We consider the second-order linear differential equation (x2 − 1)y'' + f (x)y′ + g(x)y = h(x) in the interval (−1, 1) with initial conditions or boundary conditions (Dirichlet, Neumann or mixed Dirichlet–Neumann). The ...

In previous papers [6–8,10], we derived convergent and asymptotic expansions of solutions of second order linear differential equations with a large parameter. In those papers we generalized and developed special cases not ...

The mathematical models of many short wavelength phenomena, specially wave propagation and optical diffraction, contain, as a basic ingredient, oscillatory integrals with several nearly coincident stationary phase or ...

We consider the Pearcey integral P(x; y) for large values of |x|, x, y ∈ C. We can find in the literature several convergent or asymptotic expansions in terms of elementary and special functions, with different levels ...

Convergent expansions are derived for three types of orthogonal polynomials: Charlier, Laguerre and Jacobi. The expansions have asymptotic properties for large values of the degree. The expansions are given in terms of ...

We consider the confluent hypergeometric function M(a, b; z) for z ∈ C and Rb >Ra > 0, and the confluent hypergeometric function U(a, b; z) for b ∈ C, Ra > 0, and Rz > 0. We derive two convergent expansions of M(a, b; z); ...

The effect of high-energy ball-milling on the magnetostructural properties of a Ni45Co5Mn35Sn15 alloy in austenitic phase at room temperature has been analyzed by neutron and high-resolution X-ray diffraction. The ball ...

The limit cycle of the van der Pol oscillator, x¨+ε(x2−1)x˙+x=0, is studied in the plane (x,x˙) by applying the homotopy analysis method. A recursive set of formulas that approximate the amplitude and form of this limit ...

The power series expansions of the hypergeometric functions p+1Fp (a,b1,…,bp;c1,…,cp;z) converge either inside the unit disk |z|<1 or outside this disk |z|>1. Nørlund’s expansion in powers of z/(z−1) converges in the ...

In Ferreira and López [Asymptotic expansions of the Hurwitz–Lerch zeta function. J Math Anal Appl. 2004;298(1):210–224], the authors derived an asymptotic expansion of the Lerch's transcendent Φ(z,s,a) for large |a|, valid ...