New indefinite integrals from a method using Riccati equations

CHAPTER II - LINEAR DIFFERENTIAL EQUATIONS. SUPPLEMENTARY REMARKS ON THE THEORY OF DIFFERENTIAL EQUATIONS

6.-7 - DEFINITE INTEGRALS OF SPECIAL FUNCTIONS

On the Microlocal Smoothing Effect of Dispersive Partial Differential Equations, I: Second-Order Linear Equations

Journal Article Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind Get access J. M. BORWEIN, J. M. BORWEIN Mathematics, Statistics and Computing Department, Dalhousie UniversityHalifax, Nova Scotia B3H 3J5 Search for other works by this author on: Oxford Academic Google Scholar I. J. ZUCKER I. J. ZUCKER Physics Department, King's CollegeLondon, WC2R 2LS Search for other works by this author on: Oxford Academic Google Scholar IMA Journal of Numerical Analysis, Volume 12, Issue 4, October 1992, Pages 519–526, https://doi.org/10.1093/imanum/12.4.519 Published: 01 October 1992 Article history Received: 30 July 1990 Revision received: 30 January 1991 Published: 01 October 1992

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 equations.

An integral transform $\bar f(s) = \int f(t)\varphi (s,t)dt$ can be averaged over the convex hull of $\{ s_1 , \cdots ,s_k \} $ to produce an analogous function $\bar F\{ s_1 , \cdots ,s_k \} $ of several real or complex variables. The question arises whether it is legitimate to take the average under the integral sign, so that $\bar F\{ s_1 , \cdots ,s_k \} = \int f(t)\Phi (s_1 , \cdots ,s_k ;t)dt$, where $\Phi $ is the corresponding average of $\varphi $. Conditions for the validity of this equation are of interest in the theory of special functions because the kernel $\Phi $ may be a Bessel function, elliptic integral, or other hypergeometric function when $\varphi $ is the kernel of a Laplace, Fourier, or Stieltjes transformation. Sufficient conditions of validity are established in the case of these three transformations and the inverse Laplace, Fourier, and Mellin transformations. The averages have some but not all of the operational properties of the corresponding ordinary transforms. Some examples are given involving various special functions.

6–7 - Definite Integrals 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.

Sharp Landen transformation inequalities for hypergeometric functions, with applications

Legendre hyperelliptic integrals, π new formulae and Lauricella functions through the elliptic singular moduli

In this paper, an improvement for calculating the self-inductance of circular coils of rectangular cross section is presented. An integral of elementary functions is applied to replace the Bessel and Struve functions in the integrand of the inductance expression, and the term of improper integral without the exponential factor is solved to a closed form which consists of the complete elliptic integrals of the first and second kind and generalized hypergeometric functions. Due to this method, the main difficulties of the numerical evaluations of the inductance expression are overcome. Excellent agreement with published method is confirmed by the numerical comparisons and the calculations can be accelerated to several tens to hundreds of times depending on the shape factors of the coils. Similar expressions of the special cases (thin-wall solenoids and disk coils) are also given and compared with previous methods. Furthermore, the asymptotic behaviors of the special coils which tend to be a circular ring are obtained and numerically validated.

Asymptotic formulas with error bounds are obtained for the first elliptic integral near its logarithmic singularity. It is convenient to start from the more general problem of expanding the integral over the positive t-axis of $[(t + x)(t + y)(t + z)(t + w)]^{{{ - 1} / 2}} $. The method of Mellin transforms gives an asymptotic expansion that converges uniformly if ${{0 < \max \{ x,y\} } / {\min \{ z,w\} \leqq r < 1}}$. Each term of the series contains a Legendre polynomial and the derivative of a Legendre function with respect to its degree; this derivative involves nothing worse than a logarithm. A simple bound for the relative error of the Nth partial sum is obtained from Wong’s formula for the remainder, aided by Chebyshev’s integral inequality. Error bounds are given also for more accurate asymptotic formulas containing a complete elliptic integral. Formulas for the standard elliptic integral of the first kind are obtained in the case $w = \infty $. The method of Mellin transforms and Wong’s formula are discussed in an appendix.

The contour integrals, occurring in the arbitrary-order phase-integral quantization conditions given in a previous paper, are in the first- and third-order approximations expressed in terms of complete elliptic integrals in the case that the charges of the Coulomb centers are equal. The evaluation of the integrals is facilitated by the knowledge of quasiclassical dynamics. The resulting quantization conditions involving complete elliptic integrals are solved numerically to obtain the energy eigenvalues and the separation constants of the 1sσ and 2pσ states of the hydrogen molecule ion for various values of the internuclear distance. The accuracy of the formulas obtained is illustrated by comparison with available numerically exact results.

Generalized trigonometric functions are applied to the Legendre-Jacobi standard form of complete elliptic integrals, and a new form of the generalized complete elliptic integrals of the Borweins is presented. According to the form, it can be easily shown that these integrals have similar properties to the classical ones. In particular, it is possible to establish a computation formula of the generalized $\pi$ in terms of the arithmetic-geometric mean, in the classical way as the Gauss-Legendre algorithm for $\pi$ by Salamin and Brent. Moreover, an elementary new proof of Ramanujan's cubic transformation is also given.

Analytical and numerical methods for massive two-loop self-energy diagrams