 By L. Ahlfors

A customary resource of knowledge of features of 1 complicated variable, this article has retained its large recognition during this box via being constantly rigorous with no changing into needlessly excited by complicated or overspecialized fabric. tricky issues were clarified, the ebook has been reviewed for accuracy, and notations and terminology were modernized. bankruptcy 2, advanced services, incorporates a short part at the switch of size and region less than conformal mapping, and masses 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 options are the spine of the idea. bankruptcy four, advanced Integration, now encompasses a new and easier facts of the overall type of Cauchy's theorem. there's a brief part at the Riemann zeta functionality, exhibiting using residues in a extra fascinating state of affairs than within the computation of yes integrals.

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Extra info for Complex Analysis (1979) 3rd edition

Sample text

Now go(z) and gh(Z) have the same singularities, except possibly the point Zo= -h, which may be a singularity for go(z) without being one for gh(Z). We denote by S+(h) and S-(h) the domains Iz+hl > D(h) and Iz+hl < D(h), respectively, and by C(h) their common boundary. Since gh(Z) has all its singularities in S-(h)=S-(h) u C(h), and since there is at least one singularity of gh(Z) on C(h), we see that D(h) is positive for sufficiently small h, that the intersection J(h) = S-(h) (\ S-(O) is not empty, and that all the singularities of go(z) are on l(h).

Let us now suppose thatf(s) = (A, A), uniform in PU1 ' is such that there exists a non-negative constant m with the property that to every e> 0 there corresponds a K(e, m) (=K(e, m, 0'1» such that If(s)1 0 there corresponds a K(e) = K(e, 0'1) such that in LI(a1o e) we have If(s) I ~ K(e)A(ltj).

A simple estimate of dk~~/2) yields the required result. In Theorem ILLS the starred curve r* is the exterior contour of the image of the circle Izl = 1 under z<1>(z)=z(l +z2)/2. 4. 1 is not valid if no condition is imposed on the nature of A" (for a given k). 3, but la,,1 =k is not bounded independently of k. 3. 2. A general inequality for the coefficients corresponding to finite upper density of the exponents In what follows we assume that the upper density D· of the sequence A is finite. 5 we know that aa=ac=lim sup (log lanl/An).