Representation of Elementary Functions
While the first five chapters in this book have touched upon more or less standard topics, the material of the present chapter goes in another direction. The reader will probably find it surprising. Indeed, the notions of infinite product and Green’s function, discussed in detail earlier in this volume, have customarily been included in texts on mathematical analysis and differential equations, respectively. The present chapter, in contrast, discusses an unusual idea that has never been explored in texts before. That is, a technique, reported for the first time in Melnikov (Appl. Math. Sci. 2 (2008) 81–97 and J. Math. Anal. Appl. 344 (2008) 521–534), is employed here for obtaining infinite product representations for a number of elementary functions.
KeywordsDirichlet Problem Cosine Function Hyperbolic Function Tangent Function Infinite Product
- 1.L. Euler, Introductio in Analysin Infinitorum, Vol. 1, Lausanne, Bousquet, 1748 Google Scholar
- 5.V.I. Smirnov, A Course of Higher Mathematics, Vols. 1 and 4, Pergamon, Oxford, 1964 Google Scholar
- 9.I.S. Gradstein and I.M. Ryzhik, Table of Integrals, Series and Products, Academic Press, New York, 1971 Google Scholar
- 18.M.A. Pinsky, Partial Differential Equations and Boundary-Value Problems with Applications, McGraw-Hill, Boston, 1998 Google Scholar