Bessel Functions Of Integer Order Research Articles

The transmitted impulse response of a lossy magnetoplasma by a convolution method

In this note a convolution method is used to find an exact representation in terms of special functions for the transmitted impulse response of a lossy magnetoplasma half space. For an observer deep in the plasma it is shown that the impulse response reduces to a fairly simple expression in terms of integer-order Bessel functions. Also, a comparison is made between the convolution method and the Fourier transform approach to the solution.

Legendre polynomials in irrationality proofs

It is shown that a simple trick involving Legendre polynomials readily yields the irrationality of ea, , π2, and of the zeros of Bessel functions of integer order. Generalisation of this idea yields the irrationality of ζ(3).

Analytic solutions for constant-tension coil shapes

An analytical solution of the differential equation describing the shape of a flexible filamentary conductor (incapable of supporting bending stresses) in a toroidal magnetic field has been obtained—previously only numerical solutions were available. The solution derives from a series expansion of modified Bessel functions of integer order. The characteristics of toroidal field magnets for proposed tokamak devices are obtainable by term integration of the solution series. General expressions are given for the following coil characteristics: the conductor turn length, the solenoid inductance, the area enclosed by the coil, and the coil support dimensions. For several particular cases of interest these coil chararacteritics are obtained as closed-form analytical formulas. A new type of coil, called a compound-constant-tension coil, is proposed. It is formed by selecting and matching (point and slope) segments chosen from two or more members of the one parameter family of solution curves found for the shape equation. These coils may be supported by tension members at the intersections of the solution curves or by a compression ring support and provide a unique and highly attractive solution to the toroidal field coil centering force support problem of tokamak designs.

An exact solution for the impulse response of a uniform plasma half space

A horizontally polarized impulsive plane wave is incident on a uniform half space of cold lossless plasma. The exact form of the resultant impulse response is found for any depth in the plasma in terms of two integer-order Bessel functions by inverting an appropriate Laplace transform. It is shown that as the observer's position in the plasma is increased, the frequency and relative amplitudes of the oscillations of the impulse response increase in a corresponding fashion. Numerical results are presented to illustrate these effects.

A uniform asymptotic analysis of dispersive wave motion

The signaling problem for the one dimensional Klein-Gordon equation with spatially varying coefficients is analyzed. A formal, uniformly valid, asymptotic expansion of the solution is obtained with the help of two families of rays, and involving four functions : two successive Bessel functions of integer order and two new functions which we call the diffraction functions. The validity of the expansion is established when the coefficients in the Klein-Gordon equation are constants, and the results are applied to a signaling problem for a class of acoustic wave guides.

Computational Aspects of Three-Term Recurrence Relations

Computational Aspects of Three-Term Recurrence Relations

Diffraction of light by sound film of the variable width type

Diffraction of light by strips of sound film as a means of immediate and exact Fourier analysis of the sound. Spectra obtained from various types of sound film. Occurrence of all Bessel functions of integer order. Diffraction pattern and two-dimensional Fourier analysis. Relation with conventional optical gratings. Broadening of the lines due to finite length of the strip of film. Applications and possibilities.