Advanced Calculus fi..

538aAdvanced Calculus, Fifth Editionz-planeFigure 8.2Domain and neighborhood.In general, we assume w = f (z) to be defined in a domain (open region) D inthe z-plane, as suggested in Fig. 8.2. If to is a point of D, we can then find a circularneighborhood /z - zoI < k about zo in D. If f (z) is defined in such a neighborhood,except perhaps at zo, then we writelim f (z) = woZ'ZOif for every r > 0 we can choose 6 > 0, so that. . *-2- 'If f (20) is defined and equals wo and (8.14) holds, then we call f (z) continuousat 20.THEOREM 1 The function w = f (z) is continuous at zo = xo + iyo if and onlyif U(X, y) = Re [ f (z)] and v(x, y) = Im [ f (z)] are continuous at (XO, yo).Thus w = z2 = x2 - y2 + 2ixy is continuous for all z, since u = x2 - y2 andv = 2xy are continuous for a11 (x, y). The proof of Theorem 1 is left as an exercise(Problem 5 following Section 8.4).THEOREM 2The sum, product, and quotient of continuous functions of z arecontinuous, except for division by zero; a continuous function of a continuous functionis continuous. Similarly, if the limits exist,lim [ f (z) + g(z)l = lim f (z) + lim g(Z), . . .Z'ZO Z'ZI Z'ZOThese properties are proved as for real variables. (It is assumed in Theorem 2that the functions are defined in appropriate domains.)It follows from Theorem 2 that polynomials in z are continuous for all z andthat each rational function is continuous except where the denominator is zero. FromTheorem 1 it follows thateZ = e" cosy +iex sinyis continuous for all z. Hence, by Theorem 2, so also are the functionse i ~ - e-iz ,iz + e-i~sin z = cos z =2i '2