相关论文: A remark on the matrix Airy function
We give a formula for matrix exponentials and partial fraction decompositions.
We present a systematic study of higher-order Airy-type differential equations providing the explicit form of the solutions, deriving their power series expansions and a probabilistic interpretation. Under suitable convergence hypotheses,…
In this note, we propose an integral representation for $\zeta(4)$, where $\zeta$ is the Riemann zeta function. The corresponding expression is obtained using relations for polylogarithms. A possible generalization to any even argument of…
We evaluate definite integrals involving the product of four modified Bessel functions of the first and second kind and a power function. We provide general formulas expressed in terms of the Meijer $G$-function and generalized…
Integrals arising in the Thomas-Fermi (TF) theory of atomic structure and which contain logarithms of the Airy functions have been expressed in terms of the incomplete Bell polynomials. In keeping with the spirit of TF theory closed forms…
We consider a suitable extension of the complex Airy operator, $-d^2/dx^2 + ix$, on the real line with a transmission boundary condition at the origin. We provide a rigorous definition of this operator and study its spectral properties. In…
The results presented in this paper are refinements of some results presented in a previous paper. Three such refined results are presented. The first one relaxes one of the basic hypotheses assumed in the previous paper, and thus extends…
The primary goal of this paper is to introduce and investigate generalized incomplete exponential functions with matrix parameters. Integral representation, differential formula, addition formula, multiplication formula, and recurrence…
An integral representation of the partition function for general $n$-dimensional Ising models with nearest or non-nearest neighbours interactions is given. The representation is used to derive some properties of the partition function. An…
We introduce a method for calculating individual elements of matrix functions. Our technique makes use of a novel series expansion for the action of matrix functions on basis vectors that is memory efficient even for very large matrices. We…
Let $f$ be a $r\times m$-matrix of holomorphic functions that is generically surjective. We provide explicit integral representation of holomorphic $\psi$ such that $\phi=f\psi$, provided that $\phi$ is holomorphic and annihilates a certain…
Based on a Problem and its solution published on the pages of SIAM Review, we give an interesting integral representation for the Lambert $W$ function in this short note. In particular, our result yields a new integral representation for…
An integral formula is developed which applies to an essentially arbitrary function. An application is made to the Riemann zeta function.
We present an effective formula for the Sibony function for all Reinhardt domains.
An approach to constructing an upper bound for the Riemann-Farey sum is described.
The Airy transform is an ideally suited tool to treat problem in classical and quantum optics. Even though the relevant mathematical aspects have been thoroughly investigated, the possibility it offers are wide and some aspects, as the link…
The Mellin transform of quartic products of shifted Airy functions is evaluated in a closed form. Some particular cases expressed in terms of the logarithm function and complete elliptic integrals special values are presented.
This papers discusses the leaky aquifer function considered in a recent paper by Frank Harris in the Journal of Computational and Applied Mathematics (2008). We describe properties of an integral representing this function and give details…
A new integral representation is derived using a definite integral given by Cauchy and used to evaluate a number of integrals containing the finite series of special functions.
We give an elementary characterization of rational functions among meromorphic functions in the complex plane.