions and modular e

强所

-Series,ch is almost completely devoted to .-series, it is surprising that the unorganized portion of the second notebook and the third notebook contain only a small amount of material on .-series. In fact, only five identities are to be found here (Entries 1–5 below). More interesting are the asymptotic fo

frivolous

Integrals,ghted by several elegant asymptotic expansions of integrals. In this chapter, we examine the nearly 50 results on integrals appearing in the 100 pages at the end of the second notebook, and in the 33-page third notebook.

菊花

Special Functions,e first ten entries concern the gamma function. All ten entries are either known results or can easily be derived from standard theorems on the gamma function. The next four theorems arise from the theory of Bessel functions. These four theorems also are either classical or can be simply deduced fro

Reverie

Partial Fraction Expansions,s lie in Chapters 14 and 18 and in the unorganized pages at the end of the second notebook. See our books [4] and [6] for accounts of the material in Chapters 14 and 18, respectively. In this chapter, we prove the 15 partial fraction decompositions found in the unorganized pages of the second notebo