Персона: Костин, Андрей Борисович
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Computation of Sums of Natural Powers of the Inverses of Roots of an Equation Connected with a Spectral Problem
© 2019, Springer Science+Business Media, LLC, part of Springer Nature.We consider equations arising in the oblique derivative spectral problem of the formazJv′(z)+bJv(z)=0,z∈ℂ, where ν, a, b ∈ ℂ are parameters such that |a|+|b| > 0 and Jν(z) is the Bessel function. For roots of the equation we prove summation relations. The results obtained agree with the theory of Rayleigh sums which are calculated in terms of zeros of the Bessel functions.
Enveloping of Riemann’s Zeta Function Values and Curious Approximation
In this note, by the example of approximate calculation of $ pi
Enveloping of the Values of an Analytic Function Related to the Number e
Asymptotic Behavior of Remainders of Special Number Series
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.We consider a one-parameter family of number series involving the generalized harmonic series and study asymptotic properties of the remainders. Using R(Np)≡∑n=N∞1/np as an example, we describe the typical obtained results: we obtain the integral representation, find the complete asymptotic expansion with respect to the parameter 2N − 1 as N →∞, and prove that R(N, p) is enveloped by its asymptotic series. The possibilities of the proposed approach are demonstrated by the problem of exact two-sided estimates for the central binomial coefficient.
ON TAYLOR COEFFICIENTS OF ANALYTIC FUNCTION RELATED WITH EULER NUMBER
We consider a classical construction of second remarkable limit. We pose a question on asymptotically sharp description of the character of such approximation of the number e. In view of this we need the information on behavior of the coefficients in the power expansion for the function (Formula Presented) converging in the interval (Formula Presented). We obtain a recurrent rule regulating the forming of the mentioned coefficients. We show that the coefficients form a sign-alternating sequence of rational numbers (−1)nan, where n ∈ N ∪ {0} and a0 = 1, the absolute values of which strictly decay. On the base of the Faá di Bruno formula for the derivatives of a composed function we propose a combinatorial way of calculating the numbers an as n ∈ N. The original function f(x) is the restriction of the function f(z) on the real ray x andgt; −1 having the same Taylor coefficients and being analytic in the complex plane C with the cut along (−∞, −1]. By the methods of the complex analysis we obtain an integral representation for an for each value of the parameter n ∈ N. We prove that an → 1/e as n → ∞ and find the convergence rate of the difference an − 1/e to zero. We also discuss the issue on choosing the contour in the integral Cauchy formula for calculating the Taylor coefficients (−1)nan of the function f(z). We find the exact values of arising in calculations special improper integrals. The results of the made study allows us to give a series of general two-sided estimates for the deviation e−(1+x) 1/x consistent with the asymptotics s of f(x) as x → 0. We discuss the possibilities of applying the obtained statements © Kostin A.B., Sherstyukov V.B. 2022
INTEGRAL REPRESENTATIONS OF QUANTITIES ASSOCIATED WITH GAMMA FUNCTION
We study a series of issues related with integral representations of Gamma functions and its quotients. The base of our study is two classical results in the theory of functions. One of them is a well-known first Binet formula, the other is a less known Malmsten formula. These special formulae express the values of the Gamma function in an open right half-plane via corresponding improper integrals. In this work we show that both results can be extended to the imaginary axis except for the point z = 0. Under such extension we apply various methods of real and complex analysis. In particular, we obtain integral representations for the argument of the complex quantity being the value of the Gamma function in a pure imaginary point. On the base of the mentioned Malmsten formula at the points z not equal 0 in the closed right half-plane, we provide a detailed derivation of the integral representation for a special quotient expressed via the Gamma function: D(z) Gamma(z + 1/2)/Gamma(z + 1). This fact on the positive semi-axis was mentioned without the proof in a small note by Dusan Slavic in 1975. In the same work he provided two-sided estimates for the quantity D(x) as x > 0 and at the natural points D(x) coincided with the normalized central binomial coefficient. These estimates mean that D(x) is enveloped on the positive semi-axis by its asymptotic series. In the present paper we briefly discuss the issue on the presence of this property on the asymptotic series D(z) in a closed angle vertical bar arg z vertical bar <= pi/4 with a punctured vertex. By the new formula representing D(z) on the imaginary axis we obtain explicit expressions for the quantity vertical bar D(iy)vertical bar(2) and for the set Arg D(iy) as y > 0. We indicate a way of proving the second Binet formula employing the technique of simple fractions.
Basis Property of the System of Root Functions of the Oblique Derivative Problem for the Laplace Operator in a Disk
© 2019, Pleiades Publishing, Ltd.We study the spectral oblique derivative problem for the Laplace operator in a disk D. The asymptotic properties of the eigenvalues are established, and the basis property with parentheses in the space L2(D) is proved for the system of root functions of the above problem.
Application of the Hausdorff Metric in Model Problems with Discontinuous Functions in Boundary Conditions
Using an example of the Cauchy problem for the one-dimensional heat equation, we study the approximation of the solution to the initial condition in the Hausdorff metric. The simplest discontinuous function u0(x) = sgn x is taken for the initial condition. Based on the asymptotic behavior of the Lambert W function and its modification, we obtain a two-sided estimate and an asymptotics for the Hausdorff distance between the solution given by the Poisson formula and the function u0(x). Similar results are obtained for a similar model problem for the Laplace equation in the upper half-plane.