MathieuF.jl

Julia package for Mathieu Functions with function forms similar to Mathieu related functions in Mathematica.

Mathieu functions are the eigenvalues and eigenfunction of Mathieu's equation (for more details, see NIST's Digital Library of Mathematical Functions).

Related Package: MathieuFunctions.jl

Highlights

• Support Fourier coefficients of non-integer order eigenfunctions
• Support a,q as input, see Mathieu functions toolbox
• Support output of the related Wronskian.

Examples

nu,ck=MathieuExponent(a,q)

where nu is the characteristic exponent and vector ck is the Fourier coefficients of the eigenfunction with norm(ck)≈1. Note that nu is reduced to the interval [0,2] and c0 corresponds to ck[(length(ck)-1)÷2] with the reduced nu. (For nu is real, the procedure actually reduces nu into [0,1]).

W=MathieuWron(nu,ck::Vector,index::Int)

where W is the Wronskian of the eigenfunction with and index refers to the index of c0 in ck. For example,

a=0.1;q=0.5;
nu,ck=MathieuExponent(a,q)
idx=(length(ck)-1)÷2+1
W=MathieuWron(nu,ck,idx)

In some cases, W could be negative. One can replace nu with -nu and reverse ck, i.e., reverse!(ck), to get a positive W. If one prefers a positive nu, one can further shift nu with nu+=2 and idx with idx+=1. Code example:

nu=2-nu
reverse!(ck)
idx+=1
W_new=MathieuWron(nu,ck,idx)

It can be verified that W_new==-W.

If one knows nu (not reduced) and q, one can use

W=MathieuWron(nu,q)

During my test, the result is positive with such method.