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14 Mathieu Functions
In this chapter, we will mainly derive code for the evaluation of the even
parity eigenfunctions ce n (z, q), the odd parity eigenfunctions se n (z, q) and the
corresponding eigenvalues, the functions f e n (z, q) and ge n (z, q), the non-integer
functions ce ν (z, q), se ν (z, q), and me ν (z, q), and the solution of the modified
Mathieu functions Ce ν (z, q), Se ν (z, q), F e ν (z, q), and Ge ν (z, q). Most equations
used can be found in [3] and [1].
14.1 Function Overview
The SPECFUNPHYS class mathieuFun supports the following functions:
• the computation of the eigenvalues and eigenvectors of Eq. (14.1), a n (even
parity) and b n (odd parity),
• the corresponding functions ce n (z, q), se n (z, q), and me n (z, q), n integer,
• the second solutions f e n (z, q) and ge n (z, q) of the corresponding period
functions and the related scaling factors C n (q) and S n (q),
• the non-integer eigenfunctions ce ν (z, q), se ν (z, q), and me ν (z, q), and
• the even and odd parity functions w I (z; a, q) and w I I (z; a, q) via direct
integration.
• Supported eigensolutions of Eq. (14.2) are the integer functions Ce n (z, q),
Se n (z, q), and Me n (z, q),
• the second solutions F e n (z, q) and Ge n (z, q),
• the non-integer solutions Ce ν (z, q), Se ν (z, q), and Me ν (z, q), and
• W I (z; a, q) and W I I (z; a, q).
The SPECFUNPHYS functions eiga and eigb allow the computation of the
eigenvalues a n and b n and the corresponding eigenvectors.
The SPECFUNPHYS functions [z,y]=mathieudeq(a, q, w0, zspan)
and [z,y]=modmathieudeq(a, q, w0, zspan) support the direct evaluation of the differential equations with initial condition w0.
14.2 Fundamental Equations and Computation
We will start with the solutions of Eq. (14.1) before we turn to the modified Mathieu
equation, Eq. (14.2).
14.2.1 Mathieu Differential Equation
As Eq. (14.1) is a differential equation of the second order, the most general solution
is y(z) = Ay 1 (z) + By 2 (z) with arbitrary complex constants A and B. Due to
14 Mathieu Functions
In this chapter, we will mainly derive code for the evaluation of the even
parity eigenfunctions ce n (z, q), the odd parity eigenfunctions se n (z, q) and the
corresponding eigenvalues, the functions f e n (z, q) and ge n (z, q), the non-integer
functions ce ν (z, q), se ν (z, q), and me ν (z, q), and the solution of the modified
Mathieu functions Ce ν (z, q), Se ν (z, q), F e ν (z, q), and Ge ν (z, q). Most equations
used can be found in [3] and [1].
14.1 Function Overview
The SPECFUNPHYS class mathieuFun supports the following functions:
• the computation of the eigenvalues and eigenvectors of Eq. (14.1), a n (even
parity) and b n (odd parity),
• the corresponding functions ce n (z, q), se n (z, q), and me n (z, q), n integer,
• the second solutions f e n (z, q) and ge n (z, q) of the corresponding period
functions and the related scaling factors C n (q) and S n (q),
• the non-integer eigenfunctions ce ν (z, q), se ν (z, q), and me ν (z, q), and
• the even and odd parity functions w I (z; a, q) and w I I (z; a, q) via direct
integration.
• Supported eigensolutions of Eq. (14.2) are the integer functions Ce n (z, q),
Se n (z, q), and Me n (z, q),
• the second solutions F e n (z, q) and Ge n (z, q),
• the non-integer solutions Ce ν (z, q), Se ν (z, q), and Me ν (z, q), and
• W I (z; a, q) and W I I (z; a, q).
The SPECFUNPHYS functions eiga and eigb allow the computation of the
eigenvalues a n and b n and the corresponding eigenvectors.
The SPECFUNPHYS functions [z,y]=mathieudeq(a, q, w0, zspan)
and [z,y]=modmathieudeq(a, q, w0, zspan) support the direct evaluation of the differential equations with initial condition w0.
14.2 Fundamental Equations and Computation
We will start with the solutions of Eq. (14.1) before we turn to the modified Mathieu
equation, Eq. (14.2).
14.2.1 Mathieu Differential Equation
As Eq. (14.1) is a differential equation of the second order, the most general solution
is y(z) = Ay 1 (z) + By 2 (z) with arbitrary complex constants A and B. Due to