$#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
$#C+1 = 97 : ~/encyclopedia/old_files/data/P074/P.0704200 Power function
is an integer, the power function is a particular case of a [[Rational function|rational function]].
6 KB (860 words) - 17:32, 6 January 2024
''on power series''
If a [[Power series|power series]]
7 KB (1,065 words) - 09:52, 11 November 2023
...n element of an analytic function is the circular element in the form of a power series
f (z) = \sum _ { k=0 } ^ \infty c _ {k} ( z - a ) ^ {k}
4 KB (679 words) - 18:34, 5 April 2020
1 + \sum _ {n = 1 } ^ \infty
\sum _ {n = 0 } ^ \infty {
3 KB (504 words) - 14:33, 10 March 2024
...(see [[#References|[1]]]) in connection with questions of convergence of [[power series]]. If the series
...| < 1$ 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
2 KB (369 words) - 10:19, 16 March 2023
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
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 $
6 KB (908 words) - 06:04, 12 July 2022
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
2 KB (302 words) - 10:06, 15 April 2014
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}
2 KB (262 words) - 08:02, 6 June 2020
\sum _ {m , n = 1 } ^ \infty u _ {mn} ,
S _ {mn} = \sum _ {i = 1 } ^ { m }
7 KB (1,082 words) - 19:36, 5 June 2020
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 $
4 KB (680 words) - 19:35, 5 June 2020
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 }
6 KB (828 words) - 10:58, 29 May 2020
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 ,
6 KB (901 words) - 16:08, 1 April 2020
Abel's theorem on power series: If the power series
S ( z ) \ = \ \sum _ {k = 0} ^ \infty a _ {k} ( z - b ) ^ {k} ,
6 KB (894 words) - 06:14, 26 March 2023
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
4 KB (679 words) - 19:36, 5 June 2020
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 $
5 KB (717 words) - 08:28, 6 June 2020
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
3 KB (465 words) - 08:28, 6 June 2020
...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
\sum _ { k= 0} ^ { [ n / 2 ] } ( - 1 ) ^ {k}
The ultraspherical polynomials are the coefficients of the power series expansion of the generating function
3 KB (417 words) - 07:38, 26 February 2022
such that any non-zero ideal is generated by some power of the element $ \pi $;
of formal power series in one variable $ T $
5 KB (800 words) - 19:36, 5 June 2020
\sum _ {n = 0 } ^ \infty
where $ A(t) = \sum _ {k=0 } ^ \infty a _ {k} t ^ {k} $
11 KB (1,595 words) - 17:13, 2 January 2021
and radius $ | z ^ {0} - a | = ( \sum _ {\nu = 1 } ^ {n} | z _ \nu ^ {0} - a _ \nu | ^ {2} ) ^ {1/2} $
...of points of absolute convergence (i.e. the domain of convergence) of some power series in $ z _ {1} - a _ {1} \dots z _ {n} - a _ {n} $,
4 KB (586 words) - 08:10, 6 June 2020
An arithmetical [[Fraction|fraction]] with an integral power of 10 as its denominator. The following notation has been accepted for a de
is the sum of such a series, i.e.
2 KB (331 words) - 17:32, 5 June 2020
...tic rings, cf. [[Analytic ring|Analytic ring]]), and the ring of algebraic power series (i.e. series from $ k [[ X _ {1} \dots X _ {n} ]] $
in the topologies of the local rings) coincide. Thus, the ring of algebraic power series in $ X _ {1} \dots X _ {n} $
5 KB (787 words) - 22:10, 5 June 2020
A Cartesian power $ \mathbf R ^ {n} $
\langle x, y \rangle = \sum _ {i=1 } ^ { n }
2 KB (254 words) - 08:25, 4 March 2022
''transfinite number, power in the sense of Cantor, cardinality of a set $ A $
...et of real numbers, then $ \mathsf{card}(\mathbf{R}) = \mathfrak{c} $, the power of the continuum. The set $ 2^{A} $ of all subsets of $ A $ is not equivale
9 KB (1,402 words) - 11:57, 10 April 2018
N ^ {-1} \sum _ { j=1 } ^ { N-k } x _ {j} x _ {j+k} ,\ \
r _ {k} ^ {*} + \sum _ { j=1 } ^ { q } \beta _ {j} r _ {| k - j | } ^ {*} = 0 ,\ k = 1 \do
6 KB (747 words) - 18:22, 14 January 2021
\begin{equation}a_k=\sum^n_{i=1}c_ia_{k-i}\end{equation}
...sequence $\mathbb{a}=(a_k)$ over $F$ with the [[Formal power series|formal power series]]
9 KB (1,477 words) - 09:28, 17 May 2021
...By the complexity of a diagram of functional elements one understands the sum of the weights of all functional elements that are present in this diagram.
...ts are equal to 1; or 2) the weight of a threshold element is equal to the sum of the absolute values of all coefficients $ w _ {i} $(
12 KB (1,820 words) - 11:43, 26 March 2023
} \cdot \sum _ {n \leq x } \left | {f ( n ) } \right | ^ {q} \right ) ^ { {1 / q } } .
} \cdot \sum _ {n \leq x } f ( n )
5 KB (715 words) - 10:00, 9 January 2021
The permutation is chosen so that the power of Wilcoxon's test for the given alternative is highest. The statistical di
cf. [[Rank sum test|Rank sum test]]; [[Mann–Whitney test|Mann–Whitney test]]). See also [[Van der Wa
3 KB (399 words) - 10:34, 7 February 2021
\sum _ {k = 0 } ^ { {n } - 1 } e _ {k} u _ {k} ,
\sum _ {k = 0 } ^ { {n } - 1 } w ^ {(} k) e _ {k} = 0,
5 KB (724 words) - 22:11, 5 June 2020
with all coordinates equal, but not a power of an element — a concept that is not defined in an arbitrary vector spac
\sum _ {\begin{array}{c}
2 KB (316 words) - 22:10, 5 June 2020
...problem|Goldbach problem]]); and the representation of a given number as a sum of one prime number and two squares (cf. [[Hardy–Littlewood problem|Hardy
...atic results in this field were obtained in 1748 by L. Euler, who employed power series to study the partition of integers into positive summands; in partic
10 KB (1,609 words) - 13:03, 8 February 2020
\begin{equation*} S = \sum _ { n \in A } e ^ { 2 \pi i f ( n ) }, \end{equation*}
...$f$ is a real-valued function (cf. also [[Trigonometric sum|Trigonometric sum]]). The basic problem is to show, under suitable circumstances, that $S = o
10 KB (1,528 words) - 07:11, 24 January 2024
\sum _ { i = 1 } ^ { k } \
\sum _ {\nu = 1 } ^ { {\alpha _ i } }
9 KB (1,382 words) - 08:27, 6 June 2020
...$ is summable by the Cesàro method of order $k$, or $(C,k)$-summable, with sum $S$ if
...eries is summable by the method $(C,k)$, then it is summable with the same sum by the method $(C,k')$ for $k' > k > -1$. This property does not hold for $
3 KB (519 words) - 15:44, 7 May 2012
$ \sum m _ {i} = | m | \geq 2 $
...case one says that one has [[Small denominators|small denominators]]), the power series $ x = y + \dots $
7 KB (1,085 words) - 08:13, 6 June 2020
\sum _ {r = 0 } ^ \infty
...on-central "chi-squared" distribution appears as the distribution of the sum of squares of independent random variables $ X _ {1} \dots X _ {n} $
4 KB (556 words) - 18:53, 24 January 2024
...bo looked for an algebra that is $8$-dimensional over the complex numbers, power-associative and, unlike the [[octonion]] algebra, has the [[Lie algebra|Lie
\begin{equation*} e _ { j } * e _ { k } = \sum _ { l = 1 } ^ { 8 } ( \sqrt { 3 } d _ { j k l } - f _ { j k l } ) e _ { l }
7 KB (944 words) - 10:21, 8 March 2021
...pediaofmath.org/legacyimages/b/b015/b015210/b01521051.png" />; the ordinal sum is defined uniquely up to an isomorphism, by specifying the components and
10 KB (1,447 words) - 17:26, 7 February 2011
and the Chern polynomial as an element of the formal power series ring $ H ^ {**} ( \mathop{\rm BU} _ {n} ) [ [ t ] ] $.
in other words $ c _ {k} ( \xi \oplus \eta ) = \sum _ {i} c _ {i} ( \xi ) c _ {k-i} ( \eta ) $
13 KB (1,910 words) - 19:40, 7 January 2024
''of a power series
\sum _ {k = 0 } ^ \infty
4 KB (602 words) - 19:35, 5 June 2020
\sum _ { n=0 } ^ \infty a _ {n} $$
is summable by Abel's method to a sum $ S $
11 KB (1,603 words) - 10:19, 7 May 2021
associated with a graded ring $ R = \sum _ {n=0} ^ \infty R _ {n} $(
does not contain $ \sum _ {n=1} ^ \infty R _ {n} $.
3 KB (468 words) - 09:09, 6 January 2024
denotes the algebra of all power series $ \sum a _ \alpha z _ {1} ^ {\alpha _ {1} } \dots z _ {n} ^ {\alpha _ {n} } $
$ | \alpha | = \sum \alpha _ {i} $).
8 KB (1,161 words) - 08:25, 6 June 2020
A power series which offers a complete solution to the problem of local inversion o
\sum _ {n = 1 } ^ \infty {
5 KB (731 words) - 06:29, 30 May 2020
represented by a power series
\sum _ {v = 0 } ^ \infty
5 KB (790 words) - 08:29, 20 January 2024
is represented as the sum of a convergent power series (also denoted by $ f _ {i} $
...expressed by the following properties of these series, regarded as formal power series in $ 2n $
10 KB (1,553 words) - 22:16, 5 June 2020
is the [[pointwise operation|pointwise]] sum and
...ormal Dirichlet series over $\mathbb{C}$ is isomorphic to a ring of formal power series in countably many variables.
2 KB (358 words) - 17:25, 11 November 2023
...residue formula one usually understands an integral representation for the sum of the values of a holomorphic function at all the zeros of a holomorphic m
...n - 1 ) ! } { ( 2 \pi i ) ^ {n } } \int _ { \partial D } \varphi \frac { \sum _ { k = 1 } ^ { n } ( - 1 ) ^ { k - 1 } w _ { k } d w [ k ] \wedge d f } {
7 KB (1,090 words) - 10:25, 16 March 2024
...[[Fabry theorem|Fabry theorem]] on gaps; [[Lacunary power series|Lacunary power series]].
\sum _ { j= 1} ^ { k } L _ {ij} u _ {j} ,\ 1 \leq i \leq k ,
6 KB (898 words) - 14:30, 8 January 2022
...number and a left-part number of the other, whereas the right part of the sum consists of the sums of one number and a right part of the other.
...presentations of the surreal numbers which can be given in terms of formal power series over an index "set" which is isomorphic to the surreal numbers the
8 KB (1,226 words) - 16:11, 3 July 2016
in the form of a power series $ \Pi ( z _ {1} ; r ) = \sum _ {\nu = 0 } ^ \infty c _ \nu ( z - z _ {1} ) ^ \nu $
3 KB (399 words) - 22:17, 5 June 2020
...lane (except, possibly, at the point at infinity). It can be expanded in a power series
\sum _ {k = 0 } ^ \infty a _ {k} z ^ {k} ,\ \
8 KB (1,251 words) - 20:13, 10 January 2021
X = \sum _ {i = 1 } ^ { m }
...B.L. van der Waerden, "Order tests for the two-sample problem and their power" ''Proc. Kon. Nederl. Akad. Wetensch. A'' , '''55''' (1952) pp. 453–45
3 KB (401 words) - 08:27, 6 June 2020
\sum _ { n=1 } ^ \infty a _ {n} e ^ {- \lambda _ {n} s } ,
\sum _ { n=1 } ^ \infty
11 KB (1,638 words) - 11:32, 16 April 2023
...are [[elementary symmetric function]]s and $p_k(x_1,x_2,\ldots)$ are power sum symmetric functions. The algebraic structure underlying both identities is
...ence $(x_1,x_2,\ldots)$, then the $(k+1)$-st term in $P(x^n)$ is the power sum symmetric function $x_1^n+\cdots+x_k^n$ and the $k$-th term in $P(xP(\ldots
6 KB (960 words) - 07:40, 18 November 2023
or is a power of the characteristic $ p $
[ L:K ] \geq \sum _ {i = 1 } ^ { m } e ( w _ {i} \mid v ) \cdot f ( w _ {i} \mid w ) .
5 KB (827 words) - 17:32, 5 June 2020
the coefficients of these series are convergent power series in $ q $,
\sum _{n=1} ^ \infty \left [ 2 ^ {n+1}
3 KB (379 words) - 16:31, 6 January 2024
and is called the $ p $-th Pontryagin power $ {\mathcal P} _ {p} $.
\left ( \sum _ { i= 1} ^ { p- 1 }
6 KB (870 words) - 10:04, 11 July 2022
...e function $\phi(m)/m$ is a strongly multiplicative arithmetic function, a power function $m^k$ is a totally multiplicative arithmetic function.
3 KB (419 words) - 20:15, 19 November 2017
f ( z ) = \sum _ { k=0 } ^ \infty c _ {k} ( z - a ) ^ {k}
as a power series along all possible rays from the centre $ a $
5 KB (749 words) - 19:11, 19 June 2020
th exterior power of the space $ V $.
th exterior power $ \wedge ^ {r} M $,
7 KB (1,013 words) - 19:38, 5 June 2020
$ \sum _ {i} \pi _ {i} = 1 $,
where $ a _ {i} = \sum _ {j} n _ {i.ij } $
5 KB (733 words) - 06:29, 30 May 2020
...'(0,0)$ is a natural number) has a unique solution in the form of a formal power series:
...solution $y=\xi(x)$ of equation \eqref{1} cannot be proved by any partial sum of the Taylor series of $f$ (cf. [[#References|[2]]], [[#References|[3]]]).
2 KB (376 words) - 17:27, 14 February 2020
...s an injective object, and each injective object is isomorphic to a direct sum of indecomposable injective objects; this representation is moreover unique
...pos]] an object is injective if and only if it occurs as a retract of some power-object, and injective objects are used in the study of the associated sheaf
4 KB (643 words) - 22:12, 5 June 2020
Taking the direct sum of the internal space $ S _ {k} ( U) $
and the Hilbert sum of the central spaces there results a triplet
8 KB (1,190 words) - 08:16, 20 January 2024