Page:A History of Mathematics (1893).djvu/377

 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, $\scriptstyle{\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=\Delta u=0}$, 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