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 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