“Abel's Theorem”的概念、定义、习题解题方法及翻译-数学辞典

单词

Abel's Theorem

释义

Abel's Theorem

Given a Taylor Series

(1)

where the Complex Number

has been written in the polar form

, examinethe Real and Imaginary Parts

(2)

(3)

Abel's theorem states that, if

and

are Convergent, then

(4)

Stated in words, Abel's theorem guarantees that, if a Real Power Series Convergesfor some Positive value of the argument, the Domain of Uniform Convergence extends at least up to andincluding this point. Furthermore, the continuity of the sum function extends at least up to and including this point.

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 773, 1985.

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