By Lars Ahlfors

A customary resource of knowledge of features of 1 complicated variable, this article has retained its vast reputation during this box through being constantly rigorous with out turning into needlessly serious about complicated or overspecialized fabric. tricky issues were clarified, the publication has been reviewed for accuracy, and notations and terminology were modernized. bankruptcy 2, advanced capabilities, contains a short part at the swap of size and sector below conformal mapping, and lots more and plenty of bankruptcy eight, Global-Analytic services, has been rewritten to be able to introduce readers to the terminology of germs and sheaves whereas nonetheless emphasizing that classical recommendations are the spine of the idea. bankruptcy four, complicated Integration, now incorporates a new and less complicated facts of the overall kind of Cauchy's theorem. there's a brief part at the Riemann zeta functionality, exhibiting using residues in a extra interesting state of affairs than within the computation of sure integrals.

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**Example text**

The number of poles is the same. This common number of zeros and poles is called the order of the rational function. If a is any constant, the function R(z) - a has the same poles as R(z), and consequently the same order. The zeros of R(z) - a are roots of the equation R(z) = a, and if the roots are counted as many times as the order of the zero indicates, we can state the following result: A rational function R(z) of order p has p zeros and p poles, and every equation R(z) = a has exactly p roots.

A zero of order 1 is called a simple zero and is characterized by the conditions P(a) = 0, P'(a) ~ 0. As an application we shall prove the following theorem, known as Lucas's theorem: Theorem I. If all zeros of a polynomial P(z) lie in a half plane, then all zeros of the derivative P' (z) lie in the same half plane. From (8) we obtain (9) P'(z) = _1_ P(z) z - a1 + ... +-1-· Z - an Suppose that the half plane H is defined as the part of the plane where Im (z - a)/b < 0 (see Chap. 1, Sec. 3). If ak is in H and z is not, we have then z - ak Im-b- = z - a ak - a Im-b-- Im-b- > 0.

Stereographic projection. 20 COMPLEX ANALYSIS can be written in this form. The correspondence is consequently one to one. It is easy to calculate the distance d(z,z') between the stereographic projections of z and z'. If the points on the sphere are denoted by (x1,x2,x3), (xf,x~,xD, we have first (x 1 - xD 2 + (x2 - x~) 2 + (xg - x~) 2 = 2 - 2(xlx; + X2X~ + XgX~). From (35) and (36) we obtain after a short computation X1X; + X2X~ + XgX~ (z + z)(z' + z')- + (z- z)(z'- z') (lzl 2 -1)(1z'l 2 -1) 2 2 (1 lzl )(1 lz'l ) (1 lzl 2)(1 lz'l 2 ) - 2lz(1 lzl2)(1 lz'l2) + + + + + + z'l 2 .