The Mathematical Experience

by Philip J Davis & Reuben Hersh

4.Unity within Diversity (p198)

UNIFICATION, the establishment of a relationship between seemingly diverse objects, is at once one of the great motivating forces and one of the great sources of aesthetic satisfaction in mathematics. It is beautifully illustrated by the formula of Euler which unifies the trigonometric functions with the "power" or ''exponential" functions. The trigonometric ratios and the sequences of exponential growth are both of ancient origin. Recall the legend of the magician who was to be paid in grains of wheat-one on the first square of the chessboard, with the number of grains doubling with each square. No doubt the sequence 1, 2, 4, 8, 16,. . . is the oldest exponential sequence. Now what on earth do these ideas have to do with one another?
It would be a nice piece of mathematical history to trace the growth of these notions until they fuse. We would see the extension of the sine and cosine functions to periodic functions, the switch-over to e^x as the basic exponential, where e is the mysterious transcendental number 2.718281828459. . . , the development of the theory of power series, the bold but entirely natural extension of the range of applicability of power series to admit complex variables, the derivation of the three expansions

sin x  =  x - x³/3! + x⁵/5! + . . .,

cos x = 1- x²/2! + x⁴/4! - . . .,

e^x = 1 + x + x²/2!+ x³/3! + . . .,

leading to the final unification, Euler's formula,

e^ix = cos x + i sin x, where i = Ö-1

Thus, the exponential emerges as trigonometry in disguise. Reciprocally, by solving backwards one has

cos x = ¹/₂(e^ix + e^-ix),

sin x = ¹/_2i (e^ix - e^-ix),

so that trigonometry is equally exponential algebra in disguise. The special case where x = p = 3.14159. . . leads to

e^pi = cosp + i sinp = - l, or e^pi  + 1 = 0.

There is an aura of mystery in this last equation, which links the five most important constants in the whole of analysis: 0, 1, e, p = , and i.
Within the same story one moves forward from this mystic landing to Fourier analysis, periodogram analysis, Fourier analysis over groups, differential equations, and one comes rapidly to great theories, great technological applications, and always a sense of the actual and potential unities that lurk in the corners of the universe. (See Chapter 5, Fourier Analysis.)