We are now in a position to complete the definition, according to Weierstrass, of an ana lytic function. Let Az) be analytic in a given region S. Continue f(z), if possible, analytically beyond S. Repeat this step so long as it is possible to do so. The most extended function that can thus be generated out of the given function is known as the monogenic analytic function f(z). It is uniquely determined by the values of f(z) in S. Weierstrass employed the method of overlapping circles and power series for obtaining the successive analytic con tinuations. The function is completely defined The rational functions and the elementary functions e, log z, etc., are examples of mono genic analytic functions. By an algebraic func tion is meant any monogenic analytic function which satisfies an algebraic equation, G(w, 2) = 0, where G denotes a polynomial. Every irre ducible algebraic equation in w and s defines such a function.
Two analytic functions which agree in value with each other along an arc of a curve, how ever short, lying wholly within their domain of definition, are identical throughout their whole extent.
It may happen that the singular points of an analytic function are not isolated, but exist in every internal of a curve (for example an arc of the whole circumference of a circle, the latter being the case with the function defined by the series Ivt n=1 throughout the interior of the unit circle I z 1
14. Reflection in Analytic Curves.—A curve which can be represented by the equations sf(t), where f and are both developable by Taylor's Theorem about every point to of the above in terval and where, furthermore, f' (t) and O'(t) never vanish simultaneously in the interval, is called an analytic curve.
Let S be a region bounded in part or wholly by an analytic curve l having no multiple point. if u vi= RE) is analytic in S and takes on real boundary values along then f(s) can always be con tinued analytically across I' . In particular, let T be an arc of a circle. Then the analytic continuation is effected by inverting S in this circle and assigning to f(z) in the transformed points values conjugate to those in the original points. The method is fundamental in the study of minimum surfaces, and of the elliptic modular and the automorphic functions.
15. Dirichlet's Principle.— In order to show that a simply connected region S can be mapped in a one-to-one manner and conform ally on a circle (§ 4), it is necessary to estab lish the existence of a function u which (a) is single-valued and continuous in S, (b) satisfies Laplace's Equation (§6) throughout the in terior of S, and (c) takes on preassigned values along the boundary of S. In similar cases in
mathematical physics Gauss, Thomson and Dirichlet had attempted to give a proof of the existence of such a function by making con nections with a skilfully constructed corre sponding problem of the Calculus of Variations. Riemann adopted this method in order to obtain his existence proofs, both for the present problem and for the case of algebraic functions corresponding to a preassigned Riemann's sur face. The problem of the Calculus of Varia tions which matches the present problem is as follows: To find a function u which is con tinuous in S, together with its first derivatives, takes on the prescribed boundary values, and makes the integral fsf [ (aLlix)2+ (02.1dS a minimum. If such a function exists and if, furthermore, it has continuous second deriva tives, then it will satisfy Laplace's Equation within S and thus yield the desired solution. It was assumed by the mathematicians named above that thc problem of the Calculus of Variations necessarily has a solution. This method of proof is lcnown as Dirichlet's Prin ciple. Weierstrass pointed out the insufficiency of the reasoning involved in the assumption to which we have just called attention. Other methods of proof were then devised by Schwarz and Neumann. Recently Hilbert has, with con siderable labor, partially filled the gap in the original method.
16. Weierstrass's and Mittag-Leffler's Theorems.— It is always possible to form a polynomial which has n roots situated arbi trarily in the complex plane. The question presents itself as to whether this theorem can be extended to transcendental integral func tions. This question Weierstrass answered in 1876 as follows: Let ao, ch, . . . be any set of points in the complex plane suc.h that lim an== when n =.0 Then there exists an integral transcendental function G (z) which has in each of the points a. a zero of arbitrarily preassigned order, and does not vanish for any other value of s. The most general such function is given by the formula G(z)=e- r4)01)(z), where Ch denotes a particular function of the class in question and f is a rational or tran scendental integral function. This theorem has formed the point of departure for many recent researches, chiefly of French mathematicians.