A new method to compute Mathieu functions

On spectral approximations in elliptical geometries using Mathieu functions

“…This motivates most of the algorithms for computing Mathieu functions and their eigenvalues, see, e.g., [28,1,26,36,5,3]. A package of Fortran and Matlab programs for manipulating Mathieu functions is also available (cf.…”

“…While there have been quite a few papers/books devoted to analytic, numerical and various other aspects of Mathieu functions (cf., for instance, [26,6,3,36]), but there are essentially no results on the approximation of periodic functions using Mathieu functions (in Sobolev spaces). Such results are necessary for spectral methods for elliptic geometries.…”

“…Since then, these functions have been extensively used in many areas of physics and engineering (cf., for instance, [25,23,19,4,21,5,32,8,17]), and many efforts have also been devoted to analytic, numerical and various other aspects of Mathieu functions (cf., for instance, [26,6,3,36]). A considerable amount of mathematical results of Mathieu functions are collected in the books [28,29,1].…”

Quantum mechanics of a single ion in a paul trap: motion influenced by a time dependent linear force

“…Therefore we separate (following Brown [8], cf. also [13]) these functions into modulus and phase, x ± (t) = e(t) exp { + i q~ (t)}.…”

“…The constant w is the "Wronskian" (2+ x _ -x + 2_)/(2i), which is fixed by the initial values of x_+. Choosing [13] x+=xl+_iwx2 we have cp(0)=0, c~(0)= 1 and a(0)=0 as well as ~b(0)=w. From 4 > 0 we have w > 0.…”