Introduction to Analysis of the Infinite

Abstract

This chapter explains the origin of elementary functions and the impact of Descartes’s “Géométrie” on their calculation. The interpolation polynomial leads to Newton’s binomial theorem and to the infinite series for exponential, logarithmic, and trigonometric functions. The chapter ends with a discussion of complex numbers, infinite products, and continued fractions. The presentation follows the historical development of this subject, with the mathematical rigor of the period. The justification of dubious conclusions will be an additional motivation for the rigorous treatment of convergence in Chapter III.

... our students of mathematics would profit much more from a study of Euler’s Introductio in Analysin Infinitorum,rather than of the available modern textbooks.

(André Weil 1979, quoted by J.D. Blanton 1988, p. xii)

... since the teacher was judicious enough to allow his unusual pupil (Jacobi) to occupy himself with Euler’s Introductio, while the other pupils made great efforts ....

(Dirichlet 1852, speech in commemoration of Jacobi, in Jacobi’s Werke, vol. I, p.4)