oo). Together with Example 9, we have 1 1-1+1-1+1-1+· by Borel's two methods. summable (B). 2 (B) Example 9 actually implies that the power series P(z) is The Borel summability plays an important role in the analytic theory of differential equations, because Borel's integral gives a concrete representation of the analytic continuation of an analytic function expressed in terms of a power series.

L an zn (a) of 0) is convergent for n=l Iyl <::: M. Then the inverse function is convergent in the disk Proof By formal calculation, we substitute zk = L 00 b;;:)yn n=k into the series of y : y ~ ak (E b~k)yn) ~ (~akb~k») yn and equate coefficients of like powers of y, obtaining the relations { 1 = a) b), 0 = a) bn n +L ak b;;:) (n = 2,3, ... ). k=2 Since a) of 0 and b~k) do not include bn , one can determine the coefficients bn (n = 1,2, ... ) uniquely. Thus, we have derived a formal power series.

1. Convergent and Divergent Series 29 As is seen in the history of mathematics, before the time of Cauchy, the series, even if they were divergent, had already been effectively used in developing mathematics. It was not until 19-th century that a new significance was given to divergent series and a new notion of "summation" was discovered. Consider the series (D -1) 1-1+1-1+1-1+·· . z + 2 ! Z2 - 3 ! zn. 00 +... n=O The series (D-1) is divergent and the power series (D-2) is so for Z -# O. For the n-th partial sum Sn of the series (D-1) is equal to 1 for odd nand 0 for even n, and the radius of convergence of the power series (D-2) is zero.

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