Search results for "Elliptic integral"
showing 10 items of 14 documents
Indefinite integrals involving the incomplete elliptic integrals of the first and second kinds
2016
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.
Indefinite integrals of incomplete elliptic integrals from Jacobi elliptic functions
2017
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 involving the Jacobi Zeta and Heuman Lambda functions
2017
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
2015
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…
Application of the Pontryagin maximum principle to the time-optimal control in a chain of three spins with unequal couplings
2014
We solve a time-optimal control problem in a linear chain of three coupled spins 1/2 with unequal couplings. We apply the Pontryagin maximum principle and show that the associated Hamiltonian system is the one of a three-dimensional rigid body. We express the optimal control fields in terms of the components of the classical angular momentum of the rigid body. The optimal trajectories and the minimum control time are given in terms of elliptic functions and elliptic integrals.
Analytical Solutions for the Self- and Mutual Inductances of Concentric Coplanar Disk Coils
2013
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.
Analytical solution for the solid angle subtended at any point by an ellipse via a point source radiation vector potential
2010
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 …
Calculations for a disk source and a general detector using a radiation vector potential
2008
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…
Exact solutions for the mutual inductance of circular coils and elliptic coils
2012
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…
Time-optimal selective pulses of two uncoupled spin-1/2 particles
2018
We investigate the time-optimal solution of the selective control of two uncoupled spin 1/2 particles. Using the Pontryagin Maximum Principle, we derive the global time-optimal pulses for two spins with different offsets. We show that the Pontryagin Hamiltonian can be written as a one-dimensional effective Hamiltonian. The optimal fields can be expressed analytically in terms of elliptic integrals. The time-optimal control problem is solved for the selective inversion and excitation processes. A bifurcation in the structure of the control fields occurs for a specific offset threshold. In particular, we show that for small offsets, the optimal solution is the concatenation of regular and sin…