A convolution integral equation solved by Laplace transformations

AbstractWe consider the integral equation p(t) = ∫0tK(t−τ)p(τ)dτ+r(t) where both K(t) and r(t) behave as exp(αt3) as t → ∞ (α > 0). So straightforward application of the Laplace transform technique is not possible. By introducing a complex parameter the equation is solved in the complex domain. Analytic continuation with respect to this parameter yields the desired solution. For a particular example (which arose in a statistical problem on estimating monotone densities) we describe the construction of the explicit solution of the equation

Analytical methods for an elliptic singular perturbation problem in a circle

AbstractWe consider an elliptic perturbation problem in a circle by using the analytical solution that is given by a Fourier series with coefficients in terms of modified Bessel functions. By using saddle point methods we construct asymptotic approximations with respect to a small parameter. In particular we consider approximations that hold uniformly in the boundary layer, which is located along a certain part of the boundary of the domain

Asymptotic and numerical aspects of the noncentral chi-square distribution

AbstractThe concentral χ2-distribution is related with the series e−x∑n=0∞xnn!P(μ+ n, y)=1−e−x∑n=0∞∞nn!Q(μ+n, y) where P(α, z) and Q(α, z) are incomplete gamma functions (central χ2-distributions). Another representation is in terms of Qμ(x,y)≔∫y∞zx12(μ−1)e−z−xIμ−1(2xz)dz which is also known as the generalized Marcum Q-function; Iμ(z) is the modified Bessel function. Qμ(x, y) plays a role in communication studies. From the integral representation recurrence relations for Qμ(x, y) are derived. Next, it is shown that Qμ(x, y) can be expressed in terms of the simpler integral Fμ(ξσ)≔∫ξ∞e−(σ+1)tIμ(t)dt where ξ=2xyandσ=−1+12yx+xy Two asymptotic expansions of Qμ(x, y) are derived. In one form, the function Fμ(ξ, σ) is used with μ fixed and large ξ, giving an expansion which holds uniformly with respect to σ ϵ (0, ∞). In a second expansion, both parameters ξ and μ may be large. In both asymptotic forms, an error function (the normal distribution function) is used to describe the behavior of Qμ(x, y) as y crosses the value x+μ. Series expansions in terms of incomplete gamma functions are discussed in connection with numerical evaluation of Qμ(x, y) or 1 − Qμ(x, y). It is also indicated when the asymptotic expansion can be used in order to obtain a certain relative accuracy