Scenes from the History of Real Functions by Professor Dr. Fyodor A. Medvedev (auth.)

By Professor Dr. Fyodor A. Medvedev (auth.)

To try and bring together a comparatively whole bibliography of the speculation of services of a true variable with the considered necessary bibliographical facts, to enumer­ ate the names of the mathematicians who've studied this topic, express their primary effects, and likewise comprise the main crucial biographical info approximately them, to behavior a list of the recommendations and strategies which were and stay utilized within the conception of capabilities of a true variable ... briefly, to hold out someone of those initiatives with acceptable completeness will require a separate ebook related to a corresponding quantity of labor. hence the observe essays happens within the name of the current paintings, permitting a few freedom within the collection of fabric. In justification of this feature, it truly is average to attempt to represent to a point the topic to whose background those essays are committed. the reality of the problem is this is a hopeless company if one calls for this sort of characterization to be exhaustively whole and concise. No dwelling topic may be given a last definition with out scary a few objections, often critical ones. but when we make no such claims, a characterization is feasible; and if the 1st essay of the current publication seems to be unconvincing to somebody, the reason being the private fault of the writer, and never the target necessity of the attempt.

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4]. In other words for Euler a variable quantity is not a flowing, changing quantity, as it was, for example, for Maclaurin [1, pp. 52-57, and many other places], but rather some real structure: "just as the concepts of species and genera are formed from the concepts of individuals, so a variable quantity is a genus in which all the definite quantities are contained" [1, p. 4]. Second, he writes "a variable quantity encompasses the collection of all numbers ... " Consequently Euler regards functions of one variable 24 as being defined, as we would nowadays say, on the entire real line.

578) and use it in the study of functions. It was precisely in this way, algebraically as he put it, that he obtained his series-a somewhat different universal analytic method of representing functional dependence. The opinion was sometimes expressed that it was Euler who made the concept of a function central in analysis. Thus Hankel wrote: "It was Euler who first established it and made it the foundation of all of analysis, and a new era in mathematics was thereby begun" [1, p. 23 This is not true if we interpret the opinion just quoted too literally.

4]. Second, he writes "a variable quantity encompasses the collection of all numbers ... " Consequently Euler regards functions of one variable 24 as being defined, as we would nowadays say, on the entire real line. He writes, " ... " [1, p. 4]. Moreover, "not even zero and imaginary numbers are excluded from the values of a variable quantity" (Ibid). Hence for Euler the domain of definition of a function is in general the entire plane. He treats the range of values of a function exactly the same way, for "a function of a variable quantity is itself a variable quantity" [1, p.

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