On Functions (1800-1824)

"I am convinced that the motion of the vibrating string is as exactly represented in all possible cases by trigonometric developments as by the integration which contains the arbitrary functions [...]" (Jean-Baptiste-Joseph Fourier, 1807)

"At the beginning I would ask anyone who wants to introduce a new function in analysis to clarify whether he intends to confine it to real magnitudes" (real values of the argument) and regard the imaginary values as just vestigial –or whether he subscribes to my fundamental proposition that in the realm of magnitudes the imaginary ones a+b√−1 = a+bi have to be regarded as enjoying equal rights with the real ones. We are not talking about practical utility here; rather analysis is, to my mind, a self-sufficient science. It would lose immeasurably in beauty and symmetry from the rejection of any fictive magnitudes. At each stage truths, which otherwise are quite generally valid, would have to be encumbered with all sorts of qualifications." (Carl F Gauss, [letter to Bessel,] 1811)

"In order to indicate that a quantity depends on one or several others, either by operations of any kind, or by other relations, which it is impossible to assign algebraically, but whose existence is determined by certain conditions, we call the first quantity a function of the others." (Sylvestre-François Lacroix, "An elementary treatise on the differential and integral calculus", 1816)

"If variable quantities are so joined between themselves that, the value of one of these being given, one can conclude the values of all the others, one ordinarily conceives these diverse quantities expressed by means of the one of them, which then takes the name independent variable; and the other quantities expressed by means of the independent variable are those which one calls functions of this variable." (Augustin-Louis Cauchy, "Cours d'analyse", 1821) 

"Right away, if somebody wishes to introduce a new function into analysis, I will ask himto make clear if he simply wishes to use it for real quantities (real values of the argument of the function), and at the same time will regard the imaginary values of the argument as an appendage [Gauss here spoke of a ganglion, Uberbein], or if he accedes to my principle ¨that in the domain of quantities the imaginary a + b √−1 = a + bi must be regarded as enjoying equal rights with the real. This is not a matter of utility, rather to me analysis is an independent science which, by slighting each imaginary quantity, loses exceptionally in beauty and roundness, and in a moment all truths that otherwise would hold generally, must necessarily suffer highly tiresome restrictions." (Carl FriedrichGauss, [letter to Bessel] 1821)

"The integral ∫ϕx.dx will always have the same value along two different paths if it is never the case that ϕx = ∞ in the space between the curves representing the paths. This is abeautiful theorem whose not-too-difficult proof I will give at a suitable opportunity [. . . ]. In any case this makes it immediately clear why a function arising from an integral ϕx.dx can have many values for a single value of x, for one can go round a point where ϕx = ∞ either not at all, or once, or several times. For example, if one defines logx by  dx x , starting from x = 1, one comes to logx either without enclosing the point x = 0 or by going around it once or several times; each time the constant +2πi or −2πi enters; so the multiple of logarithms of any number are quite clear." (Carl FriedrichGauss, [letter to Bessel] 1821) 

"When variable quantities are so tied to each other that, given the values of some of them, we can deduce the values of all the others, we usually conceive these various quantities expressed in terms of several of them, which then bear the name independent variables; and the remaining quantities expressed in terms of the independent variables, are what we call functions of these same variables." (Augustin-Louis Cauchy, "Cours d’analyse de l’École Royale Polytechnique", 1821)

"In general, the function f(x) represents a succession of values or ordinates each of which is arbitrary. An infinity of values being given of the abscissa x, there are an equal number of ordinates f(x). All have actual numerical values, either positive or negative or nul. We do not suppose these ordinates to be subject to a common law; they succeed each other in any manner whatever, and each of them is given as it were a single quantity." (Joseph Fourier, "Théorie analytique de la Chaleur", 1822)