On the Fundamental Theorem in the theory of Functions of a Complex Variable

Abstract

The fundamental theorem in the Theory of Functions of a Complex Variable states, as is well known, that, if dw/dz exists at all points in the interior of a circle of radius c, the function w of the complex variable z is analytic, so that, in particular, all the higher differential coefficients of w exist.There is no difficulty in translating this theorem into one which employs the language of the real variable, and the question has thus been raised as to whether it is not possible to prove the theorem without the use of the complex variable.I understand, indeed, that this very matter was the subject of conversation at a meeting of the Society held early in the present year.The progress of the Theory of Functions of a Real Variable has been recently very rapid; but, regarded as a science, in the modern sense of the term, it is but a younger brother of the Theory of Functions of a Complex Variable.This has been largely owing to the great simplicity brought about in the latter theory by precisely the theorem of which it is the question.On the other hand, from the point of view of the Theory of Functions of a Real Variable, it is certainly unsatisfactory that a theorem so important, and in any but its most recent formulations long well known, should not yet have been proved except by the use of the complex variable.

Locations

• Proceedings of the London Mathematical Society - View
• Zenodo (CERN European Organization for Nuclear Research) - View - PDF