By Raghavan Narasimhan

The unique version of this e-book has been out of print for a few years. The seem­ ance of the current moment version owes a lot to the initiative of Yves Nievergelt at japanese Washington college, and the aid of Ann Kostant, arithmetic Editor at Birkhauser. because the ebook used to be first released, numerous humans have remarked at the absence of workouts and expressed the opinion that the e-book may were extra worthwhile had routines been incorporated. In 1997, Yves Nievergelt expert me that, for a decade, he had usually taught a path at jap Washington in response to the publication, and that he had systematically compiled routines for his direction. He kindly placed his paintings at my disposal. hence, the current version looks in components. the 1st is largely only a reprint of the unique variation. i've got corrected the misprints of which i've got develop into acutely aware (including these mentioned to me via others), and feature made a small variety of different minor alterations.

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For WE D(wo, r}. ~. 1fAwo is a curve joining a to wo, then Awo· Yw is one joining a to wand is piecewise differentiable if Ywo is. Hence F(w}-F(w o} = JJdz. Yw 21 Fundamental properties of holomorphic functions This latter integral is exactly the function considered in the proof of Morera's theorem. It follows (see the proof of Morera's theorem) that § 3. Fundamental properties of hoiomorphic functions The notation which is given in the following definition is one we shall use throughout the rest of this book.

Let 8>0 and let U be an open set such that E c U and m 2 ( U - E) < 8. ) Let N"? 1 be an integer. We divide R into 4N congruent rectangles RV' v = 1, 2, ... , 4N by iterating N times the operation of dividing a rectangle into four rectangles by joining the midpoints of opposite sides as in the proof of the Cauchy-Goursat theorem (§2, Theorem 1). If Rv=[IX, p] x [y, J], we have (J - y)/(P -IX) = (d- e)/(b - a), so that A -I::; (J - y)/(P -IX)::; A. Elementary theory of holomorphic functions 50 If N is sufficiently large, we have the following: if Rv n E i=~, then Rv C V (v = 1, ...

This is equivalent to saying that f is not meromorphic on D (a, r). (The Casorati-Weierstrass Theorem) Let aEO, D*={ZE

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