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    • CommentRowNumber1.
    • CommentAuthorIan_Durham
    • CommentTimeMar 26th 2010
    Didn't have much free time, but managed to add a little bit to Stokes phenomenon and responded to Zoran's query about Birkhoff's theorem. I suggested we rename it "Birkhoff-von Neumann theorem" so as not to confuse it with other similarly titled theorems.
    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeMar 26th 2010
    • (edited Mar 26th 2010)

    I am sorry but I slightly disagree with what you exactly added here. There are many situations with different asymptotic expansions in different regions and not all are about Stokes phenomenon. The matter of jumps given by matrix faxtors at Stokes lines is one of the specifics. It is not only about property of single functions but also about whole modules of solutions of certain differential equations for example. I will see how to repair but I think the added material somewhat misleads. For now it is OK, but it will have to be more specific later, when we write more about asymptotic analysis. It would be nice to have entries on Borel summability, asymptotic expansions and Stokes sheaf.

    Unfortunately the description is still vague and can not deserve name "Details" as there is no mathematical exact detail given at all yet.

    • CommentRowNumber3.
    • CommentAuthorIan_Durham
    • CommentTimeMar 26th 2010
    But Stokes phenomenon is specific to Airy functions and functions that behave like them, which is what I thought I put.
    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeMar 26th 2010

    It is rather generic. I do not know how you define "behave like them". Only for Fuchsian equations the formal solutions are converging, as long as one has irregular singular point for linear meromorphic ODE, one deals with the Stokes phenomenon. At least this is my understanding.

    • CommentRowNumber5.
    • CommentAuthorIan_Durham
    • CommentTimeMar 27th 2010
    I was thinking about any class of functions that can be expressed in terms of Airy functions or that have some defined relation to them (e.g. Bessel functions).
    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeApr 1st 2010

    This definition (expressing in terms of Airy function) would be too narrow. At least every linear meromorphic connection with an irregular singularity exhibits the Stokes phenomenon, but there are certainly some nonlinear cases as well. For example, the Painlevé equations exhibit the "quasilinear Stokes phenomenon" which has been investigated by Kapaev and Its, and many others.

    • CommentRowNumber7.
    • CommentAuthorIan_Durham
    • CommentTimeApr 2nd 2010
    OK, I wasn't aware of that. Considering those extensions then I'll defer to you on this one.

1 to 7 of 7

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