Incomplete Elliptic Integrals Research Articles

Approximate Formulas for Sagging Deflection of an Elastic Rod under Transverse Loading

Sagging deflection of a thin elastic rod under transverse loading is expressed approximately in terms of elementary functions. Series expansions of the complete and incomplete elliptic integrals of the first and second kind in the neighborhood of k2 = 1/2 are obtained.

Real-time CNC interpolators for Bézier conics

Arbitrary conic segments can be specified in the rational Bézier form, r(ξ) for ξ∈[0,1], by control points p 0, p 1, p 2 and a scalar weight w 1. An expression for the cumulative arc length function s( ξ), amenable to accurate and efficient evaluation, is required in formulating real-time CNC interpolators capable of achieving a desired (constant or varying) feedrate V=d s/d t along such curves. For w 1=1 (a parabola), s( ξ) admits a closed-form expression that entails a single square root and natural logarithm in its evaluation. However, for w 1<1 (an ellipse) or w 1>1 (a hyperbola), complete and incomplete elliptic integrals of the first and second kind arise in s( ξ). A recursive algorithm, based on the arithmetic-geometric mean, provides a rapidly-convergent scheme to compute such integrals to machine precision in real-time applications. These methods endow CNC machines with the ability to realize time-dependent feedrates precisely along “simple” analytic curves (conics), furnishing a natural complement to the currently-available exact real-time interpolators for free-form Pythagorean-hodograph (PH) curves.

Age-redshift relation for standard cosmology

We present compact, analytic expressions for the age-redshift relation $\ensuremath{\tau}(z)$ for standard Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) cosmology. The new expressions are given in terms of incomplete Legendre elliptic integrals and evaluate much faster than by direct numerical integration.

Numerical calculation for the magnetic field in current-carrying circular arc filament

This paper introduces numerical calculations for the vector potential and the magnetic field in current-carrying circular arc filaments of arbitrary length. The analytical expressions which are derived in Cartesian coordinates for the vector potential and magnetic field consist of only complete and incomplete elliptic integrals of the first and second kinds and thus permit an efficient computation algorithm by applying arithmetic-geometric mean (AGM) numerical integrals.

Accurate Computation of Elliptic Functions

AbstractIn this note we present rapidly convergent algorithms depending on the method of arithmetic‐geometric means (AGM) for the computation of the incomplete elliptic integral of the first kind, Jacobi's Theta‐ and Zeta‐function. In particular we derive explicit a priori bounds for the error accumulation of the descending Landen transform.

The Elastic Field Resulting From Elliptical Hertzian Contact of Transversely Isotropic Bodies: Closed-Form Solutions for Normal and Shear Loading

This analysis presents the elastic field in a half-space caused by an ellipsoidal variation of normal traction on the surface. Coulomb friction is assumed and thus the shear traction on the surface is taken as a friction coefficient multiplied by the normal pressure. Hence the shear traction is also of an ellipsoidal variation. The half-space is transversely isotropic, where the planes of isotropy are parallel to the surface. A potential function method is used where the elastic field is written in three harmonic functions. The known point force potential functions are utilized to find the solution for ellipsoidal loading by quadrature. The integrals for the derivatives of the potential functions resulting from ellipsoidal loading are evaluated in terms of elementary functions and incomplete elliptic integrals of the first and second kinds. The elastic field is given in closed-form expressions for both normal and shear loading.

Elliptic integral solutions for the kinetics of autocatalytic termolecular reactions with reactive intermediate species

Abstract The kinetics of an autocatalytic termolecular reaction with overall stoichiometry of A + B → C + D was investigated. It is assumed that each of the three components A, B and C dissociate into reactive intermediate species, with equilibrium always established in all three cases. The rate-limiting step is then assumed to be the reaction between the intermediates formed from A and B, respectively, catalyzed by the reactive intermediate formed from product C. Integration of the differential equation describing the chemical kinetics of this process was performed for various special cases of initial reactant concentrations—stoichiometric and non-stoichiometric. The analytical solution in the latter case incorporates incomplete elliptic integrals of the first kind. Numerical values of reaction times may then be readily computed with the aid of various extensive compilations of elliptic integrals.

Electromagnetic Field Induced By an Infinitelysimally Thin Ring Moving in a Plane Condeser

The main idea of a wake field transformer is to induce a high (wake) field by a beam of high intensity but low energy per particle (called the drive beam) accelerating a second beam with low intensity but high energy gain per particle. A geometry suitable for a wake field accelerator is a beam widened to a ring passing a thin pill-box cavity (that is a cylinder with a circular base) (Weiland, Voss 1982). In this report a problem with the same circular symmetry was investigated. The problem consists of a ring-shaped line charge in a plane parallel to the plates of a plane condensor and moving in a direction perpendicular to the plates. The ring is at rest in front of one condensor plate until it starts suddenly to move with constant velocity; then the ring traverses the plane condensor until it reaches the second plate, where it stops. The field is derived from the potentials. The only non-zero component, the longitudinal one, of the vector potential is computed with the method of Green's function. The scalar potential is computed by using the Lorentz gauge. The field can be expressed as an infinite sum over complete and incomplete elliptic integrals and simpler terms. But at any given finite time only a finite number of terms of the infinite sum contributes. The field is evaluated numerically with the computer and shown in several figures.

Approximation Error and Error Accumulation for the Landen Transform

In this note we derive explicit a priori error bounds for the approximation error and error accumulation of the descending Landen transform. Our results apply to incomplete elliptic integral of the first kind and give the framework to calculate error bounds in the representation of Jacobi's Zeta- and Theta-function.

Numerical computation of incomplete elliptic integrals of a general form

We present an algorithm to compute the incomplete elliptic integral of a general form. The algorithm efficiently evaluates some linear combinations of incomplete elliptic integrals of all kinds to a high precision. Some numerical examples are given as illustrations. This enables us to numerically calculate the values and the partial derivatives of incomplete elliptic integrals of all kinds, which are essential when dealing with many problems in celestial mechanics, including the analytic solution of the torque-free rotational motion of a rigid body around its barycenter.

High precision computation of solenoid magnetic fields by Garrett's methods

For magnetic field calculations of solenoids two methods, due to M.W. Garrett (1967), are developed. The first one consists of the expansion of the magnetic field in a zonal harmonic series in the central region of solenoid and in outer space. For both regions the expansion coefficients are given either by simple recurrence relations, or by explicit expressions. In particular, this method is suitable for high accuracy calculations of the field within windings. In the second method, the field is expressed in terms of incomplete elliptic integrals of the first and second kinds computed by use of raising Landen transformations. Given the high precision and high computation rate of this method, it has advantages in analyzing field problems for windings consisting of many radial sections carrying different currents, as occurs during the quench of superconducting multisectioned magnets. Also, the general formulas for mutual inductance and axial force between a pair of coaxial solenoids are given.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Compact extended algorithms for elliptic integrals in electromagnetic field and potential computations. II. Elliptic integral of third kind with extended integration range

Electromagnetic field and potential computations on elements with curved contours in boundary and volume integral methods (BEM, VIM) require evaluation of a number of Jacobian complete and/or incomplete elliptic integrals of all the three kinds for the same modulus but different angles depending upon the arc length and the angle coordinate of the field point. Up to now they have been evaluated individually repeating the same algorithms a number of times. To reduce such redundant computations, as in Part I for elliptic integrals of first and second kind, a new compact algorithm based on Bartky's transformation is developed in the present paper for the elliptic integral of third kind with an extended integration range -/spl pi//spl les/a/spl les//spl pi/. Computational accuracy and time-saving are discussed. >

On the integration of incomplete elliptic integrals

We give a closed-form evaluation of Erdélyi-Kober fractional integrals, involving incomplete elliptic integrals of the first kind, F ( φ, k ), and of the second kind, E ( φ, k ), which are integrated either with respect to the modulus or the amplitude. This is made possible by representing F ( φ, k ) and E ( φ, k ) in terms of the Kampé de Fériet double hypergeometric functions. Reduction formulae for these enable us to simplify the solutions for thirteen special cases, including integrals involving complete elliptic integrals. The hypergeometric character of the incomplete integrals is useful for evaluations of other classes of integrals involving F ( φ, k ) and E ( φ, k ).

The oCLP family of triply periodic minimal surfaces

CLP surfaces with orthorhombic distortion (oCLP for short) are a family of two-parameter triply periodic embedded minimal surfaces. We show that they correspond to the Weierstrass function of the form κ/√ τ 8 -(A+B)τ 6 +(2+AB)τ 4 -(A+B)τ 2 +1 where A and B are free parameters with -2 B, τ is complex with |τ|≤1 and κ real and depends on A and B. When B=-A, the oCLP family reduces to the one-parameter CLP family with tetragonal symmetry. The Enneper-Weierstrass representation of oCLP surfaces involves pseudo-hyperelliptic integrals which can be reduced to elliptic integrals. We derive parametric equations for oCLP surfaces in terms of incomplete elliptic integrals F (O, k) alone. These equations completely avoid integration of the Weierstrass function, thus making the use of the Enneper-Weierstrass representation unnecessary in the computation of specific oCLP surfaces. We derive analytical expressions for the normalization factor and the edge-to-length ratios in terms of the free parameters. This solves the problem of finding the oCLP saddle surface inscribed in given a right tetragonal prism, crucial for the modelling of structural data using a specific surface, and enables straightorward physical applications. We have comuted exactly the coordinates of oCLP surfaces corresponding to several prescribed values of the edge-to length ratio

Axisymmetric stress distribution in the vicinity of an external crack under general surface loadings

A solution is derived of the equations of equilibrium appropriate to the axisymmetric loading on the faces of a plane crack covering the outside of a circle of radius a in an infinite isotropic elastic body. Abel transforms of stress and displacement components at an arbitrary point of the solid are known in the literature in terms of the Abel transforms of the jumps of the stress and displacement components at the crack plane. Limiting values of these expressions, as we approach the crack plane from either side, are a great help in dealing with the problem. The boundary conditions lead to Abel type integral equations which admit closed form solutions. The expressions for stress and displacement components on the crack plane are obtained explicitly in terms of the prescribed stress components on the crack surfaces. The surface tractions on the crack are symmetrical about the centre of the circle but not necessarily self-equilibrating.In order to illustrate the use of general formulae, some special cases of the loading functions are discussed in detail. In the first case the upper crack surface is subjected to a uniform normal stress acting over a circular ring of inner and outer radii ε and c, respectively, while the lower crack surface is free from tractions. In a second case the upper crack surface is subjected to a radially decaying shear load acting over a circular ring of inner and outer radii ε and c, respectively, while the lower crack surface is free from tractions. Also, we obtain the results for normal and shear concentrated ring loads by taking the limit as ε → c while the total applied load is kept fixed. The expressions for displacement components on the crack plane are in terms of complete as well as incomplete elliptic integrals of the first and second kinds. Numerical calculations for the normal components of displacement on the crack surfaces are carried out and the results are presented graphically.

The volume common to two congruent circular cylinders

This paper derives two formulae for the volume of the set of points common to the interiors of two congruent circular cylinders in general position. One formula involves incomplete elliptic integrals and the other formula involves complete elliptic integrals. A series is derived for the complete elliptic function formula. Some of the work was done on the symbol manipulator MACSYMA.

The T and CLP families of triply periodic minimal surfaces. Part 1. Derivation of parametric equations

The local Weierstrass representation of all members of the T and CLP families of triply periodic minimal surfaces involves integrals of the function R (r ) = II r~ + A r~ + (with cc ~A ~ -2 for T and 2~A~2 for CLP), which were previously evaluated only by numerical integration. We show that these integrals are pseudo-hyperelliptic, express them analytically in terms of the incomplete elliptic integral of the first kind, F (p, k ), and give explicit general parametric equations for coordinates of these minimal surfaces. The procedure completely obviates the need for numerical integration. The solutions for all three coordinates are intrinsically periodic. The well-known properties of elliptic integrals and their inverse functions provide new insights into the features of triply periodic minimal surfaces, and permit their systematic evaluation.

New integral identity for elliptic functions

Analytic solutions of what are known as the reduced equations for the amplitudes and phases of oscillations and waves in nonlinear systems are used to establish a new integral identity for elliptic functions. The consequence of this identity is a new relationship between incomplete elliptic integrals of the first and third kinds.

Thermoelastic state of a medium, containing a thermally insulated ellipsoidal cavity, when subjected to a uniform heat flow

This article examines a problem of static thermoelasticity for an isotropic medium with an ellipsoidal cavity. The orientation of the cavity relative to the heat flow is assumed to be arbitrary. The solution of the problem is constructed in closed form and is expressed through incomplete elliptic integrals of the first and second kinds. In cases when the ellipsoid degenerates into a spheroid or sphere, the stress-strain state can be written in elementary functions of the ellipsoidal coordinate.

Compact extended algorithms for elliptic integrals in electromagnetic field and potential computations. I. Elliptic integrals of the first and second kind with extended integration range

Electromagnetic field and potential computations on elements with curved contours in boundary element and volume integral methods require evaluation of a number of Jacobian complete and/or incomplete elliptic integrals of all three kinds for the same modulus but different angles, depending on the arc length and the angle coordinate of the field point. Up to now, they have been evaluated individually, repeating the same algorithms a number of times. To reduce such redundant computations, a compact algorithm based on a modified Landen transformation is developed for elliptic integrals with an extended integration range - pi <or= alpha /sub i/<or= pi . Computational accuracy and time saving are discussed.<<ETX>>