Mathieu functions for purely imaginary parameters

Periodic analytic approximate solutions for the Mathieu equation

“…It is interesting to remark that these results are similar to those obtained by C.H. Ziener et al in (12):…”

“…Thus, we are assuming from the very beginning that the series (12) are finite and sum from k = 0 up to k = n. This Ansatz is in the core of our method and, therefore, no convergence conditions should be imposed on (12). Then, with the use of trigonometric relations, we obtain an expression of the following type:…”

“…If the index m is even, all the above discussion remains valid although equation (12) should be replaced by…”

Asymptotics of the discrete spectrum for complex Jacobi matrices

“…Spectral properties of non-symmetric tridiagonal matrices and complex Jacobi matrices are investigated by several authors: Beckermann and Kaliaguine ( [1,2]), Djakov and Mityagin ([7,8]), Egorova and Golinskii ([10,11]) and others (see, e.g., [3,[12][13][14]21,23]). The connections of tridiagonal matrices with formal orthogonal polynomials on the complex plane, Mathieu equation and functions, and Bessel functions can be found in [1][2][3]7,13,21] and [25]. However, systematic research concerning spectral properties of non-selfadjoint tridiagonal operators is difficult because the structure of complex sequences can be more complicated then the structure of real sequences.…”

Lineshape of Magnetic Resonance and its Effects on Free Induction Decay and Steady-State Free Precession Signal Formation

“…e authors declare that they have no con icts of interest. Analytical solutions for this partial di erential equation can be given for di usion in a constant gradient as given in Equation (6) in terms of Airy-functions [32,33], for di usion around a two-dimensional dipole eld [55,56] as given in Equation (7) in terms of Bessel-functions [57] and Mathieufunctions [58]. For dephasing on the alveolar surface as given in Equation (9), the signal evolution can be expressed in terms of spheroidal wave functions [18].…”