A History of Mathematics/Recent Times/Theory of Functions

by Florian Cajori

 

THEORY OF FUNCTIONS.

 

We begin our sketch of the vast progress in the theory of functions by considering the special class called elliptic functions. These were richly developed by Abel and Jacobi.

Niels Henrick Abel (1802–1829) was born at Findoë in Norway, and was prepared for the university at the cathedral school in Christiania. He exhibited no interest in mathematics until 1818, when B. Holmboe became lecturer there, and aroused Abel's interest by assigning original problems to the class. Like Jacobi and many other young men who became eminent mathematicians, Abel found the first exercise of his talent in the attempt to solve by algebra the general equation of the fifth degree. In 1821 he entered the University in Christiania. The works of Euler, Lagrange, and Legendre were closely studied by him. The idea of the inversion of elliptic functions dates back to this time. His extraordinary success in mathematical study led to the offer of a stipend by the government, that he might continue his studies in Germany and France. Leaving Norway in 1825, Abel visited the astronomer, Schumacher, in Hamburg, and spent six months in Berlin, where he became intimate with August Leopold Crelle (1780–1855), and met Steiner. Encouraged by Abel and Steiner, Crelle started his journal in 1826. Abel began to put some of his work in shape for print. His proof of the impossibility of solving the general equation of the fifth degree by radicals,—first printed in 1824 in a very concise form, and difficult of apprehension,—was elaborated in greater detail, and published in the first volume. He entered also upon the subject of infinite series (particularly the binomial theorem, of which he gave in Crelle's Journal a rigid general investigation), the study of functions, and of the integral calculus. The obscurities everywhere encountered by him owing to the prevailing loose methods of analysis he endeavoured to clear up. For a short time he left Berlin for Freiberg, where he had fewer interruptions to work, and it was there that he made researches on hyperelliptic and Abelian functions. In July, 1826, Abel left Germany for Paris without having met Gauss! Abel had sent to Gauss his proof of 1824 of the impossibility of solving equations of the fifth degree, to which Gauss never paid any attention. This slight, and a haughtiness of spirit which he associated with Gauss, prevented the genial Abel from going to Göttingen. A similar feeling was entertained by him later against Cauchy. Abel remained ten months in Paris. He met there Dirichlet, Legendre, Cauchy, and others; but was little appreciated. He had already published several important memoirs in Crelle's Journal, but by the French this new periodical was as yet hardly known to exist, and Abel was too modest to speak of his own work. Pecuniary embarrassments induced him to return home after a second short stay in Berlin. At Christiania he for some time gave private lessons, and served as docent. Crelle secured at last an appointment for him at Berlin; but the news of it did not reach Norway until after the death of Abel at Froland.^[82]

At nearly the same time with Abel, Jacobi published articles on elliptic functions. Legendre's favourite subject, so long neglected, was at last to be enriched by some extraordinary discoveries. The advantage to be derived by inverting the elliptic integral of the first kind and treating it as a function of its amplitude (now called elliptic function) was recognised by Abel, and a few months later also by Jacobi. A second fruitful idea, also arrived at independently by both, is the introduction of imaginaries leading to the observation that the new functions simulated at once trigonometric and exponential functions. For it was shown that while trigonometric functions had only a real period, and exponential only an imaginary, elliptic functions had both sorts of periods. These two discoveries were the foundations upon which Abel and Jacobi, each in his own way, erected beautiful new structures. Abel developed the curious expressions representing elliptic functions by infinite series or quotients of infinite products. Great as were the achievements of Abel in elliptic functions, they were eclipsed by his researches on what are now called Abelian functions. Abel's theorem on these functions was given by him in several forms, the most general of these being that in his Mémoire sur une propriété générale d'une classe très-étendue de fonctions transcendentes (1826). The history of this memoir is interesting. A few months after his arrival in Paris, Abel submitted it to the French Academy. Cauchy and Legendre were appointed to examine it; but said nothing about it until after Abel's death. In a brief statement of the discoveries in question, published by Abel in Crelle's Journal, 1829, reference is made to that memoir. This led Jacobi to inquire of Legendre what had become of it. Legendre says that the manuscript was so badly written as to be illegible, and that Abel was asked to hand in a better copy, which he neglected to do. The memoir remained in Cauchy's hands. It was not published until 1841. By a singular mishap, the manuscript was lost before the proof-sheets were read.

In its form, the contents of the memoir belongs to the integral calculus. Abelian integrals depend upon an irrational function $\scriptstyle {y}$ which is connected with $\scriptstyle {x}$ by an algebraic equation $\scriptstyle {F(x,y)=0}$. Abel's theorem asserts that a sum of such integrals can be expressed by a definite number $\scriptstyle {p}$ of similar integrals, where $\scriptstyle {p}$ depends merely on the properties of the equation $\scriptstyle {F(x,y)=0}$. It was shown later that $\scriptstyle {p}$ is the deficiency of the curve $\scriptstyle {F(x,y)=0}$. The addition theorems of elliptic integrals are deducible from Abel's theorem. The hyperelliptic integrals introduced by Abel, and proved by him to possess multiple periodicity, are special cases of Abelian integrals whenever $\scriptstyle {p=}$ or $\scriptstyle {>3}$. The reduction of Abelian to elliptic integrals has been studied mainly by Jacobi, Hermite, Königsberger, Brioschi, Goursat, E. Picard, and O. Bolza of the University of Chicago.

Two editions of Abel's works have been published: the first by Holmboe in 1839, and the second by Sylow and Lie in 1881.

Abel's theorem was pronounced by Jacobi the greatest discovery of our century on the integral calculus. The aged Legendre, who greatly admired Abel's genius, called it "monumentum aere perennius. During the few years of work allotted to the young Norwegian, he penetrated new fields of research, the development of which has kept mathematicians busy for over half a century.

Some of the discoveries of Abel and Jacobi were anticipated by Gauss. In the Disquisitiones Arithmeticœ he observed that the principles which he used in the division of the circle were applicable to many other functions, besides the circular, and particularly to the transcendents dependent on the integral $\scriptstyle {\int {\frac {dx}{\sqrt {1-x^{4}}}}}$. From this Jacobi^[83] concluded that Gauss had thirty years earlier considered the nature and properties of elliptic functions and had discovered their double periodicity. The papers in the collected works of Gauss confirm this conclusion.

Carl Gustav Jacob Jacobi^[84] (1804–1851) was born of Jewish parents at Potsdam. Like many other mathematicians he was initiated into mathematics by reading Euler. At the University of Berlin, where he pursued his mathematical studies independently of the lecture courses, he took the degree of Ph.D. in 1825. After giving lectures in Berlin for two years, he was elected extraordinary professor at Königsberg, and two years later to the ordinary professorship there. After the publication of his Fundamenta Nova he spent some time in travel, meeting Gauss in Göttingen, and Legendre, Fourier, Poisson, in Paris. In 1842 he and his colleague, Bessel, attended the meetings of the British Association, where they made the acquaintance of English mathematicians.

His early researches were on Gauss' approximation to the value of definite integrals, partial differential equations, Legendre's coefficients, and cubic residues. He read Legendre's Exercises, which give an account of elliptic integrals. When he returned the book to the library, he was depressed in spirits and said that important books generally excited in him new ideas, but that this time he had not been led to a single original thought. Though slow at first, his ideas flowed all the richer afterwards. Many of his discoveries in elliptic functions were made independently by Abel. Jacobi communicated his first researches to Crelle's Journal. In 1829, at the age of twenty-five, he published his Fundamenta Nova Theoriœ Functionum Ellipticarum, which contains in condensed form the main results in elliptic functions. This work at once secured for him a wide reputation. He then made a closer study of theta-functions and lectured to his pupils on a new theory of elliptic functions based on the theta-functions. He developed a theory of transformation which led him to a multitude of formulæ containing $\scriptstyle {q}$, a transcendental function of the modulus, defined by the equation $\scriptstyle {q=e^{-\pi k^{'}/k}}$. He was also led by it to consider the two new functions $\scriptstyle {H}$ and $\scriptstyle {\Theta }$, which taken each separately with two different arguments are the four (single) theta-functions designated by the $\scriptstyle {\Theta _{1}}$, $\scriptstyle {\Theta _{2}}$, $\scriptstyle {\Theta _{3}}$, $\scriptstyle {\Theta _{4}}$.^[56] In a short but very important memoir of 1832, he shows that for the hyperelliptic integral of any class the direct functions to which Abel's theorem has reference are not functions of a single variable, such as the elliptic $\scriptstyle {sn}$, $\scriptstyle {cn}$, $\scriptstyle {dn}$, but functions of $\scriptstyle {p}$ variables.^[56] Thus in the case $\scriptstyle {p=2}$, which Jacobi especially considers, it is shown that Abel's theorem has reference to two functions $\scriptstyle {\lambda (u,v)}$, $\scriptstyle {\lambda _{1}(u,v)}$, each of two variables, and gives in effect an addition-theorem for the expression of the functions $\scriptstyle {\lambda (u+u^{\prime },v+v^{\prime })}$, $\scriptstyle {\lambda _{1}(u+u^{\prime },v+v^{\prime })}$, algebraically in terms of the functions $\scriptstyle {\lambda (u,v)}$, $\scriptstyle {\lambda _{1}(u,v)}$, $\scriptstyle {\lambda (u^{\prime },v^{\prime })}$, $\scriptstyle {\lambda _{1}(u^{\prime },v^{\prime })}$. By the memoirs of Abel and Jacobi it may be considered that the notion of the Abelian function of $\scriptstyle {p}$ variables was established and the addition-theorem for these functions given. Recent studies touching Abelian functions have been made by Weierstrass, E. Picard, Madame Kowalevski, and Poincaré. Jacobi's work on differential equations, determinants, dynamics, and the theory of numbers is mentioned elsewhere.

In 1842 Jacobi visited Italy for a few months to recuperate his health. At this time the Prussian government gave him a pension, and he moved to Berlin, where the last years of his life were spent.

The researches on functions mentioned thus far have been greatly extended. In 1858 Charles Hermite of Paris (born 1822), introduced in place of the variable $\scriptstyle {q}$ of Jacobi a new variable $\scriptstyle {\omega }$ connected with it by the equation $\scriptstyle {q=e^{i\pi \omega }}$, so that $\scriptstyle {\omega =ik^{\prime }/k}$, and was led to consider the functions $\scriptstyle {\phi (\omega )}$, $\scriptstyle {\psi (\omega )}$, $\scriptstyle {\chi (\omega )}$.^[56] Henry Smith regarded a theta-function with the argument equal to zero, as a function of $\scriptstyle {\omega }$. This he called an omega-function, while the three functions $\scriptstyle {\phi (\omega )}$, $\scriptstyle {\psi (\omega )}$, $\scriptstyle {\chi (\omega )}$, are his modular functions. Researches on theta-functions with respect to real and imaginary arguments have been made by Meissel of Kiel, J. Thomae of Jena, Alfred Enneper of Göttingen (1830–1885). A general formula for the product of two theta-functions was given in 1854 by H. Schröter of Breslau (1829–1892). These functions have been studied also by Cauchy, Königsberger of Heidelberg (born 1837), F. S. Richelot of Königsberg (1808–1875), Johann Georg Rosenhain of Königsberg (1816–1887), L. Schläfli of Bern (born 1818).^[85]

Legendre's method of reducing an elliptic differential to its normal form has called forth many investigations, most important of which are those of Richelot and of Weierstrass of Berlin.

The algebraic transformations of elliptic functions involve a relation between the old modulus and the new one which Jacobi expressed by a differential equation of the third order, and also by an algebraic equation, called by him "modular equation." The notion of modular equations was familiar to Abel, but the development of this subject devolved upon later investigators. These equations have become of importance in the theory of algebraic equations, and have been studied by Sohnke, E. Mathieu, L. Königsberger, E. Betti of Pisa (died 1892), C. Hermite of Paris, Joubert of Angers, Francesco Brioschi of Milan. Schläfli, H. Schröter, M. Gudermann of Cleve, Gützlaff.

Felix Klein of Göttingen has made an extensive study of modular functions, dealing with a type of operations lying between the two extreme types, known as the theory of substitutions and the theory of invariants and covariants. Klein's theory has been presented in book-form by his pupil, Robert Fricke. The bolder features of it were first published in his Ikosaeder, 1884. His researches embrace the theory of modular functions as a specific class of elliptic functions, the statement of a more general problem as based on the doctrine of groups of operations, and the further development of the subject in connection with a class of Riemann's surfaces.

The elliptic functions were expressed by Abel as quotients of doubly infinite products. He did not, however, inquire rigorously into the convergency of the products. In 1845 Cayley studied these products, and found for them a complete theory, based in part upon geometrical interpretation, which he made the basis of the whole theory of elliptic functions. Eisenstein discussed by purely analytical methods the general doubly infinite product, and arrived at results which have been greatly simplified in form by the theory of primary factors, due to Weierstrass. A certain function involving a doubly infinite product has been called by Weierstrass the sigma-function, and is the basis of his beautiful theory of elliptic functions. The first systematic presentation of Weierstrass' theory of elliptic functions was published in 1886 by G. H. Halphen in his Théorie des fonctions elliptiques et des leurs applications. Applications of these functions have been given also by A. G. Greenhill. Generalisations analogous to those of Weierstrass on elliptic functions have been made by Felix Klein on hyperelliptic functions.

Standard works on elliptic functions have been published by Briot and Bouquet (1859), by Königsberger, Cayley, Heinrich Durège of Prague (1821–1893), and others.

Jacobi's work on Abelian and theta-functions was greatly extended by Adolph Göpel (1812–1847), professor in a gymnasium near Potsdam, and Johann Georg Rosenhain of Königsberg (1816–1887). Göpel in his Theoriœ transcendentium primi ordinis adumbratio levis (Crelle, 35, 1847) and Rosenhain in several memoirs established each independently, on the analogy of the single theta-functions, the functions of two variables, called double theta-functions, and worked out in connection with them the theory of the Abelian functions of two variables. The theta-relations established by Göpel and Rosenhain received for thirty years no further development, notwithstanding the fact that the double theta series came to be of increasing importance in analytical, geometrical, and mechanical problems, and that Hermite and Königsberger had considered the subject of transformation. Finally, the investigations of G. W. Borchardt of Berlin (1817–1880), treating of the representation of Kummer's surface by Göpel's biquadratic relation between four theta-functions of two variables, and researches of H. H. Weber of Marburg, F. Prym of Würzburg, Adolf Krazer, and Martin Krause of Dresden led to broader views. Researches on double theta-functions, made by Cayley, were extended to quadruple theta-functions by Thomas Craig of the Johns Hopkins University.

Starting with the integrals of the most general form and considering the inverse functions corresponding to these integrals (the Abelian functions of $\scriptstyle {p}$ variables), Riemann defined the theta-functions of $\scriptstyle {p}$ variables as the sum of a $\scriptstyle {p}$-tuply infinite series of exponentials, the general term depending on $\scriptstyle {p}$ variables. Riemann shows that the Abelian functions are algebraically connected with theta-functions of the proper arguments, and presents the theory in the broadest form.^[56] He rests the theory of the multiple theta-functions upon the general principles of the theory of functions of a complex variable.

Through the researches of A. Brill of Tübingen, M. Nöther of Erlangen, and Ferdinand Lindemann of Munich, made in connection with Riemann-Roch's theorem and the theory of residuation, there has grown out of the theory of Abelian functions a theory of algebraic functions and point-groups on algebraic curves.

Before proceeding to the general theory of functions, we make mention of the "calculus of functions," studied chiefly by C. Babbage, J. F. W. Herschel, and De Morgan, which was not so much a theory of functions as a theory of the solution of functional equations by means of known functions or symbols.

The history of the general theory of functions begins with the adoption of new definitions of a function. With the Bernoullis and Leibniz, $\scriptstyle {y}$ was called a function of $\scriptstyle {x}$, if there existed an equation between these variables which made it possible to calculate $\scriptstyle {y}$ for any given value of $\scriptstyle {x}$ lying anywhere between $\scriptstyle {-\infty }$ and $\scriptstyle {+\infty }$. The study of Fourier's theory of heat led Dirichlet to a new definition: $\scriptstyle {y}$ is called a function of $\scriptstyle {x}$, if $\scriptstyle {y}$ possess one or more definite values for each of certain values that $\scriptstyle {x}$ is assumed to take in an interval $\scriptstyle {x_{0}}$ to $\scriptstyle {x_{1}}$. In functions thus defined, there need be no analytical connection between $\scriptstyle {y}$ and $\scriptstyle {x}$, and it becomes necessary to look for possible discontinuities. A great revolution in the ideas of a function was brought about by Cauchy when, in a function as defined by Dirichlet, he gave the variables imaginary values, and when he extended the notion of a definite integral by letting the variable pass from one limit to the other by a succession of imaginary values along arbitrary paths. Cauchy established several fundamental theorems, and gave the first great impulse to the study of the general theory of functions. His researches were continued in France by Puiseux and Liouville. But more profound investigations were made in Germany by Riemann.

Georg Friedrich Bernhard Riemann (1826–1866) was born at Breselenz in Hanover. His father wished him to study theology, and he accordingly entered upon philological and theological studies at Göttingen. He attended also some lectures on mathematics. Such was his predilection for this science that he abandoned theology. After studying for a time under Gauss and Stern, he was drawn, in 1847, to Berlin by a galaxy of mathematicians, in which shone Dirichlet, Jacobi, Steiner, and Eisenstein. Returning to Göttingen in 1850, he studied physics under Weber, and obtained the doctorate the following year. The thesis presented on that occasion, Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen complexen Grösse, excited the admiration of Gauss to a very unusual degree, as did also Riemann's trial lecture, Ueber die Hypothesen welche der Geometrie zu Grunde liegen. Riemann's Habilitationsschrift was on the Representation of a Function by means of a Trigonometric Series, in which he advanced materially beyond the position of Dirichlet. Our hearts are drawn to this extraordinarily gifted but shy genius when we read of the timidity and nervousness displayed when he began to lecture at Göttingen, and of his jubilation over the unexpectedly large audience of eight students at his first lecture on differential equations.

Later he lectured on Abelian functions to a class of three only,—Schering, Bjerknes, and Dedekind. Gauss died in 1855, and was succeeded by Dirichlet. On the death of the latter, in 1859, Riemann was made ordinary professor. In 1860 he visited Paris, where he made the acquaintance of French mathematicians. The delicate state of his health induced him to go to Italy three times. He died on his last trip at Selasca, and was buried at Biganzolo.

Like all of Riemann's researches, those on functions were profound and far-reaching. He laid the foundation for a general theory of functions of a complex variable. The theory of potential, which up to that time had been used only in mathematical physics, was applied by him in pure mathematics. He accordingly based his theory of functions on the partial differential equation, $\scriptstyle {{\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}=\Delta u=0}$, which must hold for the analytical function $\scriptstyle {w=u+iv}$ of $\scriptstyle {z=x+iy}$. It had been proved by Dirichlet that (for a plane) there is always one, and only one, function of $\scriptstyle {x}$ and $\scriptstyle {y}$, which satisfies $\scriptstyle {\Delta u=0}$, and which, together with its differential quotients of the first two orders, is for all values of $\scriptstyle {x}$ and $\scriptstyle {y}$ within a given area one-valued and continuous, and which has for points on the boundary of the area arbitrarily given values.^[86] Riemann called this "Dirichlet's principle," but the same theorem was stated by Green and proved analytically by Sir William Thomson. It follows then that $\scriptstyle {w}$ is uniquely determined for all points within a closed surface, if $\scriptstyle {u}$ is arbitrarily given for all points on the curve, whilst $\scriptstyle {v}$ is given for one point within the curve. In order to treat the more complicated case where $\scriptstyle {w}$ has $\scriptstyle {n}$ values for one value of $\scriptstyle {z}$, and to observe the conditions about continuity, Riemann invented the celebrated surfaces, known as "Riemann's surfaces," consisting of $\scriptstyle {n}$ coincident planes or sheets, such that the passage from one sheet to another is made at the branch-points, and that the $\scriptstyle {n}$ sheets form together a multiply-connected surface, which can be dissected by cross-cuts into a singly-connected surface. The $\scriptstyle {n}$-valued function $\scriptstyle {w}$ becomes thus a one-valued function. Aided by researches of J. Lüroth of Freiburg and of Clebsch, W. K. Clifford brought Riemann's surface for algebraic functions to a canonical form, in which only the two last of the $\scriptstyle {n}$ leaves are multiply-connected, and then transformed the surface into the surface of a solid with $\scriptstyle {p}$ holes. A. Hurwitz of Zürich discussed the question, how far a Riemann's surface is determinate by the assignment of its number of sheets, its branch-points and branch-lines.^[62]

Riemann's theory ascertains the criteria which will determine an analytical function by aid of its discontinuities and boundary conditions, and thus defines a function independently of a mathematical expression. In order to show that two different expressions are identical, it is not necessary to transform one into the other, but it is sufficient to prove the agreement to a far less extent, merely in certain critical points.

Riemann's theory, as based on Dirichlet's principle (Thomson's theorem), is not free from objections. It has become evident that the existence of a derived function is not a consequence of continuity, and that a function may be integrable without being differentiable. It is not known how far the methods of the infinitesimal calculus and the calculus of variations (by which Dirichlet's principle is established) can be applied to an unknown analytical function in its generality. Hence the use of these methods will endow the functions with properties which themselves require proof. Objections of this kind to Riemann's theory have been raised by Kronecker, Weierstrass, and others, and it has become doubtful whether his most important theorems are actually proved. In consequence of this, attempts have been made to graft Riemann's speculations on the more strongly rooted methods of Weierstrass. The latter developed a theory of functions by starting, not with the theory of potential, but with analytical expressions and operations. Both applied their theories to Abelian functions, but there Riemann's work is more general.^[86]

The theory of functions of one complex variable has been studied since Riemann's time mainly by Karl Weierstrass of Berlin (born 1815), Gustaf Mittag-Leffler of Stockholm (born 1846), and Poincaré of Paris. Of the three classes of such functions (viz. functions uniform throughout, functions uniform only in lacunary spaces, and non-uniform functions) Weierstrass showed that those functions of the first class which can be developed according to ascending powers of $\scriptstyle {x}$ into converging series, can be decomposed into a product of an infinite number of primary factors. A primary factor of the species $\scriptstyle {n}$ is the product $\scriptstyle {\left(1-{\frac {x}{a}}\right)e^{P_{(n)}}}$, $\scriptstyle {P_{(n)}}$ being an entire polynomial of the $\scriptstyle {n}$th degree. A function of the species $\scriptstyle {n}$ is one, all the primary factors of which are of species $\scriptstyle {n}$. This classification gave rise to many interesting problems studied also by Poincaré.

The first of the three classes of functions of a complex variable embraces, among others, functions having an infinite number of singular points, but no singular lines, and at the same time no isolated singular points. These are Fuchsian functions, existing throughout the whole extent. Poincaré first gave an example of such a function.

Uniform functions of two variables, unaltered by certain linear substitutions, called hyperfuchsian functions, have been studied by E. Picard of Paris, and by Poincaré.^[81]

Functions of the second class, uniform only in lacunary spaces, were first pointed out by Weierstrass. The Fuchsian and the Kleinian functions do not generally exist, except in the interior of a circle or of a domain otherwise bounded, and are therefore examples of functions of the second class. Poincaré has shown how to generate functions of this class, and has studied them along the lines marked out by Weierstrass. Important is his proof that there is no way of generalising them so as to get rid of the lacunæ.

Non-uniform functions are much less developed than the preceding classes, even though their properties in the vicinity of a given point have been diligently studied, and though much light has been thrown on them by the use of Riemann's surfaces. With the view of reducing their study to that of uniform transcendents, Poincaré proved that if $\scriptstyle {y}$ is any analytical non-uniform function of $\scriptstyle {x}$, one can always find a variable $\scriptstyle {z}$, such that $\scriptstyle {x}$ and $\scriptstyle {y}$ are uniform functions of $\scriptstyle {z}$.

Weierstrass and Darboux have each given examples of continuous functions having no derivatives. Formerly it had been generally assumed that every function had a derivative. Ampère was the first who attempted to prove analytically (1806) the existence of a derivative, but the demonstration is not valid. In treating of discontinuous functions, Darboux established rigorously the necessary and sufficient condition that a continuous or discontinuous function be susceptible of integration. He gave fresh evidence of the care that must be 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 $\scriptstyle {\psi (u',t)=\psi (u,t)}$, where $\scriptstyle {\psi (u,t)}$ 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.