Abel sum - Weblio 英和・和英辞典 (original) (raw)

単語を追加

× この辞書を今後表示しない

※辞書の非表示は、設定画面から変更可能

Abel sum

名詞

Abel sum (複数形 Abel sums)

1. (mathematical analysis) Given a power series f ( x ) = ∑ n = 0 ∞ a n x n {\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}x^{n}} that is convergent for real x in the open interval (0, 1), the value lim x → 1 − ∑ n = 0 ∞ a n x n {\displaystyle \lim _{x\rightarrow 1^{-}}\sum _{n=0}^{\infty }a_{n}x^{n}} , which is assigned to f ( 1 ) = ∑ n = 0 ∞ a n {\displaystyle f(1)=\sum _{n=0}^{\infty }a_{n}} by the Abel summation method (または A-method).
   • 1967, Jan Mikusiński, Operational Calculus, Cambridge University Press, page 102,
     The Abel sum of ∑ a n {\displaystyle \textstyle \sum a_{n}} is defined as the limit of the corresponding power series:
     lim x → 1 − 0 ∑ n = 0 ∞ a n x n {\displaystyle \lim _{x\rightarrow 1-0}\sum _{n=0}^{\infty }a_{n}x^{n}} .
     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.

参考

• summability method
• summation method

Abel sumのページの著作権
英和・和英辞典 情報提供元は参加元一覧 にて確認できます。

| ピン留めアイコンをクリックすると単語とその意味を画面の右側に残しておくことができます。 | | | ------------------------------------------- | |