Last edited by Narr
Thursday, July 16, 2020 | History

2 edition of On the computation of Mathieu functions. found in the catalog.

On the computation of Mathieu functions.

Gertrude K. Blanch

# On the computation of Mathieu functions.

## by Gertrude K. Blanch

Published by U.S. Dept. of Commerce, National Bureau of Standards in Washington .
Written in English

Subjects:
• Functions.

• Edition Notes

Classifications The Physical Object Other titles Mathieu functions. Series [U.S.] National Bureau of Standards. Mathematical table MT37 LC Classifications QA351 .B55 Pagination 20 p. : Number of Pages 20 Open Library OL6127810M LC Control Number 52061819 OCLC/WorldCa 8039595

The KINDLE book Accurate Computation of Mathieu Functions by Malcolm M. Bibby and Free Download PDF eBook and Read Online” Hihwfkapp Sep 6, 2 min read. G. Wolf Fachbereich Mathematik, University Duisburg-Essen, Essen, Germany.. Acknowledgements: This chapter is based in part on Abramowitz and Stegun (, Chapter 20) by G. Blanch. Notes: The main references used in writing this chapter are Arscott (), McLachlan (), Meixner and Schäfke (), and Meixner et al. ().For §§ – the main source is Magnus and Winkler ().

Mathieu Functions Mathieu functions appear frequently in physical problems involving elliptical shapes or periodic potentials. These functions were first introduced by Mathieu () when analyzing the solutions to the equation y'' + a − 2 ⁢ q ⁢ cos ⁡ 2 ⁢ z ⁢ y = 0, which arises from the separation of the 2. Toyama and K. Shogen, Computation of the value of the even and odd Mathieu functions of order 푁 for a given parameter 푆 and an argument 푋, IEEE Trans. .

Functions of Matrices at Amazon. Functions of Matrices at Eurospan. For those outside the USA. No shipping costs and 30% discount for SIAM members on quoting special code. Functions of Matrices at Google Play. A BibTeX bib entry for the book. A SIAM flyer for the book. A word cloud for the book. First edition of the first book on Mathieu functions and Spheroidal functions. McLachlan is also famous for his pioneering work on loud speakers, issued in Born in Britain, after WWII he found work teaching at the Carnegie Institute of Technology, then briefly at the University of Michigan before landing at Brown University, both teaching.

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### On the computation of Mathieu functions by Gertrude K. Blanch Download PDF EPUB FB2

The contribution of this book is documentation of successful numerical techniques for computation of Mathieu functions of the first and second kinds. The recommended (and demonstrably successful) methods are described early in a step-by-step.

Tables are included to check your software during early development, and two applications in Cited by: 6. The article presents methods for the computation of all Mathieu functions of integer order, which cover a large range of n and h; previous algorithms were limited to small values of n.

After preliminary historical remarks, Profesor McLachlan presents the theory in chapters covering such topics as computation of Mathlieu functions; computation of characteristic numbers and coefficients; numerical solution of equations (with worked examples, etc.); Hill's equation; solutions in series of Bessel functions; wave equations in elliptical coordinates; zeros of the Mathlieu and 4/5(1).

This lecture presents a modern approach for the computation of Mathieu functions. These functions find application in boundary value analysis such as electromagnetic scattering from elliptic cylinders and flat strips, as well as the analogous acoustic and optical problems, and many other applications in science and engineering.

Definition Mathieu functions. In some usages, Mathieu function refers to solutions of the Mathieu differential equation for arbitrary values of no confusion can arise, other authors use the term to refer specifically to - or -periodic solutions, which exist only for special values of and.

More precisely, for given (real) such periodic solutions exist for an infinite number of values. The article presents methods for the computation of all Mathieu functions of integer order, which cover a large range of n and h; previous On the computation of Mathieu functions.

book were limited to small values of algorithms are given in sufficient detail to enable straightforward implementation. In fact, it becomes a Mathieu equation with replaced by i».

Because a similar change of variables relates the modiﬂed Bessel functions In(x) and Kn(x) to the ordinary Bessel functions Jn(x) and Yn(x), the R equation is called the modiﬂed Mathieu equation.

Thus, if we identify a speciﬂc solution to the Mathieu equation. A continued fraction function algorithm is developed to evaluate general-order Mathieu characteristic numbers, and a new technique is presented for evaluating the Mathieu determinant which can be used to compute the order directly.

Shirts, The computation of eigenvalues and solutions of Mathieu's differential equation for noninteger order, ACM Trans. Math. Software 19 (), SM75 B. Sinha and R. MacPhie, On the computation of the prolate spheroidal radial functions of the second kind, J. Math. Phys. 16 (), TP COMPUTATION AND APPLICATIONS OF MATHIEU FUNCTIONS: A HISTORICAL PERSPECTIVE CHRIS BRIMACOMBEy, ROBERT M.

CORLESS z, AND MAIR ZAMIR Abstract. Mathieu functions of period ˇor 2ˇ, also called elliptic cylinder functions, were introduced in by Emile Mathieu together with so-called modi ed Mathieu functions, in order.

Mathieu functions of period \$π\$ or \$2π\$, also called elliptic cylinder functions, were introduced in by Émile Mathieu together with so-called modified Mathieu functions, in order to help understand the vibrations of an elastic membrane set in a fixed elliptical hoop.

These functions still occur frequently in applications today: our interest, for instance, was stimulated by a problem of. For the Mathieu equation, we consider finding eigenvalues with a given index (on the basis of oscillation theorems for the relevant difference equations), the stability of solutions to the difference equations, correct definition and calculation of eigenvalues and Mathieu functions with noninteger numbers, correct definition and calculation of the Mathieu characteristic exponent, and the.

A number of subroutines written in FORTRAN 4 to compute ordinary and modified Mathieu functions are described. A simpler and less accurate program for the computation of ordinary Mathieu functions is included.

The programs were written for use in connection with investigations on waveguides with elliptical cross sections. This article presents the outline of a set of routines that enable the computation of all Mathieu functions of integer orders for large range of the order n and the parameter h.

Algorithm Subroutines for the Computation of Mathieu Functions of Integer Orders  and a Dover reprint of a book first published in   This collection of special mathematical functions originated in Fortran programs by S.

Zhang & J. Jin, which accompany their book "Computation of Special Functions" (Wiley, ). The Matlab versions are direct machine conversions of the Fortran sources via.

Mathieu functions[ma′tyü ‚fəŋkshənz] (mathematics) Any solution of the Mathieu equation which is periodic and an even or odd function. Mathieu Functions special functions introduced by the French mathematician E. Mathieu in in the course of his work on the vibrations of an elliptical membrane.

Mathieu functions are also used in the study. MATHEMATICS OF COMPUTATION Vol NumberAprilPages – S (08) Article electronically published on Novem ON SPECTRAL APPROXIMATIONS IN ELLIPTICAL GEOMETRIES USING MATHIEU FUNCTIONS JIE SHEN AND LI-LIAN WANG Abstract.

We consider in this paper approximation properties and applica-tions of Mathieu. References  Shanjie Zhang and Jianming Jin, Computation of special functions, John Wiley, There seems to be still some contradictions, because the reference claims for the even functions to have a period of pi, where a simple plot shows that the scipy mathieu functions.

Section shows how to implement in MATLAB a Mathieu equation and simulate the roll motion produced by parametric excitation. More complicated models can be simulated in a similar manner by writing the governing equations as systems of first-order. A complete method and expressions for computing Mathieu characteristic numbers (MCNs) of integer orders are given.

The paper reviews available algorithms and their shortcomings, and then suitable methods are selected on the basis of simplicity and accuracy. () The computation of eigenvalues and solutions of Mathieu's differential equation for noninteger order. ACM Transactions on Mathematical Software (TOMS)() Chebyshev approximations for cos πχ2 and sin πχ2 with applications.Mathieu functions for the ﬁrst time.

In most books we ﬁnd a high density of equations which, from a didactic point of view, can be scary for the unfamiliar reader. and the computation of the Mathieu functions and their eigenvalues still presents some numerical difﬁculties.8,9.To avoid the complicated numerical computation of the Mathieu functions, he approximated these functions in terms of Bessel functions.

It is pointed out that the computation of Mathieu functions, or their approximate expressions can be avoided altogether if a perturbation method is applied to find the resonant frequencies of the two orthogonal.