Complex Analysis I: Entire and Meromorphic Functions by A.A. Gonchar, Viktor P. Havin, N.K. Nikolski, V.I.

By A.A. Gonchar, Viktor P. Havin, N.K. Nikolski, V.I. Rublinetskij, V. Tkachenko, M.B. Balk, A.A. Gol'dberg, B.Ya. Levin, I.V. Ostrovskii

The first a part of the quantity features a finished description of the speculation of complete and meromorphic features of 1 complicated variable and its purposes. It comprises the elemental notions, equipment and effects at the progress of whole services and the distribution in their zeros, the Rolf Nevanlinna idea of distribution of values of meromorphic features together with the inverse challenge, the speculation of thoroughly general development, the concept that of restrict units for complete and subharmonic capabilities. The authors describe the interpolation via whole capabilities, to complete and meromorphic ideas of standard differential equations, to the Riemann boundary challenge with an unlimited index and to the mathematics of the convolution semigroup of likelihood distributions. Polyanalytic features shape essentially the most ordinary generalizations of analytic features and are defined partly II. They emerged for the 1st time in aircraft elasticity thought the place they discovered very important purposes (due to Kolossof, Mushelishvili etc.). This publication incorporates a precise evaluation of modern investigations in regards to the function-theoretical pecularities of polyanalytic features (boundary behavour, price distributions, degeneration, area of expertise etc.). Polyanalytic services have many issues of touch with such fields of study as polyharmonic capabilities, Nevanlinna idea, meromorphic curves, cluster set idea, services of numerous complicated variables etc.

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Additional resources for Complex Analysis I: Entire and Meromorphic Functions Polyanalytic Functions and Their Generalizations

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Macintyre's relation combined with (3) and (4), immediately implies that K(r, f) rv ver, f) , r - 00, r ft E, dlogr < 00. L However, A. Macintyre's relation is somewhat more convenient, since it yields more precise asymptotics. Moreover, the size ofthe set, where the asymptotics are valid, turns out to be slightly larger than in (5). Some refinements and generalizations of A. Macintyre's relation were suggested by Ostrovskii in 1962 and Strelits (1962, 1972). An advantage of A. Macintyre's method is that it is not based on the representation (2) by apower series, and, as a result, it admits an extension to functions analytic in the half-plane and even to multivalued functions.

In particular, if 1/2< p < 1 and h(7r, f) = h(O, f) cos p7r (the p-trigonometric convexity of the indicator implies that the decrease of entire function f is the most fast along the negative ray), then the indicator h(

It is evident that a sequence of disks Ck(8) = {z: Iz - rkeiOkl < 8rd, 8> 0, rk i 00, cannot be covered by a Co -set. Thus, if for certain 8 > 0, E > 0 and an entire function 1 of proximate order p( r) the asymptotic inequality r-p(r) log I/(re icp )I < h(Ok, f) - E, reicp E Ck(8), k -+ 00 , holds, then 1 is not a CRG function. Azarin (1966) proved the inverse theorem: if 1 is not a CRG entire function, then there exists a sequence of disks C k (8) on which the above-indicated asymptotic inequality holds.

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