$#C+1 = 31 : ~/encyclopedia/old_files/data/A013/A.0103750 Asymptotic power series
...ries may be added, multiplied, divided and integrated just like convergent power series.
4 KB (542 words) - 06:35, 14 April 2024
==Hypotheses on the distribution of power residues and non-residues.==
(Cf. [[Power residue|Power residue]]; [[Quadratic residue|Quadratic residue]].)
3 KB (433 words) - 09:08, 2 January 2021
The derivative of a polynomial, rational function or formal power series, which can be defined purely algebraically (without using the concep
\sum _ {i = 0 } ^ { n }
2 KB (246 words) - 19:39, 5 June 2020
Power series in one complex variable $ z $.
s(z) \ = \ \sum _ { k=0 } ^ \infty b _ {k} (z-a) ^ {k} ,
16 KB (2,404 words) - 13:34, 4 November 2023
...e of series of complex numbers, used often to determine the convergence of power series at the radius of convergence
If $\sum a_n$ is a convergent series of real numbers and $\{b_n\}$ is a bounded mono
2 KB (382 words) - 12:44, 10 December 2013
$#C+1 = 97 : ~/encyclopedia/old_files/data/P074/P.0704200 Power function
is an integer, the power function is a particular case of a [[Rational function|rational function]].
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''on power series''
If a [[Power series|power series]]
7 KB (1,065 words) - 09:52, 11 November 2023
...n element of an analytic function is the circular element in the form of a power series
f (z) = \sum _ { k=0 } ^ \infty c _ {k} ( z - a ) ^ {k}
4 KB (679 words) - 18:34, 5 April 2020
1 + \sum _ {n = 1 } ^ \infty
\sum _ {n = 0 } ^ \infty {
3 KB (504 words) - 14:33, 10 March 2024
...(see [[#References|[1]]]) in connection with questions of convergence of [[power series]]. If the series
...| < 1$ the sum $\phi(x)$ of the series \eqref{eq1} can be represented as a power series:
2 KB (368 words) - 07:06, 29 March 2024
...t $e \neq 0$, then $A$ can be decomposed according to Peirce into a direct sum of vector subspaces:
<TR><TD valign="top">[1]</TD> <TD valign="top"> A.A. Albert, "Power-associative rings" ''Trans. Amer. Math. Soc.'' , '''64''' (1948) pp. 552
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The sum of the power series
F ( x , w ) = \sum _ { n= 0} ^ { \infty} a _ {n} ( x) w^ {n}
3 KB (451 words) - 08:25, 16 March 2023
\right ) + \sum
\right ) = \sum _
4 KB (524 words) - 04:11, 6 June 2020
1) If for a power series
f ( z) = \sum _ {n=1} ^ \infty a _ {n} z ^ {\lambda _ {n} }
3 KB (438 words) - 12:39, 6 January 2024
and if the power series
\sum _ {k = 0 } ^ \infty
4 KB (577 words) - 19:43, 5 June 2020
...\alpha = P / Q$, that is, $\alpha$ is equal to the formal expansion of $P \sum _ { n = 0 } ^ { \infty } ( Q - 1 ) ^ { n }$. For instance, if $K \subseteq
...with coefficients in $A$, and let $A ( ( X ) )$ denote the set of Laurent power series, that is,
5 KB (828 words) - 11:51, 24 December 2020
u \cdot v = \sum _ {i = 1 } ^ { n } u _ {i} v _ {i} .
\sum _ {j = 0 } ^ { {n } - \nu } \left ( \begin{array}{c}
6 KB (890 words) - 04:11, 6 June 2020
A generalization of a [[power series]] in non-negative integral powers of the difference $ z - a $
\sum _ {k = - \infty } ^ {+\infty }
8 KB (1,203 words) - 10:36, 20 January 2024
...ial estimates of trigonometric sums (cf. [[Trigonometric sum|Trigonometric sum]]) of the form
S( f ) = \sum _ {1 \leq x \leq P } e ^ {2 \pi i f( x) } ,
5 KB (759 words) - 08:29, 6 June 2020
are two formal power series, then, by definition,
The set $A[[T_1,\ldots,T_N]]$ of all formal power series forms a ring under these operations.
6 KB (1,093 words) - 08:26, 16 March 2023