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Abel's Theorem
In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel. Theorem Let the Taylor series G (x) = \sum_^\infty a_k x^k be a power series with real coefficients a_k with radius of convergence 1. Suppose that the series \sum_^\infty a_k converges. Then G(x) is continuous from the left at x = 1, that is, \lim_ G(x) = \sum_^\infty a_k. The same theorem holds for complex power series G(z) = \sum_^\infty a_k z^k, provided that z \to 1 entirely within a single ''Stolz sector'', that is, a region of the open unit disk where , 1-z, \leq M(1-, z, ) for some fixed finite M > 1. Without this restriction, the limit may fail to exist: for example, the power series \sum_ \frac n converges to 0 at z = 1, but is unbounded near any point of the form e^, so the value at z = 1 is not the limit as z tends to 1 in the whole open disk. Note that G(z) is continuous on the real clo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Series
In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a constant. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations, ''c'' (the ''center'' of the series) is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form \sum_^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + \dots. Beyond their role in mathematical analysis, power series also occur in combinatorics as generating functions (a kind of formal power series) and in electronic engineering (under the name of the Z-transform). The familiar decimal notation for real numbers can also be viewed as an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be ''finite'', as in these examples, or ''infi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Complex Analysis
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Real Analysis
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divergent Series
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. However, convergence is a stronger condition: not all series whose terms approach zero converge. A counterexample is the harmonic series :1 + \frac + \frac + \frac + \frac + \cdots =\sum_^\infty\frac. The divergence of the harmonic series was proven by the medieval mathematician Nicole Oresme. In specialized mathematical contexts, values can be objectively assigned to certain series whose sequences of partial sums diverge, in order to make meaning of the divergence of the series. A ''summability method'' or ''summation method'' is a partial function from the set of series to values. For example, Cesàro summation assigns Grandi's divergent ser ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tauberian Theorem
In mathematics, Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber. The original examples are Abel's theorem showing that if a series converges to some limit then its Abel sum is the same limit, and Tauber's theorem showing that if the Abel sum of a series exists and the coefficients are sufficiently small (o(1/''n'')) then the series converges to the Abel sum. More general Abelian and Tauberian theorems give similar results for more general summation methods. There is not yet a clear distinction between Abelian and Tauberian theorems, and no generally accepted definition of what these terms mean. Often, a theorem is called "Abelian" if it shows that some summation method gives the usual sum for convergent series, and is called "Tauberian" if it gives conditions for a series summable by some method that allows it to be summable in the usual sense. In the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Summation By Parts
In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformation, named after Niels Henrik Abel who introduced it in 1826. Statement Suppose \ and \ are two sequences. Then, :\sum_^n f_k(g_-g_k) = \left(f_g_ - f_m g_m\right) - \sum_^n g_(f_- f_). Using the forward difference operator \Delta, it can be stated more succinctly as :\sum_^n f_k\Delta g_k = \left(f_ g_ - f_m g_m\right) - \sum_^ g_\Delta f_k, Summation by parts is an analogue to integration by parts: :\int f\,dg = f g - \int g\,df, or to Abel's summation formula: :\sum_^n f(k)(g_-g_)= \left(f(n)g_ - f(m) g_m\right) - \int_^n g_ f'(t) dt. An alternative statement is :f_n g_n - f_m g_m = \sum_^ f_k\Delta g_k + \sum_^ g_k\Delta f_k + \sum_^ \Delta f_k \Delta g_k which is analogous to the integration by parts formula for semimartingales. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galton–Watson Process
The Galton–Watson process is a branching stochastic process arising from Francis Galton's statistical investigation of the extinction of family names. The process models family names as patrilineal (passed from father to son), while offspring are randomly either male or female, and names become extinct if the family name line dies out (holders of the family name die without male descendants). This is an accurate description of Y chromosome transmission in genetics, and the model is thus useful for understanding human Y-chromosome DNA haplogroups. Likewise, since mitochondria are inherited only on the maternal line, the same mathematical formulation describes transmission of mitochondria. The formula is of limited usefulness in understanding actual family name distributions, since in practice family names change for many other reasons, and dying out of name line is only one factor. History There was concern amongst the Victorians that aristocratic surnames were becoming e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability-generating Function
In probability theory, the probability generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable. Probability generating functions are often employed for their succinct description of the sequence of probabilities Pr(''X'' = ''i'') in the probability mass function for a random variable ''X'', and to make available the well-developed theory of power series with non-negative coefficients. Definition Univariate case If ''X'' is a discrete random variable taking values in the non-negative integers , then the ''probability generating function'' of ''X'' is defined as http://www.am.qub.ac.uk/users/g.gribakin/sor/Chap3.pdf :G(z) = \operatorname (z^X) = \sum_^p(x)z^x, where ''p'' is the probability mass function of ''X''. Note that the subscripted notations ''G''''X'' and ''pX'' are often used to emphasize that these pertain to a particular random variable ''X'', and to its ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series, the ''formal'' power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series; definitions and examples are given below. Every sequence in principle has a generating function of each type (except ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation can be restricted to the affine algebraic plane curve of equation . These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered. More generally, an algebraic curve is an algebraic variety of dimension one. Equivalently, an algebraic curve is an algebraic variety that is birationally equivalent to an algebraic plane curve. If the curve is contained in an affine space or a projective space, one can take a projection for such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Series
In mathematics, the binomial series is a generalization of the polynomial that comes from a binomial formula expression like (1+x)^n for a nonnegative integer n. Specifically, the binomial series is the Taylor series for the function f(x)=(1+x)^ centered at x = 0, where \alpha \in \Complex and , x, 0 and diverges to +\infty if \operatorname\alpha<0. If , then converges if and only if the sequence converges , which is certainly true if but false if : in the latter case the sequence is dense , due to the fact that diverges and converges to zero). Summation of the binomial series The usual argument to compute the sum of the binomial series goes as follows.[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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