AN ASYMPTOTIC EXPANSION FORMULA FOR BERNSTEIN POLYNOMIALS ON A TRIANGLE

In this paper,an asymptotic expansion formula for approximation to a continuous function by Bernsteinpolynomials on a triangle is obtained.     

A Laurent series expansion formula and its applications

This paper gives a certain Laurent series expansion for a generalized Rodrigues type formula. The main result finds many applications which are enumerated briefly.     

An asymptotic expansion for the incomplete beta function

A new asymptotic expansion is derived for the incomplete beta function , which is suitable for large , small and . This expansion is of the form

where is the incomplete Gamma function ratio and . This form has some advantages over previous asymptotic expansions in this region in which depends on as well as on and .

     

On the asymptotic expansion of Bergman kernel

We study the asymptotics of the Bergman kernel and the heat kernel of the ${\mathrm{spin}}^c$ Dirac operator on high tensor powers of a line bundle. To cite this article: X. Dai et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

An asymptotic expansion related to the Dickman function

We prove a conjecture of Broadhurst on asymptotic expansions of certain polylogarithm type functions related to the Dickman function.     

An asymptotic expansion in wavelet analysis and its application to accurate numerical wavelet decomposition

We consider a dilation operatorT admitting a scaling function with compact support as fixed point. It is shown that the adjoint operatorT*admits a sequence of polynomial eigenfunctions and that a smooth functionf admits an expansion in these eigenfunctions, which reveals the asymptotic behavior ofT* forn.Due to this asymptotic expansion, an extrapolation technique can be applied for the accurate numerical computation of the integrals appearing in the wavelet decomposition of a smooth function. This extrapolation technique fits well in a multiresolution scheme.     

A systematic formula for the asymptotic expansion of singular integrals

Asymptotic theories like the lifting-line, the slender body or the slender ship lead to lineintegrals with singular kernels. Sometimes these integrals are improper, that is to say that they are defined only by their Finite Part. To find asymptotic expansions of these integrals, the Matched Asymptotic Expansion Method is widely used along with other more specific methods depending on the kernel type. The first method is laborious and not systematic, and the other methods are sometimes too much specific to treat general cases. Moreover, all of them are not well adapted to deal with Finite Part integrals.Here, a new method is proposed to avoid the previous difficulties. This method is systematic for homogeneous kernels and gives approximations up to any order, as long as the derivative of the weight function exists at this given order. Moreover the occurrence of logarithmic terms in the expansion is explained and easily predictable. An elliptic integral and the classical lifting-line theory are treated to illustrate the ease of this method.

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Résumé Les théories asymptotiques telles que la ligne portante, le corps élancé ou le navire de grand allongement conduisent à des intégrales curvilignes à noyaux singuliers. Parfois, ces intégrales sont impropres c'est à dire qu'elles sont définies en Parties Finies. Différentes méthodes ont été mises au point pour trouver les développements asymptotiques de ces intégrales. Généralement elles dépendent fortement de la nature du noyau, et c'est finalement la méthode des développements raccordés qui est utilisées quand le noyau est trop compliqué. Cependant, cette méthode est laborieuse et comme les précèdentes non adaptée aux intégrales défines par leur Partie Finie.Une nouvelle méthode est proposée pour surmonter ces difficultés. Cette méthode est systématique pour les noyaux homogènes et donne les approximations à tout ordre pourvu que les dérivées de la fonction poids existent jusqu'à cet ordre. De plus la présence de termes logarithmiques dans le développement est expliquée et aisément prédictible.Une intégrale elliptique, ainsi que la fameuse théorie de la ligne portante sont traités pour illustrer les possibilités de la méthode.

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Nomenclature D domain of integration - f(x) weight function - FP Finite Part - h(x) weight function - I ,I _o bounded intervals - j, J integers - K(x, ) singular kernel - L, L integers - M integer defining the approximation order - P _k (x) Legendre polynomial - R set of real numbers - R(ß) equals 1 if is an integer and 0 if not - R _{f, J} ,R _{K, L} remainders of Taylor developments - S () equals either 1 or the sign function:sgn() - t, u, v, x variable of integration - , real numbers - homogeneity order of the kernel - F () Euler's integral (gamma function) - small parameter - [.] integer part of     

The asymptotic expansion for n! and the Lagrange inversion formula

We obtain an explicit simple formula for the coefficients of the asymptotic expansion for the factorial of a natural number, $$n!=n^n\sqrt{2\pi n}\mbox{e}^{-n}\biggl\{1+\frac{a_1}{n}+\frac{a_2}{n^2}+\frac{a_3}{n^3}+\cdots\biggr\},$$ in terms of derivatives of powers of an elementary function that we call normalized left truncated exponential function. The unique explicit expression for the a _k that appears to be known is that of Comtet in (Advanced Combinatorics, Reidel, 1974), which is given in terms of sums of associated Stirling numbers of the first kind. By considering the bivariate generating function of the associated Stirling numbers of the second kind, another expression for the coefficients in terms of them follows also from our analysis. Comparison with Comtet??s expression yields an identity which is somehow unexpected if considering the combinatorial meaning of the terms. It suggests by analogy another possible formula for the coefficients, in terms of a normalized left truncated logarithm, that in fact proves to be true. The resulting coefficients, as well as the first ones are identified via the Lagrange inversion formula as the odd coefficients of the inverse of a pair of formal series. This in particular leads to the identification of a couple of simple implicit equations, which permits us to obtain also some recurrences related to the a _k ??s.     

New asymptotic expansion for the Gamma function

Using a series transformation, the Stirling-De Moivre asymptotic series approximation to the Gamma function is converted into a new one with better convergence properties. The new formula is being compared with those of Stirling, Laplace, and Ramanujan for real arguments greater than 0.5 and turns out to be, for equal number of “correction” terms, numerically superior to all of them. As a side benefit, a closed-form approximation has turned up during the analysis which is about as good as 3rd order Stirling’s (maximum relative error smaller than 1e − 10 for real arguments greater or equal to 24).     

An asymptotic series for an integral

The Ramanujan Journal - Let $$f(z)=\sum _{n=0}^{\infty }a_{n}{\mathbf {e}}(nz),g(z)=\sum _{n=0}^{\infty }b_{n}{\mathbf {e}}(nz)\ ({\mathbf {e}}(z)=e^{2\pi \sqrt{-1}z})$$ be holomorphic modular...     

An asymptotic formula for the t-core partition function and a conjecture of Stanton

For a positive integer t, a partition is said to be a t-core if each of the hook numbers from its Ferrers-Young diagram is not a multiple of t. In 1996, Granville and Ono proved the t-core partition conjecture, that at(n), the number of t-core partitions of n, is positive for every nonnegative integer n as long as t?4. As part of their proof, they showed that if p?5 is prime, the generating function for ap(n) is essentially a multiple of an explicit Eisenstein Series together with a cusp form. This representation of the generating function leads to an asymptotic formula for ap(n) involving L-functions and divisor functions. In 1999, Stanton conjectured that for t?4 and n?t+1, at(n)?at+1(n). Here we prove a weaker form of this conjecture, that for t?4 and n sufficiently large, at(n)?at+1(n). Along the way, we obtain an asymptotic formula for at(n) which, in the cases where t is coprime to 6, is a generalization of the formula which follows from the work of Granville and Ono when t=p?5 is prime.     

The t-coefficient method II: A new series expansion formula of theta function products and its implications

By means of Jacobi?s triple product identity and the t  -coefficient method, we establish a general series expansion formula with five free parameters for the product of arbitrary two Jacobi theta functions. It embodies the triple, quintuple, sextuple and septuple theta function product identities and the generalized Schröter formula. As further applications, we also set up a series expansion formula for the product of three theta functions. It not only generalizes Ewell?s and Chen–Chen–Huang?s octuple product identities, but also contains three cubic theta function identities due to Farkas–Kra and Ramanujan respectively and the Macdonald identity for the root system _A2$A_2$ as special cases. In the meantime, many other new identities including a new short expression of the triple theta series of Andrews are also presented.     

Eisenstein series and an asymptotic for the K-Bessel function

The Ramanujan Journal - We produce an estimate for the K-Bessel function $$K_{r + i t}(y)$$ with positive, real argument y and of large complex order $$r+it$$ where r is bounded and $$t = y sin...     

An asymptotic formula for the ranks of homotopy groups

Let X be a finite simply connected CW complex of dimension n. The loop space homology H_∗(ΩX;Q) is the universal enveloping algebra of a graded Lie algebra LX isomorphic with π∗−1(X)⊗Q. Let QX⊂LX be a minimal generating subspace, and set .Theorem: If dimLX=∞ and , then

     

An asymptotic formula for the mean value of the V. I. Arnold function

An asymptotic formula for the mean value of the V. I. Arnold function A(n) = \(\tfrac{{\sigma (n)}}{{\tau (n)}}\) is obtained, here σ(n) = \(\mathop \Sigma \limits_{d|n} \) d is the sum of all divisors of the number n, τ (n) = \(\mathop \Sigma \limits_{d|n} \) 1 is their quantity.     

A uniform asymptotic expansion for the incomplete gamma function

We describe a new uniform asymptotic expansion for the incomplete gamma function Γ(a,z) valid for large values of z. This expansion contains a complementary error function of an argument measuring transition across the point z=a (which is different from that in the well-known uniform expansion for large a of Temme), with easily computable coefficients that do not involve a removable singularity at z=a. Our expansion is, however, valid in a smaller domain of the parameters than that of Temme. Numerical examples are given to illustrate the accuracy of the expansion.