The above is an example of a loop integral, i.e., a definite integral extended over a certain loop-circuit. Such integrals have been gener alized, being extended over double circuits, and these latter integrals are an aid in the study of the solutions of certain linear differential equations.
20. Functions of Several Variables.—In each of n complex planes let a region S n be given, and let be an arbitrary point of S Then the set of values (zi,...,zn) is called from analogy with the case ts=1 a point. The totality of these points constitutes the region (S). In every point of S let a function f(z.,...") be uniquely defined. The idea of the limit and of continuity can be extended at once to such a function. f zn) is said to be analytic in (S) if it is continuous in (S) and if, further more, when all the but one,— zni, let us say, — are held fast, f is then analytic in S.; this condition to be satisfied for m =1,..., n. For brevity let n=2. Cauchy's Integral Formula becomes f f 1 r(111 o., h)di, .figi,s0— C, The function can furthermore be developed by Taylor's Theorem as follows: Let (a., a') be an interior point of (S), and let IC x be the largest circle that can be drawn in SK about as centre, containing in its interior no boundary point of S K. Then Z c 1st— ai)ifn on=o where 1 r4) limi a ast" (ar as) 1 di& f Ph, t's)dts = --ch)it c, and is an arbitrary interior point of the circle We have, as in the case n 1, where I — I ----ri, Its— ri.
The idea and the method of analytic con tinuation can be extended without difficulty to functions of several variables, and thus the monogenic analytic function is completely de fined. For implicit functions and analytic con
figurations lying in hyperspace the reader is referred to the author's report in the Encyklo piidie, §§ 44 and 47 (see below).
The following theorem is due to Weierstrass : Let (b, an) be an interior point of a region (S) in which f(w, en) is analytic.
Let f vanish at this point; but f(w, a n) shall not vanish identically. Then f(w, ,sn) win + + . + 0, where 41) (w, zi,..., sn) is analytic throughout a certain neighborhood of the point (b, an ) and does not vanish there, while each of the coefficients ..., an) , . . ., A in (za, . • • , as) is sii.gle-valued and analytic throughout a re gion including the point (a,, ..., an) in its in terior. The excluded case f (w, a,, ..., a is also treated by Weierstrass.
21. Bibliography.— Burkhardt, (Theory of Functions of a Complex Variable' (New York 1913) ; Forsyth, 'Theory of Functions' (1893) ; Goursat, 'Cours d'analyse) (Vol. II, 2d ed., Paris 1910-13, has been translated by Hed rick) ' • Harkness and Morley, 'Treatise on the Theory of Functions) (1893), and 'Intro duction to Analytic Functions' (1898) ; Osgood, 'Lehrbuch der Funktionentheorie' (Leipzig 1905-07) ; Picard, (Traite d'analyse) (Vol. II, 2d ed., Paris 1901-08). For a comprehensive report on the theory of functions, including numerous historical and bibliographical refer ences, see the author's article: "Allgemeine Theorie der Funktionen a) einer and b) melt rerer komplexer Grifissen,p in der mathematischen Wissenschaften,' Vol. II, B. 1.