Abel sum

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After Norwegian mathematician Niels Henrik Abel (1802-1829).


Abel sum (plural Abel sums)

  1. (mathematical analysis) Given a power series that is convergent for real x in the open interval (0, 1), the value , which is assigned to by the Abel summation method (or A-method).
    • 1967, Jan Mikusiński, Operational Calculus, Cambridge University Press, page 102,
      The Abel sum of is defined as the limit of the corresponding power series:
      The existence of the Abel sum is ascertained when the series in question is known to be summable (C, r) for some value of r.
    • 2005, Bulletin of the American Mathematical Society, page 81:
      Jacobi in his Vorlesungen über Dynamik [1884] had used Abel sums to separate variables in the Hamilton-Jacobi equation in connection with the geodesic flow on the surface of a 3-dimensional ellipsoid, etc.
    • 2012, Peter L. Duren, Invitation to Classical Analysis, American Mathematical Society, page 180:
      Also, Abel's theorem guarantees that the Abel sum of a convergent series exists and is equal to the ordinary sum.

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