1,093 Results Found

In this paper, we consider the simultaneous approximation of tuples of analytic functions by tuples of shifts of Lerch zeta-functions with arbitrary parameters. We prove that there exists a closed set of tuples of functions analytic in the right-hand...

For $j=1,\dots ,r$, let ${\mathcal{P}}_j$ be a system of generalized prime numbers, ${\mathcal{N}}_{{\mathcal{P}}_j}$ the corresponding system of generalized integers, and ${\zeta }_{{\mathcal{P}}_j}\left(s\right)$, $s=\sigma +it$, the Beurling zeta-function. In the paper, we consider simultaneous approximation of a collection...

The Lerch zeta function is defined by a Dirichlet series depending on two fixed parameters. In the paper, we consider the approximation of analytic functions by discrete shifts of the Lerch zeta function, and we prove that, for arbitrary parameters a...

For $j=1,\dots ,r$, ${\alpha }_j\in (0,1)$ are such that the set $\{\log (m+{\alpha }_j):m\in {\mathbb{N}}_0,j=1,\dots ,r\}$ is linearly independent over $\mathbb{Q}$. In the paper, we consider simultaneous approximation of a tuple of analytic functions by shifts $(\zeta (s+i\tau ,$...

We introduce a modern methodology for constructing global analytical approximations of special functions over their entire domains. By integrating the traditional method of matching asymptotic expansions—enhanced with Padé approximants&m...

We found analytic approximations for the Bessel function of the first kind $J_{\nu }\left(x\right)$, valid for any real value of x and any value of $\nu$ in the interval (−1/2, 3/2). The present approximation is exact for $\nu =-1/2$, $\nu =1/2$, and $\nu =3/2$...

In this contribution, we define a new operator sequence which contains analytic functions. Using approximation techniques found by Korovkin, some results are derived. Moreover, a generalization of this operator sequence called Kantorovich type genera...

A global, uniform two-variable analytic approximation for the modified Bessel function $I_{\nu }\left(x\right)$ is presented, valid for all real x and for orders $-1/2\le \nu \le 1/4$. The approximation is constructed using a two-variable multipoint quasi-ration...

In this paper, we obtain approximation theorems of classes of analytic functions by shifts $L(\lambda ,\alpha ,s+i\tau )$ of the Lerch zeta-function for $\tau \in [T,T+H]$ where $H\in [T^{27/82},T^{1/2}]$. The cases of all parameters, $\lambda ,\alpha \in ($...

Let $t_{\tau }$ be a solution to the equation $\theta \left(t\right)=(\tau -1)\pi$, $\tau >0$, where $\theta \left(t\right)$ is the increment of the argument of the function ${\pi }^{-s/2}\mathsf{\Gamma }(s/2)$ along the segment connecting points $s=1/2$ and $s=1/2+it$. $t_{\tau }$ is called...

Let $\theta \left(t\right)$ denote the increment of the argument of the product ${\pi }^{-s/2}\Gamma (s/2)$ along the segment connecting the points $s=1/2$ and $s=1/2+it$, and $t_n$ denote the solution of the equation $\theta \left(t\right)=(n-1)\pi$, $n=0,1,\dots$. The numbe...

In this paper, a theorem is obtained on the approximation in short intervals of a collection of analytic functions by shifts $(\zeta (s+i{t}_k^{{\alpha }_1}),\dots ,\zeta (s+i{t}_k^{{\alpha }_r}))$ of the Riemann zeta function. Here, $\{t_k:k\in \mathbb{N}\}$ is the sequence of G...

In the paper, we obtain universality theorems for compositions of some classes of operators in multidimensional space of analytic functions with a collection of periodic zeta-functions. The used shifts of periodic zeta-functions involve the sequence...

Let $\mathfrak{a}=\{a_m:m\in \mathbb{N}\}$ be a periodic multiplicative sequence of complex numbers and $L(s;\mathfrak{a})$, $s=\sigma +it$ a Dirichlet series with coefficients $a_m$. In the paper, we obtain a theorem on the approximation of non-vanishing analytic functions defined in the s...

Since 1975, it has been known that the Hurwitz zeta-function has a unique property to approximate by its shifts all analytic functions defined in the strip $\mathfrak{D}=\{s=\sigma +it:1/2<\sigma <1\}$. However, such an approximation causes efficiency problems, and apply...

This paper is devoted to studying approximations of symmetric continuous functions by symmetric analytic functions on a Banach space X with a symmetric basis. We obtain some positive results for the case when X admits a separating polynomial using a...

Let $\mathfrak{a}=\{a_m:m\in \mathbb{N}\}$ be a multiplicative periodic sequence of complex numbers. In this paper, we consider the approximation of analytic functions defined in the strip $\{s=\sigma +it:1/2<\sigma <1\}$ by shifts $\zeta (s+i\tau ;\mathfrak{a})$ of the zeta-function...

The paper is devoted to Epstein zeta-function ${\zeta }_3\left(s\right)$, $s=\sigma +it$, of a quadratic form of three variables. We obtain a probabilistic characterization of the function ${\zeta }_3\left(s\right)$ in terms of weak convergence of probability measures in the space of...

In this paper, we prove the theorems on the simultaneous approximation of a pair of analytic functions by discrete shifts $(\zeta (s+ik{h}_1),\zeta (s+ik{h}_2,\alpha ))$, $h_1>0$, $h_2>0$ of the Riemann zeta function $\zeta \left(s\right)$ and Hurwitz zeta function $\mathrm{\&zeta}$...

This study is divided into two important axes; for the first one, a new symmetric analytical (approximate) solution to the Duffing–Helmholtz oscillatory equation in terms of elementary functions is derived. The obtained solution is compared wit...

In the paper, we consider the approximation of analytic functions by shifts from the wide class $\tilde{S}$ of L-functions. This class was introduced by A. Selberg, supplemented by J. Steuding, and is defined axiomatically. We prove the so-called joint...

In this paper, we first establish an evaluation formula to calculate Wiener integrals of functionals on Wiener space. We then apply our evaluation formula to carry out easy an calculation for the analytic Fourier–Feynman transform of the functi...

We consider the Beurling zeta function ${\zeta }_{\mathbb{P}}\left(s\right)$, $s=\sigma +it$, of the system of generalized prime numbers $\mathbb{P}$ with generalized integers m satisfying the condition ${\sum }_{m⩽x}1=ax+O\left(x^{\delta }\right)$, $a>0$, $0⩽\delta <1$, and suppose that ${\zeta }_{}$...

Let $\mathcal{P}$ be a system of generalized prime numbers, and ${\mathcal{N}}_{\mathcal{P}}$ the corresponding system of generalized integers. Assuming that ${\displaystyle \underset{m\in {\mathcal{N}}_{\mathcal{P}}}{{\displaystyle \sum _{m⩽x}}}}1-ax\ll x^{\beta }$ with $a>0$ and $0⩽\beta <1$, we consider the Beurling zeta-function ${\mathrm{\&zeta}}_{}$...

This paper studies the uniqueness of the bounded solution to a new Cauchy problem of the fractional nonlinear partial integro-differential equation based on the multivariate Mittag–Leffler function as well as Banach’s contractive principl...

The stabilization of a planar premixed flame front on a lattice (porous) burner is considered. The developed model captures all the important features of the phenomenon, while also admitting qualitative analytical investigation. It has been rigorousl...

In this work, we present a modified generalized Mittag–Leffler function method (MGMLFM) and Laplace Adomian decomposition method (LADM) to get an analytic-approximate solution for nonlinear systems of partial differential equations (PDEs) of fr...

In this paper, we derive an analytical and explicit approximation for the Bessel function $J_2\left(x\right)$ for positive real x with a maximum absolute error of approximately $0.004$, which refines some new published analytic approximation. The absolute errors for...

We obtain an accurate analytic approximation for the Bessel function $J_2\left(x\right)$ using an improved multipoint quasirational approximation technique (MPQA). This new approximation is valid for all real values of the variable x, with a maximum absolute error...

The current paper addresses the lack of explicit analytical solutions for stress evaluations in variable-thickness flywheels by proposing an approximate formulation for conical profiles, where thickness varies linearly along the radius. The main obje...

We show that neural networks with an absolute value activation function and with network path norm, network sizes and network weights having logarithmic dependence on $1/\epsilon$ can $\epsilon$-approximate functions that are analytic on certain region...

Approximate analytical methods have been frequently used in geotechnical engineering to estimate the reliability of geotechnical structures due to their efficiency and simplicity. The main spirit of these methods is using the moments of the limit sta...

A new simple and accurate expression to approximate the modified Bessel function of the first kind $I_1(x)$ is presented in this work. This new approximation is obtained as an improvement of the multi-point quasi-rational approximation technique, MPQA....

This paper develops two novel arcsine-type analytic approximation formulas for $\mathrm{arcsl}\left(x\right)$, the Gauss lemniscate sine function, each equipped with a monotonic and bounded remainder term of order $x^{29}$ as $x\to 0$, which demonstrates the high accuracy of t...

The Z-function is the real function given by $Z\left(t\right)=e^{i\theta \left(t\right)}\zeta \left({\displaystyle \frac{1}{2}}+it\right)$, where $\zeta \left(s\right)$ is the Riemann zeta function, and $\theta \left(t\right)$ is the Riemann–Siegel theta function. The function, central to the study of the Riemann hypothesis (RH), has...

The comprehensive system of information security of an enterprise includes both tactical aspects of information and strategic priorities, reflecting the information policy and information strategy of the enterprise. Ensuring a given level of cybersec...

A solution of the governing equation representing the drawdown in a horizontal confined aquifer, where groundwater flow is unsteady, was first provided by Theis and is famously known as the Theis solution. This solution was given in terms of an expon...

This Special Issue contains 14 articles from the MDPI journal Symmetry on the general subject area of “Special Functions and Polynomials”, written by scholars belonging to different countries of the world. A similar number of submitted ar...

Using geometric considerations, we provided a clear derivation of the integral representation for the error function, known as the Craig formula. We calculated the corresponding power series expansion and proved the convergence. The same geometric me...

In this paper we provide approximations for the error function using the Padé approximation method and the Fourier series method. These approximations have simple forms and acceptable bounds for the absolute error. Then we use them in diffusio...

The paper considers the problem of representation and extension of Appell’s hypergeometric functions by a special family of functions—branched continued fractions. Here, we establish new symmetric domains of the analytical continuation of...

The paper considers the generalized hypergeometric function ${}_{3}F_2,$ which is important in various fields of mathematics, physics, and economics. The method is used, according to which the domains of the analytical continuation of the special f...

This paper considers the problem of approximating some Appell’s hypergeometric functions $F_2$ by their branched continued fraction expansions. Using the formula for the difference of two approximants of a branched continued fraction, we establish...

The paper considers the problem of the analytical extension of the ratios of generalized hypergeometric functions ${}_{3}F_2$ A new domain of analytic continuation for these ratios under certain conditions to parameters is established. In this case, the doma...

This paper considers the Horn’s hypergeometric function $H_3$, which is closely related to other hypergeometric functions and has various mathematical or physical applications. The problem of analytical extension of this function is solved using a...

In this paper, we will consider the Hyers-Ulam stability for the second order inhomogeneous linear differential equation, $u^{″}\left(x\right)+\alpha u^{\prime }\left(x\right)+\beta u\left(x\right)=r\left(x\right)$ , with constant coefficients. More pre...

The practical use of computational thermo-fluid dynamics (CFD) for food thermal process calculations still appears very premature due to both the high costs and the inhomogeneity and anisotropy of foods. Therefore, the traditional formula method with...

For the calculation of thermal processes of canned food, the original formula method of Ball is still widely used for its accuracy and safety. However, it requires the consultation of tables that Ball prepared and the relative interpolation of the da...

The velocity of air that crosses the canopy of tree crops when using orchard sprayers is a variable that affects pesticide dispersion in the environment. Therefore, having an equation to describe air velocity decay through the canopy is of interest....

A spline-based integral approximation is utilized to define a sequence of approximations to the error function that converge at a significantly faster manner than the default Taylor series. The real case is considered and the approximations can be im...