5 - Indefinite Integrals of Special Functions

Summarized is the recent progress of the new methods to compute Legendre's complete and incomplete elliptic integrals of all three kinds and Jacobian elliptic functions. Also reviewed are the entirely new methods to (i) compute the inverse functions of complete elliptic integrals, (ii) invert a general incomplete elliptic integral numerically, and (iii) evaluate the partial derivatives of the elliptic integrals and functions recursively. In order to avoid the information loss against small parameter and/or characteristic, newly introduced are the associate complete and incomplete elliptic integrals. The main techniques used are (i) the piecewise approximation for single variable functions, and (ii) a systematic utilization of the half and double argument transformations and the truncated Maclaurin series expansions for the others. The new methods are of the errors of 5 ulps at most without any chance of cancellation against small input arguments. They run significantly faster than the existing methods: (i) slightly faster than Bulirsch's procedure for the incomplete elliptic integral of the first kind, (ii) 1.5 times faster than Bulirsch's procedure for Jacobian elliptic functions, (iii) 2.5 times faster than Cody's and Bulirsch's procedures for the complete elliptic integrals, and (iv) 3.5 times faster than Carlson's procedures for the incomplete elliptic integrals of the second and third kind. Their Fortran programs are available at https://www.researchgate.net/profile/Toshio_Fukushima

Based on the computerized symbolic computation, a new rational expansion method using the Jacobian elliptic function was presented by means of a new general ansätz and the relations among the Jacobian elliptic functions. The results demonstrated an effective direction in terms of a uniformed construction of the new exact periodic solutions for nonlinear differential–difference equations, where two representative examples were chosen to illustrate the applications. Various periodic wave solutions, including Jacobian elliptic sine function, Jacobian elliptic cosine function and the third elliptic function solutions, were obtained. Furthermore, the solitonic solutions and trigonometric function solutions were also obtained within the limit conditions in this paper.

Numerical inversion of a general incomplete elliptic integral

This volume is a basic introduction to certain aspects of elliptic functions and elliptic integrals. Primarily, the elliptic functions stand out as closed solutions to a class of physical and geometrical problems giving rise to nonlinear differential equations. While these nonlinear equations may not be the types of greatest interest currently, the fact that they are solvable exactly in terms of functions about which much is known makes up for this. The elliptic functions of Jacobi, or equivalently the Weierstrass elliptic functions, inhabit the literature on current problems in condensed matter and statistical physics, on solitons and conformal representations, and all sorts of famous problems in classical mechanics. The lectures on elliptic functions have evolved as part of the first semester of a course on theoretical and mathematical methods given to first- and second-year graduate students in physics and chemistry at the University of North Dakota. They are for graduate students or for researchers who want an elementary introduction to the subject that nevertheless leaves them with enough of the details to address real problems. The style is supposed to be informal. The intention is to introduce the subject as a moderate extension of ordinary trigonometry in which the reference circle is replaced by an ellipse. This entre depends upon fewer tools and has seemed less intimidating that other typical introductions to the subject that depend on some knowledge of complex variables. The first three lectures assume only calculus, including the chain rule and elementary knowledge of differential equations. In the later lectures, the complex analytic properties are introduced naturally so that a more complete study becomes possible.

5 - INDEFINITE INTEGRALS OF SPECIAL FUNCTIONS

In this work, we give approximate expressions for Jacobian and elliptic Weierstrass functions and their inverses by means of the elementary trigonometric functions, sine and cosine. Results are reasonably accurate. We show the way the obtained results may be applied to solve nonlinear ODEs and other problems arising in nonlinear physics. The importance of the results in this work consists on giving easy and accurate way to evaluate the main elliptic functions cn, sn, and dn, as well as the Weierstrass elliptic function and their inverses. A general principle for solving some nonlinear problems through elementary functions is stated. No similar approach has been found in the existing literature.

3.1 Introduction In addition to the gamma function, there are numerous other special functions whose primary definition involves an integral. Some of these functions were introduced in Chap. 2 along with the gamma function, and in this chapter we consider several others. The error function derives its name from its importance in the theory of errors, but it also occurs in probability theory and in certain heat conduction problems on infinite domains. The closely related Fresnel integrals, which are fundamental in the theory of optics, can be derived directly from the error function. A special case of the incomplete gamma function (Sec. 2.5) leads to the exponential integral and related functions—the logarithmic integral, which is important in analysis and number theory, and the sine and cosine integrals, which arise in Fourier transform theory. Elliptic integrals first arose in the problems associated with computing the arclength of an ellipse and a lemniscate (a curve in the shape of a figure eight). Some early results concerning elliptic integrals were discovered by L. Euler and J. Landen, but virtually the whole theory of these integrals was developed by Legendre over a period spanning 40 years. The inverses of the elliptic integrals, called elliptic functions, were independently introduced in 1827 by C. G. J. Jacobi (1802–1859) and N. H. Abel (1802–1829). Many of the properties of elliptic functions, however, had already been developed as early as 1809 by Gauss. Elliptic functions have the distinction of being doubly periodic, with one real period and one imaginary period. Among other areas of application, the elliptic functions are important in solving the pendulum problem (Sec. 3.5.2).

6.-7 - DEFINITE INTEGRALS OF SPECIAL FUNCTIONS

12 - Elliptic Integrals and Functions

The field of elliptic functions, apart from its own mathematical beauty, has many applications in physics in a variety of topics, such as string theory or integrable systems. This book, which focuses on the Weierstrass theory of elliptic functions, aims at senior undergraduate and junior graduate students in physics or applied mathematics. Supplemented by problems and solutions, it provides a fast, but thorough introduction to the mathematical theory and presents some important applications in classical and quantum mechanics. Elementary applications, such as the simple pendulum, help the readers develop physical intuition on the behavior of the Weierstrass elliptic and related functions, whereas more Interesting and advanced examples, like the n=1 Lame problem-a periodic potential with an exactly solvable band structure, are also presented.

In this paper, we investigate the variable coefficients Sasa-Satsuma model, which can describe the propagation of a light pulse in a cylindrical fiber. We study this model and obtain rich solutions using two separate methods. We obtain analytical Weierstrass elliptic function solutions using the Weierstrass elliptic function expansion method. Some Jacobi elliptic function solutions are obtained using the modified Jacobi elliptic function expansion method. When the Jacobi elliptic function degenerates, we obtain the corresponding trigonometric, hyperbolic function solutions and periodic solutions. We also try to take the coefficients of the equation as some functions and obtain some more complicated exact solutions, which have not appeared in previous studies. Finally, we simulate some waveform diagrams of the solutions using the computer software Mathematica and obtain periodic waves, bright and dark soliton, double solitons and some complex periodic waves. With these waveform diagrams, we can observe the dynamical behavior of the solutions more clearly.

Optical solitons in birefringent fibers with four-wave mixing for quadratic–cubic nonlinearity by F-expansion

Direct and inverse interpolation for Jacobian elliptic functions, zeta function of Jacobi and complete elliptic integrals of the second kind

Only two years after the publication of their volume of integrals and sums of elementary functions [1], reviewed in [2], the authors now present a 749 page collection of formulas for integrals and sums of special functions. The main part of the volume is divided into five chapters, and each chapter is divided into many sections and subsections. As in [1], the notation is standard, and knowledge of Russian (required for the few short sections of text) is not essential. Chapter 1 (54 pages) deals with indefinite integrals, including integrands involving incomplete gamma functions, the exponential integral, the error functions, Fresnel integrals, different types of Bessel and Hankel functions, and orthogonal polynomials; as well as products of these functions with powers, logarithms, and exponential functions. The very long Chapter 2 (562 pages) consists of definite integrals. This chapter is very impressive, and many of the formulas it contains seem to have been compiled for the first time. It is divided into sections for integrands containing the gamma function, the psi function, the Riemann zeta function, the exponential integral, ordinary and hyperbolic sines and cosines, error functions, Fresnel integrals, incomplete gamma functions, parabolic cylinder functions, ordinary and modified Bessel functions (very extensive), Hankel functions, and Legendre, Laguerre, Hermite, Gegenbauer, and Jacobi polynomials. Many integrands contain several of these functions, often in combination with elementary functions, and depend on a certain number of parameters. Results are frequently given as infinite series or as hypergeometric functions pFq, which may limit their practical applicability in certain cases. Chapter 3 (18 pages) contains double, triple, and some multiple integrals; in particular, many involving products of Bessel functions with exponential functions or powers. Chapter 4 (11 pages) gives finite sums; in particular of (ordinary and modified) Bessel functions, and of Legendre, Laguerre, Hermite, Gegenbauer, and Jacobi polynomials. Sums involving products of these polynomials are also given. Chapter 5 (73 pages) contains an extensive compilation of infinite series; in particular, series involving incomplete gamma functions, the Riemann zeta function,

The Weierstrass elliptic function expansion method and its applications in nonlinear wave equations