$#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.
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==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} ,
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...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
$#C+1 = 97 : ~/encyclopedia/old_files/data/P074/P.0704200 Power function
is an integer, the power function is a particular case of a [[Rational function|rational function]].
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''on power series''
If a [[Power series|power series]]
7 KB (1,065 words) - 09:52, 11 November 2023
...n element of an analytic function is the circular element in the form of a power series
f (z) = \sum _ { k=0 } ^ \infty c _ {k} ( z - a ) ^ {k}
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1 + \sum _ {n = 1 } ^ \infty
\sum _ {n = 0 } ^ \infty {
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...(see [[#References|[1]]]) in connection with questions of convergence of [[power series]]. If the series
...| < 1$ the sum $\phi(x)$ of the series \eqref{eq1} can be represented as a power series:
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...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}
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\right ) + \sum
\right ) = \sum _
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1) If for a power series
f ( z) = \sum _ {n=1} ^ \infty a _ {n} z ^ {\lambda _ {n} }
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and if the power series
\sum _ {k = 0 } ^ \infty
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...\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,
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u \cdot v = \sum _ {i = 1 } ^ { n } u _ {i} v _ {i} .
\sum _ {j = 0 } ^ { {n } - \nu } \left ( \begin{array}{c}
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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) } ,
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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.
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Thus, the generating function $ \sum _ {h \geq 0 } a _ {h} X ^ {h} $
are given by a generalized power sum $ a _ {h} = a ( h ) = \sum _ {i = 1 } ^ {m} A _ {i} ( h ) \alpha _ {i} ^ {h} $ ($ h = 0,1, \dots $
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For functions represented by a power series, a majorant is e.g. the sum of a power series with positive coefficients which are not less than the absolute valu
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is summable by the Hölder method $(H,k)$ to sum $s$ if
...mable to a sum $s$ by the method $(H,k)$, it will also be summable to that sum by the method $(H,k')$ for any $k'>k$. For any $k$ the method $(H,k)$ is eq
2 KB (280 words) - 13:46, 14 February 2020
Consider a complex power series
\sum _ {k = 0 } ^ \infty
4 KB (625 words) - 15:35, 4 June 2020
The formula for the expansion of an arbitrary positive integral power of a [[Binomial|binomial]] in a polynomial arranged in powers of one of the
\sum _ {k = 0 } ^ { m } \left ( \begin{array}{c}
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\sum _ {m , n = 1 } ^ \infty u _ {mn} ,
S _ {mn} = \sum _ {i = 1 } ^ { m }
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and if the sequence of partial sums of a series $ \sum _ {n = 1 } ^ \infty b _ {n} ( x) $
may take complex values), then the series $ \sum _ {n = 1 } ^ \infty a _ {n} ( x) b _ {n} ( x) $
1 KB (212 words) - 07:38, 1 November 2023
$#C+1 = 41 : ~/encyclopedia/old_files/data/D032/D.0302740 Direct sum
...[[Abelian category|Abelian category]]. In the non-Abelian case the direct sum is usually called the discrete direct product. Let $ \mathfrak A $
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is summable by means of the Euler summation method ($(E,q)$-summable) to the sum $S$ if
...eries. Thus, the series $\sum_{n=0}^\infty z^n$ is $(E,q)$-summable to the sum $1/(1-z)$ in the disc with centre at $-q$ and of radius $q+1$.
2 KB (358 words) - 17:36, 14 February 2020
A power series of the form
\sum _ { n=0 } ^ \infty
3 KB (470 words) - 08:17, 26 March 2023
\sum _ { s=0 } ^ { n }
\sum _ { s=0 } ^ { n-1 }
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th exterior power $ \wedge ^ {r} V $.
The direct sum of the spaces of skew-symmetric $ r $-
1 KB (155 words) - 19:38, 5 June 2020
\sum _ {k=0} ^ \infty u _ {k} $$
is summable by the Lindelöf summation method to the sum $ s $
3 KB (417 words) - 08:17, 6 January 2024
f ( x + h ) = \sum _ {n = 0 } ^ \infty
P ( x + \xi h ) = \sum _ {\nu = 0 } ^ { m } P _ \nu ( x , h ) \xi ^ \nu ,
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Abel's theorem on power series: If the power series
S ( z ) \ = \ \sum _ {k = 0} ^ \infty a _ {k} ( z - b ) ^ {k} ,
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be its power function (cf. [[Power function of a test|Power function of a test]]), which gives for every $ \theta $
the corresponding sequence of power functions $ \{ \beta _ {n} ( \theta ) \} $
7 KB (902 words) - 17:46, 4 June 2020
...r any $x_0\in I$ there is a neighborhood $J$ of $x_0$ and a power series $\sum a_n (x-x_0)^n$ such that
An analytic function is infinitely differentiable and its power expansion coincides with the [[Taylor series]]. Namely, the coefficients $a
6 KB (1,048 words) - 21:19, 14 January 2021
..., where $ \lambda $ is a limit ordinal number and $ n $ is an integer, the sum being understood in the sense of addition of [[Order type|order types]].
...omega $ is the least initial ordinal number. The initial ordinal number of power $ \tau $ is denoted by $ \omega(\tau) $. The set $ \{ \omega(\delta) \mid \
9 KB (1,404 words) - 18:33, 4 December 2017
...uestion must be posed, not what the sum is equal to, but how to define the sum of a divergent series, and he found an approach to the solution of this pro
\sum _ {n = 0 } ^ \infty
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th symmetric power of $ E $(
th exterior power of the module $ E $(
4 KB (590 words) - 07:45, 7 January 2024
...\in\mathbf Z$, $0<x<1$. For integers $c$ and $d$, with $c>0$, the Dedekind sum $S(d,c)$ is the rational number defined by
...dratic reciprocity law]]). This relation resembles the reciprocity law for power-residue symbols. Several elementary proofs of this relation can be found in
3 KB (398 words) - 21:39, 23 December 2015
''integro-power series''
The finite sum of Volterra terms (of all types) of degree $ m $
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A divided power structure on $ R $(
A divided power sequence in a co-algebra $ ( C, \mu ) $
4 KB (663 words) - 08:28, 20 January 2024
...eries completions of $ A $ and $ L $, i.e., $ \widehat{A} $ is the ring of power series in the associative but non-commutative variables $ u $ and $ v $, an
...e Campbell-Hausdorff formula provides an expression for $ u \circ v $ as a power series in $ u $ and $ v $:
6 KB (1,020 words) - 17:41, 4 May 2017
...ivisor of $n$ if and only if every prime factor of $d$ appears to the same power in $d$ as in $n$.
The sum of unitary divisors function is denoted by $\sigma^*(n)$. The sum of the $k$-th powers of the unitary divisors is denoted by $\sigma_k^*(n)$.
2 KB (317 words) - 19:43, 17 November 2023
1 + \sum _ {1 \leq i \leq \infty }
} \sum
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...differentiable at $x_0$, its Taylor series at $x_0$ is the [[Power series|power series]] given by
...function $f$ defined in a neighborhood of $x_0$ there is a power series $\sum a_n (x-x_0)^n$ which converges to the values of $f$, then such series coinc
4 KB (710 words) - 06:13, 13 June 2022
\sum _ {n=0 } ^ \infty
f (x) = \sum _ {n=0 } ^ { N }
4 KB (660 words) - 19:29, 13 April 2024
\sum _ { i } {\mathcal H} ^ { i } ( X ; h ^ {n-i} ( \mathop{\rm pt} ) \otim
is isomorphic to the ring of formal power series $ \Omega _ {u} ^ {*} [ [ u ] ] $,
5 KB (632 words) - 11:51, 21 March 2022
h _ {n} = \sum _ {k = 1 } ^ { n } f _ {k} B _ {n,k } ( g _ {1} \dots g _ {n - k + 1 } )
Y _ {n} ( g _ {1} \dots g _ {n} ) = \sum _ {k = 1 } ^ { n } B _ {n,k } ( g _ {1} \dots g _ {n - k + 1 } ) .
12 KB (1,714 words) - 10:58, 29 May 2020