In this paper we prove existence and uniqueness of classical solutions for the non-autonomous inhomogeneous Cauchy problem d dt u(t) = A(t)u(t) + f(t), 0 ≤ s ≤ t ≤ T, L(t)u(t) = Φ(t)u(t) + g(t), 0 ≤ s ≤ t ≤ T, u(s) = x. The solution to this problem is obtained by a variation of constants formula.

Abstract This paper studies the problem of construction of the optimal quadrature formula in the sense of Sard in (2) 2 ( 1,1) L − S.L.Sobolev space for approximate calculation of the Cauchy type singular integral. Using the discrete analogue of the operator 4 4 / d dx we obtain new optimal quadrature formulas. Furthermore, explicit formulas of the optimal coefficients are obtained. Finally, in...

This paper concerns the fourth order differential equation x′′′′ + ax′′′ + f(x′′) + g(x′) + h(x) = p(t). Using the Cauchy formula for the particular solution of non-homogeneous linear differential equations with constant coefficients, we prove that the solution and its derivatives up to order three are bounded.

We consider a nonsymmetric algebraic matrix Riccati equation arising from transport theory. The nonnegative solutions of the equation can be explicitly constructed via the inversion formula of a Cauchy matrix. An error analysis and numerical results are given. We also show a comparison theorem of the nonnegative solutions.

By applying a classical method, already employed by Cauchy, to a terminating curious summation by one of the authors, a new curious bilateral q-series identity is derived. We also apply the same method to a quadratic summation by Gessel and Stanton, and to a cubic summation by Gasper, respectively, to derive a bilateral quadratic and a bilateral cubic summation formula.

This article concerns the fourth order differential equation x + ax′′′ + bx′′ + g(x′) + h(x) = p(t). Using the Cauchy formula for the particular solution of non-homogeneous linear differential equations with constant coefficients, we prove that the solution and its derivatives up to order three are bounded.

We define the Stirling numbers for complex values and obtain extensions of certain identities involving these numbers. We also show that the generalization is a natural one for proving unimodality and monotonicity results for these numbers. The definition is based on the Cauchy integral formula and can be used for many other combinatorial numbers.

THE CAUCHY DOUBLE ALTERNANT AND DIVIDED DIFFERENCES WENCHANG CHU Dedicated to my friend Pier Vittorio Ceccherini for his 65th birthday Abstract. As an extension of Cauchy’s double alternant, a general determinant evaluation formula is established. Several interesting determinant identities are derived as consequences by means of divided differences.