|
Abel's Summation Formula
In mathematics, Abel's summation formula, introduced by Niels Henrik Abel, is intensively used in analytic number theory and the study of special functions to compute series. Formula Let (a_n)_^\infty be a sequence of real or complex numbers. Define the partial sum function A by :A(t) = \sum_ a_n for any real number t. Fix real numbers x . Then: :\sum_ a_n\phi(n) = A(y)\phi(y) - A(x)\phi(x) - \int_x^y A(u)\phi'(u)\,du. The formula is derived by applying integration by parts for a Riemann–Stieltjes integral to the functions A and \phi. Variations Taking the left endpoint to be -1 gives the formula :\sum_ a_n\phi(n) = A(x)\phi(x) - \int_0^x A(u)\phi'(u)\,du. If the sequence (a_n) is indexed starting at n = 1, then we may formally define a_0 = 0. The previous formula becomes :\sum_ a_n\phi(n) = A(x)\phi(x) - \int_1^x A(u)\phi'(u)\,du. A common way to apply Abel's summation formula is to take the limit of one of these formulas as x \to \infty. The resulting formulas are :\b ... [...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] |
Riemann–Stieltjes Integral
In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was first published in 1894 by Stieltjes. It serves as an instructive and useful precursor of the Lebesgue integral, and an invaluable tool in unifying equivalent forms of statistical theorems that apply to discrete and continuous probability. Formal definition The Riemann–Stieltjes integral of a real-valued function f of a real variable on the interval ,b/math> with respect to another real-to-real function g is denoted by :\int_^b f(x) \, \mathrmg(x). Its definition uses a sequence of partitions P of the interval ,b/math> :P=\. The integral, then, is defined to be the limit, as the mesh (the length of the longest subinterval) of the partitions approaches 0 , of the approximating sum :S(P,f,g) = \sum_^ f(c_i)\left g(x_) - g(x_) \right/math> where c_i is in the i-th subinterval _,x_/math> ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Undergraduate Texts In Mathematics
Undergraduate Texts in Mathematics (UTM) () is a series of undergraduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are small yellow books of a standard size. The books in this series tend to be written at a more elementary level than the similar Graduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level. There is no Springer-Verlag numbering of the books like in the Graduate Texts in Mathematics series. The books are numbered here by year of publication. List of books # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integration By Parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be thought of as an integral version of the product rule of differentiation; it is indeed derived using the product rule. The integration by parts formula states: \begin \int_a^b u(x) v'(x) \, dx & = \Big (x) v(x)\Biga^b - \int_a^b u'(x) v(x) \, dx\\ & = u(b) v(b) - u(a) v(a) - \int_a^b u'(x) v(x) \, dx. \end Or, letting u = u(x) and du = u'(x) \,dx while v = v(x) and dv = v'(x) \, dx, the formula can be written more compactly: \int u \, dv \ =\ uv - \int v \, du. The former expression is written as a definite integral and the latter is written as an indefinite ... [...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] |
Mertens Function
In number theory, the Mertens function is defined for all positive integers ''n'' as : M(n) = \sum_^n \mu(k), where \mu(k) is the Möbius function. The function is named in honour of Franz Mertens. This definition can be extended to positive real numbers as follows: : M(x) = M(\lfloor x \rfloor). Less formally, M(x) is the count of square-free integers up to ''x'' that have an even number of prime factors, minus the count of those that have an odd number. The first 143 ''M''(''n'') values are The Mertens function slowly grows in positive and negative directions both on average and in peak value, oscillating in an apparently chaotic manner passing through zero when ''n'' has the values :2, 39, 40, 58, 65, 93, 101, 145, 149, 150, 159, 160, 163, 164, 166, 214, 231, 232, 235, 236, 238, 254, 329, 331, 332, 333, 353, 355, 356, 358, 362, 363, 364, 366, 393, 401, 403, 404, 405, 407, 408, 413, 414, 419, 420, 422, 423, 424, 425, 427, 428, ... . Because the Möbius function only ta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius Function
The Möbius function \mu(n) is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most often appears as part of its namesake the Möbius inversion formula. Following work of Gian-Carlo Rota in the 1960s, generalizations of the Möbius function were introduced into combinatorics, and are similarly denoted \mu(x). Definition The Möbius function is defined by :\mu(n) = \begin 1 & \text n = 1 \\ (-1)^k & \text n \text k \text \\ 0 & \text n \text > 1 \end The Möbius function can alternatively be represented as : \mu(n) = \delta_ \lambda(n), where \delta_ is the Kronecker delta, \lambda(n) is the Liouville function, Prime omega function, \omega(n) is the number of distinct prime divisors of n, and Prime omega function, \Omega(n) is the number of prime factors of n, counted with multiplicity. Another characterization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Series
In mathematics, a Dirichlet series is any series of the form \sum_^\infty \frac, where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in analytic number theory. The most usually seen definition of the Riemann zeta function is a Dirichlet series, as are the Dirichlet L-functions. Specifically, the Riemann zeta function ''ζ(s)'' is the Dirichlet series of the constant unit function ''u(n)'', namely: \zeta(s) = \sum_^\infty \frac = \sum_^\infty \frac = D(u, s), where ''D(u, s)'' denotes the Dirichlet series of ''u(n)''. It is conjectured that the Selberg class of series obeys the generalized Riemann hypothesis. The series is named in honor of Peter Gustav Lejeune Dirichlet. Combinatorial importance Dirichlet series can be used as generating series for counting weighted sets of objects with respect to a weight which is combined multiplicatively when taking Cartesian product ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Residue (complex Analysis)
In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities. (More generally, residues can be calculated for any function f\colon \mathbb \setminus \_k \rightarrow \mathbb that is holomorphic except at the discrete points ''k'', even if some of them are essential singularities.) Residues can be computed quite easily and, once known, allow the determination of general contour integrals via the residue theorem. Definition The residue of a meromorphic function f at an isolated singularity a, often denoted \operatorname(f,a), \operatorname_a(f), \mathop_f(z) or \mathop_f(z), is the unique value R such that f(z)- R/(z-a) has an analytic antiderivative in a punctured disk 0<\vert z-a\vert<\delta. Alternatively, residues can be calculated by finding |
Pole (complex Analysis)
In complex analysis (a branch of mathematics), a pole is a certain type of singularity (mathematics), singularity of a complex-valued function of a complex number, complex variable. It is the simplest type of non-removable singularity of such a function (see essential singularity). Technically, a point is a pole of a function if it is a zero of a function, zero of the function and is holomorphic function, holomorphic (i.e. complex differentiable) in some neighbourhood (mathematics), neighbourhood of . A function is meromorphic function, meromorphic in an open set if for every point of there is a neighborhood of in which at least one of and is holomorphic. If is meromorphic in , then a zero of is a pole of , and a pole of is a zero of . This induces a duality between ''zeros'' and ''poles'', that is fundamental for the study of meromorphic functions. For example, if a function is meromorphic on the whole complex plane plus the point at infinity, then the sum of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Zeta Function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic continuation elsewhere. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics. Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers. This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that many mathematicians consider th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Number
In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers: H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac. Starting from , the sequence of harmonic numbers begins: 1, \frac, \frac, \frac, \frac, \dots Harmonic numbers are related to the harmonic mean in that the -th harmonic number is also times the reciprocal of the harmonic mean of the first positive integers. Harmonic numbers have been studied since antiquity and are important in various branches of number theory. They are sometimes loosely termed harmonic series, are closely related to the Riemann zeta function, and appear in the expressions of various special functions. The harmonic numbers roughly approximate the natural logarithm function and thus the associated harmonic series grows without limit, albeit slowly. In 1737, Leonhard Euler used the divergence of the harmonic series to provide a new proof of the infinity of prime numbers. His work was extended into the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
