IDEAS home Printed from https://ideas.repec.org/h/spr/sprchp/978-1-4612-1588-2_10.html
   My bibliography  Save this book chapter

Analytic Continuation

In: Handbook of Complex Variables

Author

Listed:
  • Steven G. Krantz

    (Washington University in St. Louis, Department of Mathematics)

Abstract

Suppose that V is a connected, open subset of $$ \mathbb{C} $$ and that f1: V → $$ \mathbb{C} $$ and f2: V → $$ \mathbb{C} $$ are holomorphic functions. If there is an open, non-empty subset U of V such that f1 ≡ f2 on U, then f1 ≡ f2 on all of V (see §§3.2.3). Put another way, if we are given an f holomorphic on U, then there is at most one way to extend f to V so that the extended function is holomorphic. [Of course there might not even be one such extension: if V is the unit disc and U the disc D(3/4, 1/4), then the function f(z) = 1/z does not extend. Or if U is the plane with the non-positive real axis removed, then again no extension from U to V is possible.

Suggested Citation

  • Steven G. Krantz, 1999. "Analytic Continuation," Springer Books, in: Handbook of Complex Variables, chapter 0, pages 123-141, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-1588-2_10
    DOI: 10.1007/978-1-4612-1588-2_10
    as

    Download full text from publisher

    To our knowledge, this item is not available for download. To find whether it is available, there are three options:
    1. Check below whether another version of this item is available online.
    2. Check on the provider's web page whether it is in fact available.
    3. Perform a
    for a similarly titled item that would be available.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:sprchp:978-1-4612-1588-2_10. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.