14
Mathieu Functions
Abstract
Topic of this chapter are the solutions of the (angular) Mathieu equation and
the radial or modified Mathieu equation in the complex domain. The Mathieu
differential equations are related to the Helmholtz equation in elliptic cylindrical
coordinates. The periodic Mathieu functions of integer order and the Mathieu
functions of non-integer (complex) order are discussed. The second solution
of the corresponding differential equation is as well derived. Similar to the
modified Mathieu differential equation, first and second solutions are discussed,
and computational methods for evaluating all functions are presented. The
corresponding code and, in addition, code for direct integration of the differential
equations are available for download.
From the Helmholtz equation in elliptic cylindrical coordinates, Eq. (23.19), we get
the (angular) Mathieu equation, see Fig. 14.2, with parameters (a,q),
d 2
dz 2 + a − 2 q cos(2z)
y(z) = 0,
(14.1)
and the radial or modified Mathieu equation, see Fig. 14.2,
d 2
dz 2 − a + 2 q cosh(2z)
y(z) = 0.
(14.2)
The Mathieu equation is an example of a Hill equation and was derived from
Mathieu in the context of the free oscillations of an elliptic membrane. The Mathieu
equation plays, e.g., an important role in applications involving elliptic geometries
or in problems involving periodic motions. An example about dynamical trapping
of ions in a Paul trap can be found in [4].
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
W. Schweizer, Special Functions in Physics with MATLAB,
https://doi.org/10.1007/978-3-030-64232-7_14
165
Mathieu Functions
Abstract
Topic of this chapter are the solutions of the (angular) Mathieu equation and
the radial or modified Mathieu equation in the complex domain. The Mathieu
differential equations are related to the Helmholtz equation in elliptic cylindrical
coordinates. The periodic Mathieu functions of integer order and the Mathieu
functions of non-integer (complex) order are discussed. The second solution
of the corresponding differential equation is as well derived. Similar to the
modified Mathieu differential equation, first and second solutions are discussed,
and computational methods for evaluating all functions are presented. The
corresponding code and, in addition, code for direct integration of the differential
equations are available for download.
From the Helmholtz equation in elliptic cylindrical coordinates, Eq. (23.19), we get
the (angular) Mathieu equation, see Fig. 14.2, with parameters (a,q),
d 2
dz 2 + a − 2 q cos(2z)
y(z) = 0,
(14.1)
and the radial or modified Mathieu equation, see Fig. 14.2,
d 2
dz 2 − a + 2 q cosh(2z)
y(z) = 0.
(14.2)
The Mathieu equation is an example of a Hill equation and was derived from
Mathieu in the context of the free oscillations of an elliptic membrane. The Mathieu
equation plays, e.g., an important role in applications involving elliptic geometries
or in problems involving periodic motions. An example about dynamical trapping
of ions in a Paul trap can be found in [4].
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
W. Schweizer, Special Functions in Physics with MATLAB,
https://doi.org/10.1007/978-3-030-64232-7_14
165