User:Boris Tsirelson/Sandbox1

Birth and infancy of the idea

Some tables compiled by ancient Babylonians may be treated now as tables of some functions. Also, some arguments of ancient Greeks may be treated now as integration of some functions. Thus, in ancient times some functions were used (implicitly). However, they were not recognized as special cases of a general notion.

Further progress was made in the 14th century. Two "schools of natural philosophy", at Oxford (William Heytesbury, Richard Swineshead) and Paris (Nicole Oresme), trying to investigate natural phenomena mathematically, came to the idea that laws of nature should be formulated as functional relations between physical quantities. The concept of function was born, including a curve as a graph of a function of one variable, and a surface — for two variables. However, the new concept was not yet widely exploited either in mathematics or in its applications. Linear functions were well understood, but nonlinear functions remained intractable, except for few isolated marginal cases.

The name "function" was assigned to the new concept later, in 1698, by Johann Bernoulli and Gottfried Leibniz, and published by Bernoulli in 1718.

Power series

The sum of the geometric series

$1+x+x^{2}+x^{3}+\dots ={\frac {1}{1-x}}$

was calculated by Archimedes, but only for x=1/4, since only this value was needed, and of course not written in this form, since algebraic notation appeared only in the 16th century. New wonderful formulas with infinite sums were discovered (and repeatedly rediscovered) in the 14–17 centuries: for arctangent,

$\arctan x=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-\dots$

(Madhava of Sangamagramma, around 1400; James Gregory, 1671); for logarithm,

$\log(1+x)=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-\dots$

(Nicholas Mercator, 1668); and many others (Isaac Barrow, Isaac Newton, Gottfried Leibniz, ...) Nonlinear functions, desperately needed for the study of motion (Johannes Kepler, Galileo Galilei) and geometry (Pierre Fermat, René Descartes), became tractable via such infinite sums now called power series.

Newton understood by analysis the investigation of equations by means of infinite series. In other words, Newton's basic discovery was that everything had to be expanded in infinite series.(Arnold, page 35)

These studies [on power series] stand in the same relation to algebra as the studies of decimal fractions to ordinary arithmetic. (Newton)

Power series became a de facto standard of function, since on one hand, all functions needed in applications were successfully developed into power series, and on the other hand, only functions developed into power series were tractable in the theory. It was not unusual, to claim a theorem for an arbitrary function, and then, in the proof, to consider its development into a power series.

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Further progress appears in the 17th century from the study of motion (Johannes Kepler, Galileo Galilei) and geometry (P. Fermat, R. Descartes). A formulation by Descartes (La Geometrie, 1637) appeals to graphic representation of a functional dependence and does not involve formulas (algebraic expressions):

If then we should take successively an infinite number of different

values for the line y, we should obtain an infinite number of values for the line x, and therefore an infinity of different points, such as C, by means of which the required curve could be

drawn.

The term function is adopted by Leibniz and Jean Bernoulli between 1694 and 1698, and disseminated by Bernoulli in 1718:

One calls here a function of a variable a quantity composed in any manner whatever of this variable and of constants.

This time a formula is required, which restricts the class of functions. However, what is a formula? Surely, y = 2x² - 3 is allowed; what about y = sin x? Is it "composed of x"? "In any manner whatever" is now interpreted much more widely than it was possible in 17th century.

... little by little, and often by very subtle detours, various

transcendental operations, the logarithm, the exponential, the trigonometric functions, quadratures, the solution of differential equations, passing to the limit, the summing of series, acquired the

right of being quoted. (Bourbaki, p. 193)

Surely, sin x is not a polynomial of x. However, it is the sum of a power series:

$\sin x=x-{\frac {x^{3}}{6}}+{\frac {x^{5}}{120}}-\dots$

which was found already by James Gregory in 1667. Many other functions were developed into power series by him, Isaac Barrow, Isaac Newton and others. Moreover, all these formulas appeared to be special cases of a much more general formula found by Brook Taylor in 1715.

...

But on the first stage the notion of an algebraic expression is quite restrictive. More general, possibly ill-behaving functions have to wait for the 19th century.