16 - Series and Products for Elementary Functions

Summary

Preliminary Remarks

Euler was the first mathematician to give a systematic and coherent account of the elementary functions, although earlier mathematicians had certainly paved the way. These functions are comprised of the circular or trigonometric, the logarithmic, and the exponential functions. Euler's approach was a departure from the prevalent, more geometric, point of view. On the geometric perspective, the elementary functions were defined as areas under curves, lengths of chords, or other geometric conceptions. Euler's 1748 Introductio in Analysin Infinitorum defined the elementary functions arithmetically and algebraically, as functions.

At that time, infinite series were regarded as a part of algebra, though they had been obtained through the use of calculus. The general binomial theorem was considered an algebraic theorem. So in his Introductio, Euler used the binomial theorem to produce new derivations of the series for the elementary functions. Interestingly, in this book, where he avoided using calculus, Euler gave no proof of the binomial theorem itself; perhaps he had not yet found any arguments without the use of calculus. In a paper written in the 1730s, Euler derived the binomial theorem from the Taylor series, as Stirling had done in 1717. It was only much later, in the 1770s, that Euler found an argument for the binomial theorem depending simply on the multiplication of series.