Series Of Bessel Functions Research Articles

On Coulomb wave functions

AbstractThe irregular radial Coulomb wave function in repulsive fields is expressed, for any angular momentum value, in series of Bessel functions of arbitrary order. Previous expansions, obtained for angular momenta zero and one, can now be obtained as special cases. The expansion can be most conveniently used for high-energy particles.

Slow-wave structures for M-type devices

M-type devices such as magnetron oscillators, magnetron amplifiers and backward-wave oscillators always contain a sole plate to provide the static electric field which, together with a transverse static magnetic field, provides electron focusing. The sole plate unavoidably becomes part of the RF structure. It is the purpose of this paper to show the effect of the sole plate on the RF characteristics of the more common slow-wave structures. The results of the study indicate that no slow-wave structure containing a sole plate can be truly a backward-wave structure. The fundamental space harmonic must contain a region wherein the phase and group velocities are of the same sign. These results are predicted by a field analysis of the uniform-vane, strapped magnetron anode system. Such an analysis is made possible by two simplifying assumptions. The first assumption is that the interaction space between anode and sole can be replaced by an equivalent linear structure. In this way, the derived expressions for admittance become infinite series of exponential functions rather than infinite series of untabulated higher order Bessel functions. The second assumption is that the fields peculiar to the straps can be described by TEM waves. This assumption allows one to formulate a strap admittance which is in parallel with the admittance of the interaction space and the side cavity. The resultant dispersion equation can be solved graphically for the frequency-phase shift characteristic of the composite system. This analysis predicts that the straps may be either capacitive or inductive depending upon mode number, frequency and geometry. It shows that the frequency-phase shift characteristic is not simply derivable from the unstrapped anode system by the addition of a constant phase shift. It shows further that the fundamental space harmonic must contain a region wherein the phase and group velocities are of the same sign, i.e., the anode system is not, and cannot be, a simple backward-wave structure. These results are verified in detail by a careful and complete experimental study on a standard strapped magnetron anode structure. They are further verified by a second experimental study of the very common interdigital line structure.

Effective resistance and reactance of a solid cylindrical conductor placed in a semi-closed slot

The paper is concerned with the investigation of the current distribution over the cross-section of a solid cylindrical conductor lying in a slot with a narrow opening. It is found that the current density at any point is expressible as the sum of an infinite series of Bessel functions, each including an arbitrary constant which is determinable by making a reasonable assumption regarding the boundary conditions. By means of this result it is possible to determine the complex impedance of the conductor, and hence its equivalent resistance and internal reactance per unit length.

A note on the Summation of some Series of Bessel Functions

A note on the Summation of some Series of Bessel Functions

Summation of Some Series of Bessel Functions

Summation of Some Series of Bessel Functions

Expansion of Hypergeometric Functions in Series of Other Hypergeometric Functions

In a previous paper [1] one of us developed an expansion for the confluent hypergeometric function in series of Bessel functions. A different expansion of the same kind given by Buchholz [2] was also studied. Since publication of [1], it was found that Rice [3] has also developed an expansion of this type, and yet a fourth expansion of this kind can be deduced from some recent work by Alavi and Wells [4]. In this note, we first deduce a multiplication formula for the Gaussian hypergeometric function which generalizes a statement of Chaundy, see (11), page 187 of [5], and includes a multiplication theorem for the confluent hypergeometric functions due to Erdélyi, see (7), page 283 of [5]. Our principal result is specialized to give an expansion of the confluent hypergeometric function in series of Bessel functions which includes the four above as special cases. With the aid of the Laplace transform, the latter result is used to derive an expansion of the Gaussian hypergeometric function in series of functions of the same kind with changed argument. This is advantageous since, throughout most of the unit disc, the change in argument leads to more rapidly converging series. For special values of the parameters, the expansion degenerates into known quadratic transformations.

Expansion of hypergeometric functions in series of other hypergeometric functions

In a previous paper [1] one of us developed an expansion for the confluent hypergeometric function in series of Bessel functions. A different expansion of the same kind given by Buchholz [2] was also studied. Since publication of [1], it was found that Rice [3] has also developed an expansion of this type, and yet a fourth expansion of this kind can be deduced from some recent work by Alavi and Wells [4]. In this note, we first deduce a multiplication formula for the Gaussian hypergeometric function which generalizes a statement of Chaundy, see (11), page 187 of [5], and includes a multiplication theorem for the confluent hypergeometric functions due to Erdélyi, see (7), page 283 of [5]. Our principal result is specialized to give an expansion of the confluent hypergeometric function in series of Bessel functions which includes the four above as special cases. With the aid of the Laplace transform, the latter result is used to derive an expansion of the Gaussian hypergeometric function in series of functions of the same kind with changed argument. This is advantageous since, throughout most of the unit disc, the change in argument leads to more rapidly converging series. For special values of the parameters, the expansion degenerates into known quadratic transformations.

Self-Energy of a Helical Dislocation

Kr\"oner's energy expression is used in this theoretical calculation. The helical dislocation is assumed to have a uniform shape with the Burgers vector along its axis. The axial length of the helix is large compared to its radius and the radius is large compared to the dislocation "cross section," which is of the order of a Burgers vector. For a helix of many turns and arbitrary pitch an expansion in a Fourier cosine series is used. The self energy is found in terms of elementary functions and Kapteyn series of Bessel functions. In the limiting cases of a tightly wound helix (small pitch) and a nearly straight helix (large pitch) simple expressions result, which have a plausible physical explanation. For a tightly wound helix the dominant term represents the contribution from the cylindrical part of the helix, the first-order terms represent the influence of the size of the dislocation cross section and the second order terms represent the effect of the axial component of the helix. For the nearly straight helix the dominant terms represent the contribution from the straight screw part and the second-order terms are taken to give the interaction between the turns of the helix. Finally the correction in the self-energy when a return loop is present is considered.

Expansion of the Confluent Hypergeometric Function in Series of Bessel Functions

Expansion of the Confluent Hypergeometric Function in Series of Bessel Functions

Maximally-Flat Time Delay Ladders

The driving-point impedance for a maximally-flat time delay response is derived. The impedance is synthesized as an infinite low-pass LC ladder that starts with an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\epsilon G/\omega_0</tex> -farad shunt capacitor The ladder elements rapidly taper toward a capacitance of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2G/\omega_0</tex> farads and an inductance of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2R/\omega_0</tex> henries. The impulse and step responses of the impedance are derived as a series of Bessel functions. A three-terminal maximally-flat time delay transfer impedance is also considered. The conditions for a smoothly-tapering ladder structure are given. The transfer impedance is synthesized as an infinite low-pass LC ladder whose first two shunt capacitors are <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">32G/(9\omega_0)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\epsilon^{3}G/(9\omega_0)</tex> farads, respectively. The impulse and step responses of the transfer impedance are also derived.

Expansion of the confluent hypergeometric function in series of Bessel functions

An expansion of the confluent hypergeometric function Φ ( a , c , z ) \Phi (a,c,z) in series of functions of the same kind has been given by Buchholz [1]. By specialization of some quantities, there is obtained an expansion in series of modified Bessel functions of the first kind, I υ ( z ) {I_\upsilon }(z) , where v v depends on the parameter a a . Tricomi [2, 3] has developed two expansions of similar type where both the order and argument of the Bessel functions depend on the parameters a a and c c . In the present paper, we derive an expansion in series of Bessel functions of integral order whose argument is independent of a a and c c . Our expansion is advantageous for many purposes of computation since the parameters and variable of Φ ( a , c , z ) \Phi (a,c,z) appear in separated form. Also, for desk calculation, extensive tables of I n ( z ) {I_n}(z) are available, while for automatic computation Bessel functions are easy to generate [4]. Special cases of the confluent function, such as the incomplete gamma function, are also studied. For the class of transcendents known as the error functions, including the Fresnel integrals, it is shown that our expansion coincides with that of Buchholz [1]. By specializing a parameter and passing to a limit, we derive expansions for the exponential integral and related functions. Other expansions for the error and exponential integrals are derived on altogether different bases. Finally, some numerical examples are presented to manifest the efficiency of our formulas.

THEORETISGHE UNTERSUCHUNGEN ÜBER DEN EINFLUSS DER VERWITTERUNGSSCHICHT AUF DAS SPEKTRUM ELASTISCHER WELLEN IN DER REFLEXIONSSEISMIK

AbstractThe following assumptions are made in the mathematical treatment of the problem. Below a plane earth's surface there is a three‐layered elastic medium the interfaces of which are parallel to the earth's surface. The uppermost layer represents the weathered layer in which the velocity of propagation of seismic waves increases linearly with depth. The two lower layers, the so‐called intermediate layer and the substratum each have a constant velocity. The surface of the earth is acted on simultaneously by a normal pressure N in the form of a Heaviside pulse. The seismic wave thus generated is propagated through the elastic media.The aim of the investigation is to study the shape of the wave1) in the intermediate layer, after the wave has entered it the first time2) at the earth's surface, after the wave has been reflected once at the interface between the intermediate layer and the substratum.The mathematical solutions can in both cases be expressed as series of Bessel functions. Some numerical examples illustrate the quasi‐periodic nature of the solutions. The pseudo‐frequency is determined by the gradient of velocity in the uppermost layer; it assumes a value of approximately 50 c.p.s. for a gradient of appr.

Corner reflector antennas with arbitrary dipole orientation and apex angle

This paper is concerned with the determination of corner reflector characteristics for reflectors of arbitrary apex angle excited by any infinitesimal dipole source which is tangent to a circular cylinder having the reflector axis as its axis. Use is made of the dyadic Green's functions for the perfectly conducting wedge recently given by Tai, and the results are obtained as infinite series of Bessel functions. The quantities of interest are the relative phase and magnitude of vertical and horizontal components of electric field, the directive gain of the antenna, and the radiation resistance of the dipole source. These have been computed for a continuous range of apex angles from 30\deg to 180\deg for dipole-to-axis spacings up to about a wavelength. The results of this general analysis are useful not only for the determination of reflector characteristics for apex angles not subject to image analysis, but also for the obtaining of certain limiting forms for small apex angles which are not readily apparent from the image analyses.

The diffraction of light by two perpendicular ultrasonic waves II

Referring to our previous paper, we deduce solutions in the form of a series of Bessel functions for the systems of difference-differential equations, appearing there. After giving rigorous relations between the coefficients we are able to give explicit expressions for the intensities of the different spectra in the case of small widths of the ultrasonic field. Investigating the extremum of these intensities for variable angles of incidence, we find that they become maximum when the angle of incidence is equal to the Bragg angle. So we have proved the complete analogy between the diffraction of light by two perpendicular supersonic waves and the X-ray diffraction in crystals.

The wave functions in Coulomb fields

In this paper a detailed mathematical study of the confluent hypergeometric functions is carried out. Exact analytic expansions in series of Bessel functions are developed. These expansions are found to be useful for the computation of the wave functions in Coulomb fields. They have the advantage over power series expansions in the clearance of imaginaries from the wave functions for repulsive Coulomb fields.

A SOLUTION OF TRANTER'S DUAL INTEGRAL EQUATIONS PROBLEM

Tranter (1, 2) has given the solution of the pair of integral equations (1) and (2) below in the form of a Neumann series of Bessel functions whose coefficients are determined by means of an infinite series of linear equations. This paper gives the formal solution as an integral, containing an unknown function which is determined by means of an integral equation of Fredholm's type. This solution appears to have both theoretical and practical advantages over Tranter's solution in that it is (in a sense) in closed form; moreover it does not depend on certain parameters being specified as large or small in order to complete a practical solution. Applications are given to several potential problems involving circular disks, some of these problems being believed to be new.

On the sums of certain series of Bessel functions

Though many forms of series of Bessel functions of increasing order appear in the literature, they all seem to involve a term by term increase in the order by unity, possibly by 1/2, or by some integer greater than unity, usually 2. InWatson's “Bessel Functions” (1) no case appears in which the increase in the order is not an integer or 1/2. The purpose of this note is to give the sums of series of the form $$\mathop \Sigma \limits_{n = 0}^\infty J{}_s(z) and \mathop \Sigma \limits_{n = 0}^\infty ( - 1)^n J{}_s(z)$$ and similar series involvingI s , wheres is of the form 2kn+v andk is any positive real number. Such series have occurred in the theory of heat conduction and fluid flow (2). The method is to replace the function by a contour integral of Schlafli type.

On the radiation pattern of a paraboloid of revolution

The radiation pattern of a paraboloid of revolution is determined by using the mathematical form of Huygens’ principle where the field intensities in a given domain are expressed in their tangential components on the boundary. These tangential components are supposed to have the values which follow from the reflection of a plane wave at an infinite metal plate, coinciding with the tangent plane to the surface of the paraboloid. Expressions involving infinite series of Bessel functions are obtained which give the intensities of the distant field when an electric dipole is placed at the focus. Approximations are made for the main lobe in the planes perpendicular to and through the dipole. Finally Zuhrt’s method leads to the same results as the approximations given in the present paper.

Expansions of generalized Whittaker functions

ABSTRACTIn this paper I give a direct proof of a new expansion of the Whittaker function M(a; b; x) as a series of Bessel functions, and some generalizations of this result for the functions 2F2(x) and 4F4(x).

Current Flow in Cylinders

The mixed boundary value problem arising from the flow of current into a right circular cylinder of radius a through a perfectly conducting coaxial disk electrode at one end is solved approximately. The boundary conditions are met rigorously except that the electrode is not quite plane. A solution valid for all values of the disk radius is given for which in the worst case the deviation from flatness is nearly 0.01a. More exact solutions are worked out for disk radii ¼a, ½a, and ¾a and the first forty coefficients in the Bessel function series for the potential in the cylinder are tabulated. A cylinder whose diameter is eight times its length is also treated for disk radii ¼a and ½a and the first forty coefficients in the potential series are tabulated. The probable errors in the resistance formulas derived from these solutions are given.