Integrals Of Elementary Functions Research Articles

Exact solutions for the magnetic fields of axisymmetric solenoids and current distributions

An analytical method originally developed for fluid mechanics has been applied to the analogous problem of calculating the magnetic field induced by axisymmetric current sheets, disks, and solenoids. The magnetic fields of a generalized axisymmetric current distribution are given either in closed form as elliptic integral expressions or as integrals of elementary functions over a finite range. The paper concentrates on the analysis of the fields generated by specified current distributions, though the method can also be applied to solve boundary value problems.

Efficient calculation of lattice sums for free-space periodic Green's function

An efficient method to calculate the lattice sums is presented for a one-dimensional (1-D) periodic array of line sources. The method is based on the recurrence relations for Hankel functions and the Fourier integral representation of the zeroth-order Hankel function. The lattice sums of arbitrary high order are then expressed by an integral of elementary functions, which is easily computed using a simple scheme of numerical integration. The calculated lattice sums are used to evaluate the free-space periodic Green's function. The numerical results show that the proposed method provides a highly accurate evaluation of the Green's function with far less computation time, even when the observation point is located near the plane of the array.

Green's functions for the isotropic or transversely isotropic space containing a circular crack

Green's functions for an infinite three-dimensional elastic solid containing a circular crack are derived in terms of integrals of elementary functions. The solid is assumed to be either isotropic or transversely isotropic (with the crack being parallel to the plane isotropy).

Band structure and interband optical transitions in the sawtooth superlattice.

The band structure and interband matrix elements of the ${\mathrm{Al}}_{\mathrm{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$As sawtooth superlattice are analyzed. We have shown that the envelope wave functions may be expressed as integrals of elementary functions. Numerical results are given for a superlattice with maximal mole fraction ${x}_{\mathrm{max}}$=0.3, and period equal to 14 nm. A comparison of our results with those obtained via the pseudopotential method is given. Also, the possibility of determining the conduction-band offset \ensuremath{\Delta}${E}_{c}$ from optical measurements, based on strong dependence of the transition matrix elements on \ensuremath{\Delta}${E}_{c}$, is discussed.

Analytical representation of transient ultrasonic phased-array near- and far-fields

The present contribution provides a method for the derivation of analytical near- and far-field expressions for the pressure step-function response of nonuniformly excited ultrasonic transducers modeled as pistons of arbitrary shape in an infinite rigid baffle. Explicit inversion of the Laplace transformed ultrasonic field using Cagniard's idea results in elementary functions or definite integrals of elementary functions, depending on the assumed nonuniform aperture distribution. The basic signal distortion is illustrated in terms of three-dimensional space-time plots of the pressure unit step-function response of a rectangular uniform aperture. The corresponding results for a phased-array transducer are also given. In addition, we investigate the pressure profile on the axis perpendicular to a uniform rectangular piston and its horizontal radiation pattern for different bandwidths of the exciting impulse.

On the Integrals of Elementary Functions

On the Integrals of Elementary Functions

On the integrals of elementary functions

On the integrals of elementary functions