Exact Fourier expansion in cylindrical coordinates for the three-dimensional Helmholtz Green function

A new method is presented for Fourier decomposition of the Helmholtz Green Function in cylindrical coordinates, which is equivalent to obtaining the solution of the Helmholtz equation for a general ring source. The Fourier coefficients of the Helmholtz Green function are split into their half advanced+half retarded and half advanced-half retarded components. Closed form solutions are given for these components in terms of a Horn function and a Kampe de Feriet function, respectively. The systems of partial differential equations associated with these two-dimensional hypergeometric functions are used to construct a fourth-order ordinary differential equation which both components satisfy. A s…

Mutual inductance of thick coils for arbitrary relative orientation and position

An exact solution method has been developed recently which gives the mutual inductance of two thin cylindrical coils in terms of line integrals of a new kind of vector potential, induced by the primary coil, around the two circular edges of the secondary coil. This paper describes the extension of this method to thick coils, by wrapping two radial integrations around these line integrals. Results are presented for two pairs of conventional coils and a combination of a superconducting coil and a Bitter coil. Excellent agreement with existing results for non coaxial coils was obtained. The trade-off between accuracy and computing time is also examined.

New special function recurrences giving new indefinite integrals

ABSTRACTSequences of new recurrence relations are presented for Bessel functions, parabolic cylinder functions and associated Legendre functions. The sequences correspond to values of an integer variable r and are generalizations of each conventional recurrence relation, which correspond to r=1. The sequences can be extended indefinitely, though the relations become progressively more intricate as r increases. These relations all have the form of a first-order linear inhomogeneous differential equation, which can be solved by an integrating factor. This gives a very general indefinite integral for each recurrence. The method can be applied to other special functions which have conventional …

Indefinite integrals from Wronskians and related linear second-order differential equations

Many indefinite integrals are derived for Bessel functions and associated Legendre functions from particular transformations of their differential equations which are closely linked to Wronskians. A large portion of the results for Bessel functions is known, but all the results for associated Legendre functions appear to be new. The method can be applied to many other special functions. All results have been checked by differentiation using Mathematica.

Indefinite integrals involving the incomplete elliptic integrals of the first and second kinds

ABSTRACTA substantial number of indefinite integrals are presented for the incomplete elliptic integrals of the first and second kinds. The number of new results presented is about three times the total number to be found in the current literature. These integrals were obtained with a Lagrangian method based on the differential equations which these functions obey. All results have been checked numerically with Mathematica. Similar results for the incomplete elliptic integral of the third kind will be presented separately.

Geometric efficiency for a parallel-surface source and detector system with at least one axisymmetric surface

Abstract An exact and numerically friendly method is given to calculate the geometric efficiency G of a planar radiation source and cosine detector system. Either the source or the detector, but not necessarily both, must have axial symmetry. For two non-coaxial disks the results are in exact agreement with a recent generalization of Ruby's formula for G. Detailed formulas and sample numerical results are given for a disk combined with rectangles and triangles. A disk and a general polygon can be solved by dividing the polygon into triangles. The method can also be applied to electrical inductance calculations and a solution recently given for the inductance of circular and elliptic loops c…

Non coaxial force and inductance calculations for bitter coils and coils with uniform radial current distributions

Recently the Bessel function approach to calculating the magnetic fields of coils has been used to calculate the mutual inductance and the force between two non coaxial thick cylindrical coils with parallel axes and uniform radial current distributions. This method can also be applied to calculate the force and inductance between an ordinary coil and a Bitter coil, or between two bitter coils, not necessarily coaxial. Bitter coils give a simpler case of the method, and it is possible to solve analytically for the magnetic field of a bitter disk.

Analytical and Semi-Analytical Solutions for the Force Between Circular Loops in Parallel Planes

Closed-form solutions are presented for the force between noncoaxial coplanar circular current loops. A semi-analytical solution is given for the case where the loops lie in parallel planes. Numerical results are given which cross check these solutions against each other and against an independently developed method. The closed form solution for the force between a circular loop and a coaxial circular arc segment is also given.

New indefinite integrals from a method using Riccati equations

ABSTRACTAn earlier method for obtaining indefinite integrals of special function from the second-order linear equations which define them has been reformulated in terms of Riccati equations, which ...

Noncoaxial Inductance Calculations Without the Vector Potential for Axisymmetric Coils and Planar Coils

This paper presents an exact method for calculating the mutual inductance between a general axisymmetric coil and a second planar coil consisting of either a disk coil or a planar loop of essentially arbitrary shape. The approach is based directly on the magnetic field rather than the vector potential . The paper gives detailed results for two circular loops, a circular loop and an elliptic loop, and a circular loop and an annular disk coil. The method can be extended to cover the cases where all these loops and coils are extruded in the axial direction to give the corresponding solenoids. The method is also applicable to calculations for nuclear radiation detectors.

Indefinite integrals of Lommel functions from an inhomogeneous Euler–Lagrange method

ABSTRACTA method given recently for deriving indefinite integrals of special functions which satisfy homogeneous second-order linear differential equations has been extended to include functions which obey inhomogeneous equations. The extended method has been applied to derive indefinite integrals for the Lommel functions, which obey an inhomogeneous Bessel equation. The method allows integrals to be derived for the inhomogeneous equation in a manner which closely parallels the homogeneous case, and a number of new Lommel integrals are derived which have well-known Bessel analogues. Results will be presented separately for other special functions which obey inhomogeneous second-order linear…

Indefinite integrals of products of special functions

ABSTRACTA method is given for deriving indefinite integrals involving squares and other products of functions which are solutions of second-order linear differential equations. Several variations of the method are presented, which applies directly to functions which obey homogeneous differential equations. However, functions which obey inhomogeneous equations can be incorporated into the products and examples are given of integrals involving products of Bessel functions combined with Lommel, Anger and Weber functions. Many new integrals are derived for a selection of special functions, including Bessel functions, associated Legendre functions, and elliptic integrals. A number of integrals o…

Calculations for a disk source and a general detector using a radiation vector potential

A closed form expression for a radiation vector potential is derived for a generalized disk radiation source. By applying Stokes's theorem the surface integral for the radiation flux into a general detector is converted into a much simpler line integral of the vector potential around the edge of the detector. This line integral can be easily evaluated for general detector geometry and general location and angular orientation relative to the disk source. For a number of cases the line integral reduces to integrals of Bessel functions which give various generalizations of Ruby's formula. Explicit formulas and numerical results for the geometric efficiency are given for circular and elliptical…

Inductance Calculations for Circular Coils of Rectangular Cross Section and Parallel Axes Using Bessel and Struve Functions

A simple method for calculating the mutual and self inductances of circular coils of rectangular cross section and parallel axes is presented. The method applies to non-coaxial as well as coaxial coils, and self inductance can be calculated by considering two identical coils which coincide in space. It is assumed that current density is homogeneous in the coil windings. The inductances are given in terms of one-dimensional integrals involving Bessel and Struve functions, and an exact solution is given for one of these integrals. The remaining terms can be evaluated numerically to great accuracy using computer packages such as Mathematica. The method is compared with other exact methods for …

Indefinite integrals of quotients of Gauss hypergeometric functions

A method recently applied to obtain indefinite integrals involving quotients of some common special functions is applied to obtain indefinite integrals of some quotients of Gauss hypergeometric fun...

Indefinite integrals involving the Jacobi Zeta and Heuman Lambda functions

ABSTRACTJacobian elliptic functions are used to obtain formulas for deriving indefinite integrals for the Jacobi Zeta function and Heuman's Lambda function. Only sample results are presented, mostly obtained from powers of the twelve Glaisher elliptic functions. However, this sample includes all such integrals in the literature, together with many new integrals. The method used is based on the differential equations obeyed by these functions when the independent variable is the argument u of elliptic function theory. The same method was used recently, in a companion paper, to derive similar integrals for the three canonical incomplete elliptic integrals.

Indefinite integrals of some special functions from a new method

A substantial number of indefinite integrals of special functions are presented, which have been obtained using a new method presented in a companion paper [Conway JT. A Lagrangian method for deriving new indefinite integrals of special functions. Integral Transforms Spec Funct. 2015; submitted to]. The method was originally derived from the Euler–Lagrange equations but an elementary proof is also presented in [Conway JT. A Lagrangian method for deriving new indefinite integrals of special functions. Integral Transforms Spec Funct. 2015; submitted to]. Sample results are presented here for Bessel functions, Airy functions and hypergeometric functions. More extensive results are given for th…

Extended two-body problem for rotating rigid bodies

A new technique that utilizes surface integrals to find the force, torque and potential energy between two non-spherical, rigid bodies is presented. The method is relatively fast, and allows us to solve the full rigid two-body problem for pairs of spheroids and ellipsoids with 12 degrees of freedom. We demonstrate the method with two dimensionless test scenarios, one where tumbling motion develops, and one where the motion of the bodies resemble spinning tops. We also test the method on the asteroid binary (66391) 1999 KW4, where both components are modelled either as spheroids or ellipsoids. The two different shape models have negligible effects on the eccentricity and semi-major axis, but…

A third integrating factor for indefinite integrals of special functions

An integrating factor f ~ x is presented involving the terms in y ′ ′ x and q x y x of the general homogenous second-order linear ordinary differential equation. The new integrating factors obey se...

Indefinite integrals involving the exponential integral function

Indefinite integrals involving complete elliptic integrals of the third kind

ABSTRACTA method developed recently for obtaining indefinite integrals of functions obeying inhomogeneous second-order linear differential equations has been applied to obtain integrals with respect to the modulus of the complete elliptic integral of the third kind. A formula is derived which gives an integral involving the complete integral of the third kind for every known integral for the complete elliptic integral of the second kind. The formula requires only differentiation and can therefore be applied for any such integral, and it is applied here to almost all such integrals given in the literature. Some additional integrals are derived using the recurrence relations for the complete …

More indefinite integrals from Riccati equations

ABSTRACTTwo new methods for obtaining indefinite integrals of a special function using Riccati equations are presented. One method uses quadratic fragments of the Riccati equation, the solutions of...

Dynamics of asteroid systems post-rotational fission

Asteroid binaries found amongst the Near-Earth objects are believed to have formed from rotational fission. In this paper, we aim to study the dynamical evolution of asteroid systems the moment after fission. The initial condition is modelled as a contact binary, similar to that of Boldrin et al. (2016). Both bodies are modelled as ellipsoids, and the secondary is given an initial rotation angle about its body-fixed $y$-axis. Moreover, we consider six different cases, three where the density of the secondary varies, and three where we vary its shape. The simulations consider 45 different initial tilt angles of the secondary, each with 37 different mass ratios. We start the dynamical simulat…

Indefinite integrals involving the incomplete elliptic integral of the third kind

ABSTRACTA substantial number of new indefinite integrals involving the incomplete elliptic integral of the third kind are presented, together with a few integrals for the other two kinds of incomplete elliptic integral. These have been derived using a Lagrangian method which is based on the differential equations which these functions satisfy. Techniques for obtaining new integrals are discussed, together with transformations of the governing differential equations. Integrals involving products combining elliptic integrals of different kinds are also presented.

A generalized integration formula for indefinite integrals of special functions

An integration formula for generating indefinite integrals which was presented in Conway JT [A Lagrangian method for deriving new indefinite integrals of special functions. Integral Transforms Spec...

Analytical Solutions for the Self- and Mutual Inductances of Concentric Coplanar Disk Coils

In this paper, closed-form solutions are presented for the self- and mutual inductances of disk coils which lie concentrically in a plane. The solutions are given as generalized hypergeometric functions which are closely related to elliptic integrals. The method used is a Legendre polynomial expansion of the inductance integral, which renders all integrations straightforward. Excellent numerical agreement with previous studies is obtained. An asymptotic formula for the approach to the ring coil limit is also derived and numerically validated. The methods presented here can be applied to noncoaxial and noncoplanar cases.

Indefinite integrals for some orthogonal polynomials obtained using integrating factors

A method has been presented recently for deriving integrals of special functions using two kinds of integrating factor for the homogeneous second-order linear differential equations which many spec...

Forces Between Thin Coils With Parallel Axes Using Bessel Functions

A method based on Bessel functions is presented for calculating the forces between combinations of thin coils with parallel axes. The coaxial case is solved in closed form in terms of elliptic integrals, whereas for the noncoaxial case the force components are expressed both as integrals of Bessel functions and as integrals of complete elliptic integrals. The results for the coaxial case have been compared with calculations in the literature with excellent agreement. The numerical results presented for the noncoaxial have been cross-checked by comparing the two methods. These methods can also be applied to current loops, disk coils, thick noncoaxial cylindrical coils, and various combinatio…

MUTUAL INDUCTANCE FOR AN EXPLICITLY FINITE NUMBER OF TURNS

Non coaxial mutual inductance calculations, based on a Bessel function formulation, are presented for coils modelled by an explicitly flnite number of circular turns. The mutual inductance of two such turns can be expressed as an integral of a product of three Bessel functions and an exponential factor, and it is shown that the exponential factors can be analytically summed as a simple geometric progression, or other related sums. This allows the mutual inductance of two thin solenoids to be expressed as an integral of a single analytical expression. Sample numerical results are given for some representative cases and the approach to the limit where the turns are considered to be smeared ou…

Plenary talk - non coaxial force and inductance calculations for bitter coils and coils with uniform radial current distributions

Recently the Bessel function approach to calculating the magnetic fields of coils has been used to calculate the mutual inductance and the force between two non coaxial thick cylindrical coils with parallel axes and uniform radial current distributions. This method can also be applied to calculate the force and inductance between an ordinary coil and a Bitter coil, or between two bitter coils, not necessarily coaxial. Bitter coils give a simpler case of the method, and it is possible to solve analytically for the magnetic field of a bitter disk.

Geometric efficiency for a circular detector and a linear source of arbitrary orientation and position

A new axisymmetric radiation vector potential which is singular along its entire axis of symmetry is derived for a spherically symmetric point radiation source. This potential and a previously given non-singular point source potential are integrated to give radiation vector potentials for a straight linear source of constant strength. Analytical solutions are given for the geometric efficiency G of a line source and a circular disk detector when the line source is parallel to the detector axis. The analytical solution is also given for the case where the line source is parallel to the disk surface, such that the source axis and the detector axis intersect. All other cases are given as simpl…

Analytical solution for the solid angle subtended at any point by an ellipse via a point source radiation vector potential

An axially symmetric radiation vector potential is derived for a spherically symmetric point source. This vector potential is used to derive a line integral for the solid angle subtended at a point source by a detector of arbitrary shape and location. An equivalent line integral given previously by Asvestas for optical applications is derived using this formulation. The line integral can be evaluated in closed form for important cases, and the analytical solution for the solid angle subtended by an ellipse at a general point is presented. The solution for the ellipse was obtained by considering sections of a right elliptic cone. The general solution for the ellipse requires the solution of …

Fourier series for elliptic integrals and some generalizations via hypergeometric series

Fourier series are derived for generalizations of the three canonical Legendre incomplete elliptic integrals using a hypergeometric series approach. The Fourier series for the incomplete Epstein–Hubbell integrals are obtained as special cases of the generalization of the Legendre integrals of the first and second kinds. The Fourier series for the integrals of the first and second kinds, and those for the Epstein–Hubbell integrals, were obtained recently using a different approach, but the series obtained for the generalization of the incomplete integral of the third kind is new. All cases of the integral of the third kind are given, with the modulus and the parameter being complex variables…

The planar two-body problem for spheroids and disks

We outline a new method suggested by Conway (2016) for solving the two-body problem for solid bodies of spheroidal or ellipsoidal shape. The method is based on integrating the gravitational potential of one body over the surface of the other body. When the gravitational potential can be analytically expressed (as for spheroids or ellipsoids), the gravitational force and mutual gravitational potential can be formulated as a surface integral instead of a volume integral, and solved numerically. If the two bodies are infinitely thin disks, the surface integral has an analytical solution. The method is exact as the force and mutual potential appear in closed-form expressions, and does not invol…

Indefinite integrals of incomplete elliptic integrals from Jacobi elliptic functions

Integration formulas are derived for the three canonical Legendre elliptic integrals. These formulas are obtained from the differential equations satified by these elliptic integrals when the indep...

Indefinite integrals of quotients of special functions

ABSTRACTA new method is presented for deriving indefinite integrals involving quotients of special functions. The method combines an integration formula given previously with the recursion relations obeyed by the function. Some additional results are presented using an elementary method, here called reciprocation, which can also be used in combination with the new method to obtain additional quotient integrals. Sample results are given here for Bessel functions, Airy functions, associated Legendre functions and the three complete elliptic integrals. All results given have been numerically checked with Mathematica.

Indefinite integrals involving Jacobi polynomials from integrating factors

A method was presented recently for deriving integrals of special functions using two kinds of integrating factor for the homogeneous second-order linear differential equations which many special f...

Indefinite integrals of special functions from hybrid equations

Elementary linear first and second order differential equations can always be constructed for twice differentiable functions by explicitly including the function's derivatives in the definition of ...

Indefinite integrals of special functions from inhomogeneous differential equations

A method is presented for deriving integrals of special functions which obey inhomogeneous second-order linear differential equations. Inhomogeneous equations are readily derived for functions sati...

Exact solutions for the mutual inductance of circular coils and elliptic coils

An exact solution is presented for the mutual inductance between general noncoaxial thin circular and elliptic coils with parallel axes. The thin coil solution is given as an angular integral of an elliptic integral expression. In addition, for the coaxial case, an exact solution is given for the mutual inductance of a thick circular coil and a thick elliptic coil. The elliptic coil is such that the coil thickness is the same along both elliptic semi-axes. The thick coil solution is given as an integral of an expression involving Bessel and Struve functions. Extensive numerical results for sample geometries are given for both solutions, which are cross checked against each other in the limi…

Indefinite integrals of special functions from integrating factors

Some general integrals are presented which were obtained from two integrating factors f(x) and fˆ(x) for the first two and last two terms, respectively, of the second-order linear ordinary differen...

A Lagrangian method for deriving new indefinite integrals of special functions

A new method is presented for obtaining indefinite integrals of common special functions. The approach is based on a Lagrangian formulation of the general homogeneous linear ordinary differential equation of second order. A general integral is derived which involves an arbitrary function, and therefore yields an infinite number of indefinite integrals for any special function which obeys such a differential equation. Techniques are presented to obtain the more interesting integrals generated by such an approach, and many integrals, both previously known and completely new are derived using the method. Sample results are given for Bessel functions, Airy functions, Legendre functions and hype…