Theory of Functions

w = u + i v of z = x + i y . It had been proved by Dirichlet that (for a plane) there is always one, and only one, function of x and y , which satisfies Δ u = 0 , and which, together with its differential quotients of the first two orders, is for all values of x and 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 $w$ is uniquely determined for

all points within a closed surface, if $u$ is arbitrarily given for all points on the curve, whilst $v$ is given for one point within the curve. In order to treat the more complicated case where $w$ has $n$ values for one value of $z$, and to observe the conditions about continuity, Riemann invented the celebrated surfaces, known as "Riemann's surfaces," consisting

of $n$ coincident planes or sheets, such that the passage from one sheet to another is made at the branch-points, and that the $n$ sheets form together a multiply-connected surface, which can be dissected by cross-cuts into a singly-connected surface. The $n$-valued function $w$ 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 $n$ leaves are multiply-connected, and then transformed the surface into the surface of a solid with $p$ 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