# Functions

The concept of *function* is one of the most important in mathematics. However, its history is relatively short. M. Kline credits [Kline, p. 338] Galileo (1564-1642) with the first statements of dependency of one quantity on another, e.g., "The times of descent along inclined planes of the same height, but of different slopes, are to each other as the lengths of these slopes." In a 1673 manuscript Leibniz used the word "function" to mean any quantity varying from point to point of a curve, like the length of the tangent or the normal. The curve itself was said to be given by an equation. But in 1714, he already used the word "function" to mean quantities that depend on a variable. The notation f(x) was introduced by Euler in 1734. Still, in the 1930s, a well known Russian mathematician N. Luzin wrote:

The function concept is one of the most fundamental concepts of modern mathematics. It did not arise suddenly. It arose more than two hundred years ago out of the famous debate on the vibrating string and underwent profound changes in the very course of that heated polemic. From that time on this concept has deepened and evolved continuously, and this twin process continues to this very day. That is why no single formal definition can include the full content of the function concept. This content can be understood only by a study of the main lines of the development that is extremely closely linked with the development of science in general and of mathematical physics in particular.

Functions, especially of the numeric variety, are often confused with formulas by means of which they are defined. In one of the discrete mathematics textbooks, the authors fling a particularly inept remark to the effect that "Whereas classical mathematics is about formulas, discrete mathematics is as much about algorithms as about formulas." Charitably, I interpret the maxim as the authors' attempt to emphasize the importance of functions in mathematics in general and discrete mathematics in particular. In their view, I believe, the efficiency of function computations gains prominence when it comes to practical matters. In mathematics, the function of two variables $f(x, y) = x^{2} - y^{2}$ can be equally well defined as $f(x, y) = (x - y)(x + y).$ In algorithmic mathematics there is an important difference between the two definitions: one requires two multiplications and one addition (with the sign minus), the other needs one multiplication and two additions. The latter is faster! But the authors, of course, might have had their own reasons.

That is a fact, however, that the definition we currently use has been introduced by Johann Peter Gustav Lejeune Dirichlet (1805-1859). The turning point in the common perception of *function* as associated with an analytic curve - the curve whose shape in any small region defines its shape everywhere else - has occurred with the 1807 publication by Joseph Fourier (1768-1830) of his solution to the wave equation. Fourier represented his solution as (what is now called) Fourier series:

$\displaystyle f(x)=\frac{a_{0}}{2}+\sum_{n=1}^{\infty}(a_{n}\cos nx+b_{n}\sin nx),$

where $\displaystyle a_{n}=\frac{1}{\pi}\int_{0}^{2\pi}f(t)\cos nt\,dt$ and $\displaystyle b_{n}=\frac{1}{\pi}\int_{0}^{2\pi}f(t)\sin nt\,dt.$ The crucial argument for reconsidering the notion of function was the realization that Fourier series converges pointwise for a wide range of functions, not necessarily analytic, but, for example, defined piece-wise.

### References

- M. Kline,
*Mathematical Thought From Ancient to Modern Times I*, Oxford University Press, 1972 - N. Luzin,
__Function__,*Mathematical Evolutions*, A. Shenitzer and J. Stillwell (eds.), MAA, 2002

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