Global Analysis in Linear Differential Equations by Mitsuhiko Kohno (auth.) PDF

By Mitsuhiko Kohno (auth.)

ISBN-10: 9401059462

ISBN-13: 9789401059466

ISBN-10: 9401146055

ISBN-13: 9789401146050

Since the initiative works for international research of linear differential equations by way of G.G. Stokes and B. Riemann in 1857, the ethereal functionality and the Gauss hypergeometric functionality turned an important and the best useful exact capabilities, that have quite a few purposes to mathematical technological know-how, physics and engineering. The cffcctivity of those features is largely because of their "behavior within the huge" . for example, the ethereal functionality performs a simple position within the asymptotic research of many services bobbing up as strategies of differential equations in numerous difficulties of utilized math­ ematics. In case of the employment of its habit, one must always be aware of the Stokes phenomenon. nonetheless, as is recognized, the Gauss hypergeometric functionality arises in all fields of arithmetic, e.g., in quantity concept, within the thought of teams and in research itself. it's not an excessive amount of to claim that every one energy sequence are distinct or prolonged situations of the hypergeometric sequence. For the total use of its houses, one wishes connection formulation or contiguous relations.

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In fact, we shall show that F(z) represents an analytic function in a more extended domain V than the disk of convergence. The domain V consists of points ~ such that the circle with the segment 0 ~ as its diameter has no singularities of P(z) and its analytic continuation inside it. ,. be singularities of P(z) and its analytic continuation. Draw a line passing through a singularity aj and perpendicular to the segment Oaj. Then we have a polygonal domain including the origin. This is the domain V.

Draw a line passing through a singularity aj and perpendicular to the segment Oaj. Then we have a polygonal domain including the origin. This is the domain V. 1. 1: Domain of Borel's analytic continuation Of course, by the principle of analytic continuation, P(z) is considered to be analytic in the domain V. We now insist that F(z) is exactly the expression of the analytic continuation of P(z) in the domain V. Let ~ be any point of V . , the circle C : Iz - ~ ~I = ~ I~I + {j for a sufficiently small {j > O.

Proof Let f(O) == 0 (J'(O) # 0) and lJ(z)1 > Iyl on z E r : Izl == r. 1. Convergent and Divergent Series 25 In the considered case, the last integral is calculated as ~ 27r~ r lr (¢(())n d( = (n 1 (n - 1) ! [ dn- 1 (¢(Z)t] dz n - 1 . z=O Concerning other extensions of the Lagrange formula by H. G. T. N. Watson [117]. Lastly, we consider the inversion of the convergent power series y = zq (aq + a q+1 z + . . ) ( aq of 0; q ::: 2 ) near z = O. Then we have I yq = Z + al z + ... ) (ao ( ao of 0) for Izl :S: r, r being a sufficiently small positive number.

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Global Analysis in Linear Differential Equations by Mitsuhiko Kohno (auth.)


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