CodalSearch this book — or all of Codal…⌘K
Projects
Sign inStart a book
nydus/A History of MathematicsPublic

This text examines the transition from the Middle Ages to the Modern era, highlighting how the fall of Constantinople and the invention of the printing press catalyzed a revival of classical learning. It traces the shift toward scientific inquiry through the rise of pure mathematics and astronomy, detailing the intellectual struggle against established scholastic and ecclesiastical authority.

Page 158 of 219
Table of Contents

Theory of Functions

exercised in the use of series by giving an example of a series always convergent and continuous, such that the series formed by the integrals of the terms is always convergent, and yet does not represent the integral of the first series.87

The general theory of functions of two variables has been investigated to some extent by Weierstrass and Poincaré.

H. A. Schwarz of Berlin (born 1845), a pupil of Weierstrass,

has given the conform representation (Abbildung) of various surfaces on a circle. In transforming by aid of certain substitutions a polygon bounded by circular arcs into another also bounded by circular arcs, he was led to a remarkable differential equation ψ(u′,t)=ψ(u,t), where ψ(u,t) is the expression which Cayley calls the "Schwarzian derivative,"

and which led Sylvester to the theory of reciprocants.

Schwarz's developments on minimum surfaces, his work on hypergeometric series, his inquiries on the existence of solutions

to important partial differential equations under prescribed conditions, have secured a prominent place in mathematical literature.

The modern theory of functions of one real variable was first worked out by H. Hankel, Dedekind, G. Cantor, Dini, and

Heine, and then carried further, principally, by Weierstrass,

Schwarz, Du Bois-Reymond, Thomae, and Darboux. Hankel

established the principle of the condensation of singularities;

Dedekind and Cantor gave definitions for irrational numbers;

definite integrals were studied by Thomae, Du Bois-Reymond,

and Darboux along the lines indicated by the definitions of such integrals given by Cauchy, Dirichlet, and Riemann. Dini

wrote a text-book on functions of a real variable (1878), which was translated into German, with additions, by J. Lüroth and

A. Schepp. Important works on the theory of functions are

the Cours de M. Hermite, Tannery's Théorie des Fonctions

d'une variable seule, A Treatise on the Theory of Functions by James Harkness and Frank Morley, and Theory of Functions of

a Complex Variable by A. R. Forsyth.

158