Download PDF by Professor Dr. Fyodor A. Medvedev (auth.): Scenes from the History of Real Functions
By Professor Dr. Fyodor A. Medvedev (auth.)
To try to assemble a comparatively entire bibliography of the idea of features of a true variable with the considered necessary bibliographical info, to enumer ate the names of the mathematicians who've studied this topic, convey their basic effects, and likewise comprise the main crucial biographical facts approximately them, to behavior a listing of the recommendations and strategies which were and stay utilized within the idea of features of a true variable ... briefly, to hold out a person of those initiatives with applicable completeness will require a separate publication concerning a corresponding volume of labor. as a result the be aware essays happens within the identify of the current paintings, permitting a few freedom within the choice of fabric. In justification of this option, it really is moderate to aim to symbolize to a point the topic to whose heritage those essays are dedicated. the reality of the problem is this is a hopeless firm if one calls for this kind of characterization to be exhaustively entire and concise. No residing topic will be given a last definition with out upsetting a few objections, frequently severe ones. but when we make no such claims, a characterization is feasible; and if the 1st essay of the current publication appears to be like unconvincing to somebody, the reason being the non-public fault of the writer, and never the target necessity of the attempt.
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Additional info for Scenes from the History of Real Functions
He writes as follows: "Let the ordinate be" [3, pp. 116, 135], "Let the ordinate be ~: Jbz + zz" (pp. 116, 135), and the like. And to use Newton's expression, there are "an infinite number" (pp. 119, 138) of such "ordinates," including those of the type mentioned above, where the expression for the function contains a variable in the exponent that is a parameter. Analytic images of this type become impossible to study by the old synthetic methods. Therefore, in calculating the integrals of such "ordinates" Newton writes candidly to Leibniz: "I would not, of course, have been able to obtain any of these results until 1 renounced the contemplation of figures and reduced the whole matter to the study of ordinates alone" (pp.
We remark only that in  Euler restricted himself to real-valued functions and usually functions of a real variable; but in other works he also made a broad study of functions of a complex variable with complex values 25 ; in  he not only stated the concept of a function as an analytic expression, he also developed it profoundly in many diverse directions, summarizing what had been done by his predecessors and adding much of his own. 26 And it is the study of these objects that forms the subject of his book .
Apparently until the publication of Euler's book  it was customary to think of functional dependence primarily either in a purely geometric way, as some line, surface, or solid, or as some mixed geometric-kinematic structure, when lines, surfaces, and solids were thought of as generated by the motion of points, lines, and surfaces. As evidence of what has just been said we refer to the books of Reyneaux [1, p. 151], Agnesi [1, p. 2], and others. , p. 142. 21 Quoted by Yushkevich [4, p. 143].
Scenes from the History of Real Functions by Professor Dr. Fyodor A. Medvedev (auth.)