Multiple Contour Integrals Research Articles

Fractional Derivatives of Certain Generalized Hypergeometric Functions of Several Variables

The main object of the present paper is to derive a number of key formulas for the fractional derivatives of the multivariable H-function (which is defined by a multiple contour integral of Mellin-Barnes type). Each of these formulas can be shown to yield interesting new results for various classes of generalized hypergeometric functions of several variables. Some of these applications of the key formulas provide potentially useful generalizations of known results in the theory of fractional calculus.

Convergence regions for multiple hypergeometric inegrals representing h-function in several variables

Convergence regions for multiple Mellin-Barnes contour integrals represening hypergeometric H-functions in several variables is described

Correlators in the theory of D=1 random surfaces

The common expression for multi-point correlators in the theory of random surfaces embedded in D=1 is obtained in terms of multiple contour integrals.

Scattering of an ultrasonic beam from a curved interface

We calculate the wavefield scattered from a circular fluid-solid interface when it is struck by a well collimated ultrasonic beam, emitted from a transducer in the fluid. Both convex and concave interfaces are considered. The radius of curvature of the interface is assumed to be sufficiently large that only reflection, transmission and the excitation of a leaky Rayleigh wave need be considered; the incident beam does not graze the interface. The incident beam is modeled as a two-dimensional wavefield whose initial profile is either Gaussian or rectangular. In both cases the aperture is assumed to be many wavelengths across and the interface close enough to the aperture that it is struck by a wavefield that has not yet evolved into a cylindrical wave. The scattered wavefields are represented as multiple contour integrals and are evaluated asymptotically.

A class of convolution integral equations

A new class of convolution integral equations whose kernels involve an H-function of several variables, which is defined by a multiple contour integral of the Mellin-Barnes type, is solved. It is also indicated how the main theorem can be specialized to derive a number of (known or new) results on convolution integral equations involving simpler special functions of interest in problems of applied mathematics and mathematical physics.

The reflexion of a pulse within a sphere

A pulse sets out from an interior point of a uniform compressible liquid sphere. The formal solution for the velocity at a surface point at any subsequent time is derived by operational methods. This solution, having the form of an infinite series of Legendre polynomials, with coefficients which are the ratios of Bessel functions, converges too slowly to be useful. We use a modification of a method of van der Pol & Bremmer, in which the first part of the series is transformed by Watson’s device into a series of multiple contour integrals, which are evaluated at saddlepoints. The expressions thus derived are shown to correspond term by term to the pulses whose times of arrival can be calculated by geometrical ray theory. The shape of each arriving pulse, not obtainable by ray theory, is given by the present method. It is shown that stationary time pulses which are extremals are related to stationary time pulses which are not extremals as Fourier integrals to the allied Fourier integrals. In particular, it is shown that the two singly-reflected pulses which travel within the shorter arc of a great circle (see figure 14) are so related. If the initial disturbance is given by a Heaviside unit function, the minimum time pulse arrives as a unit function, but the stationary time pulse which is not extremal arrives gradually and rises to a logarithmic infinity. This suggests an explanation of the difference observed on seismograms between the sharp arrivals ofpPand the blunt arrivals ofPP.