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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1913 Excerpt: ...these derivatives are regular functions. If instead of the expression "regular function of a complex argument," we use only its meaning in terms of real functions, this theorem is stated as follows: ViIa. If u and v are two real functions of x, y which are continuous in a given domain and which have continuous first derivatives satisfying the differential equations Bu dv dv----du dx By' Bx By it follows at once from this that they have an unlimited number of successive derivatives continuous within this domain. With the aid of these results the theorems deduced in § 34 may be supplemented at important places. If w = u.+-iv is a regular function of z = x + iy, then the functional determinant Bu By By du--/du. Idv'l--dw Bx By Bx By BxJ Bx J dz' ) and hence is not negative. Since we have proved (VII) the continuity of the derivatives which appear, we can apply Theorem IV of § 27 and conclude that: VIII. Ifw=f(z) is a function which is regular and singlevalued in a domain B and which has a derivative different from zero everywhere in this domain, then the values which w takes on in B cover once without gaps a definite region C of the w-plane. Since the value of the limit--is the reciprocal of the value dw j of the limit--, it follows further that: dz IX. z is also a function of w regular within the region C. Besides: X. If the function w = f(z) satisfies the provisions of Theorem VIII and if W= tfw) is a function of w regular in C, then W=f(f(z)) is also a function of z regular within B. dw dlV For, from the existence of the limits--and----we infer dz dw dW the existence of the limit as with functions of real variables. dz Finally, the following theorems are proved just as if the variables were restricted to real values (A. A. I, II, § ...
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