a) A-> + oo, z on the real interval [z0, zx], b) a (z) twice continuously diff erentiable, and b (z) continuously differentiate on [z0, *J, and c) a(z)>0 on [*o,*J. Convergent expansions in I/A were obtained for a transformed equation. The inadequacies of Liouville's result foreshadow in embryo form the problems that have given birth to the future development of the theory. Permitting a (z) to have one or more zeros in the domain of discussion leads to the immense complications of turning point theory, while replacing the closed interval [z0, zt] by a complex

An historical survey of ordinary linear differential equations with a large parameter and turning points

http://link.springer.com/content/pdf/10.1007/978-94-009-0003-5_13.pdf

Smooth Actions of Non-Compact Semi-Simple Lie Groups

We consider complex manifolds that admit actions by holomorphic transformations of classical simple real Lie groups and classify all such manifolds in a natural situation. Under our assumptions, which require the group at hand to be dimension-theoretically large with respect to the manifold on which it is acting, our classification result states that the manifolds which arise are described precisely as invariant open subsets of certain complex flag manifolds associated to the complexified groups.

Classical Symmetries of Complex Manifolds

with v -2 a positive real constant. The complex variable z is to be confined to a bounded neighborhood of the irregular singular point zO. The analogous problem for a class of nth order differential equations, containing (1) for integral v >6, has been discussed by Hurd [2]. His analysis is somewhat complicated, and it appears that a clearer and more direct investigation of the asymptotic solutions of (1) than is attainable by the methods applied in [2] is possible. This is accomplished largely by adapting the formal structure of the analysis [3] used in the discussion of (1) for v = 2. The case v> 2 discussed here resembles, in some respects, that when v <2. In order to contrast the case v <2 with the present one, we first outline the nature of our results. Suitable changes of the variables in (1) reduce it to the form

/pdf/asymptotic-solution-with-respect-to-a-parameter-of-a-2ym8dojs2o.pdf

Asymptotic solution with respect to a parameter of a differential equation having an irregular singular point

1. G. D. Birkhoff, On the asymptotic character of the solution of certain differential equations containing a parameter, Trans. Amer. Math. Soc. vol. 9 (1908) pp. 214-231. 2. C. C. Hurd, Asymptotic theory of linear differential equations singular in the variable of differentiation and in a parameter, T6hoku Math. J. vol. 44 (1938) pp. 243274. 3. N. D. Kazarinoff and R. W. McKelvey, Asymptotic solution of differential equations in a domain containing a regular singular point, Canadian Journal of Mathematics vol. 8 (1956). 4. R. E. Langer, On the asymptotic solution of differential equations, with reference to the Stokes' phenomenon about a singular point, Trans. Amer. Math. Soc. vol. 37 (1935) pp. 397-416.

/pdf/on-stieltjes-integration-1jva59erd9.pdf

Tomoji Tomuro

Papers

Smooth ${\rm SL}(n,{\bf H})$-, ${\rm Sp}(n,{\bf C})$-actions on $(4n-1)$-manifolds