The Mathieu equation is a linear second-order differential equation with periodic coefficients. The French mathematician, E. Léonard Mathieu, first introduced this family of differential equations, nowadays termed Mathieu equations, in his “Memoir on vibrations of an elliptic membrane” in 1868. "Mathieu functions are applicable to a wide variety of physical phenomena, e.g., diffraction, amplitude distortion, inverted pendulum, stability of a floating body, radio frequency quadrupole, and vibration in a medium with modulated density"
Elliptic-cylinder wavelets This is a wide family of wavelet system that provides a multiresolution analysis. The magnitude of the detail and smoothing filters corresponds to first-kind Mathieu functions with odd characteristic exponent. The number of notches of these filters can be easily designed by choosing the characteristic exponent. Elliptic-cylinder wavelets derived by this method possess potential application in the fields of optics and electromagnetism due to its symmetry.
Mathieu differential equations Mathieu's equation is related to the wave equation for the elliptic cylinder. In 1868, the French mathematician Émile Léonard Mathieu introduced a family of differential equations nowadays termed Mathieu equations. Given a ∈ R , q ∈ C {\displaystyle a\in \mathbb {R} ,q\in \mathbb {C} } , the Mathieu equation is given by
d 2 y d w 2 + ( a − 2 q cos 2 w ) y = 0. {\displaystyle {\frac {d^{2}y}{dw^{2}}}+(a-2q\cos 2w)y=0.}
The Mathieu equation is a linear second-order differential equation with periodic coefficients. For q = 0, it reduces to the well-known harmonic oscillator, a being the square of the frequency. The solution of the Mathieu equation is the elliptic-cylinder harmonic, known as Mathieu functions. They have long been applied on a broad scope of wave-guide problems involving elliptical geometry, including:
analysis for weak guiding for step index elliptical core optical fibres power transport of elliptical wave guides evaluating radiated waves of elliptical horn antennas elliptical annular microstrip antennas with arbitrary eccentricity ν {\displaystyle \nu } ) scattering by a coated strip.
Mathieu functions: cosine-elliptic and sine-elliptic functions In general, the solutions of Mathieu equation are not periodic. However, for a given q, periodic solutions exist for infinitely many special values (eigenvalues) of a. For several physically relevant solutions y must be periodic of period π {\displaystyle \pi } or 2 π {\displaystyle 2\pi } . It is convenient to distinguish even and odd periodic solutions, which are termed Mathieu functions of first kind. One of four simpler types can be considered: Periodic solution ( π {\displaystyle \pi } or 2 π {\displaystyle 2\pi } ) symmetry (even or odd). For q ≠ 0 {\displaystyle q\neq 0} , the only periodic solutions y corresponding to any characteristic value a = a r ( q ) {\displaystyle a=a_{r}(q)} or a = b r ( q ) {\displaystyle a=b_{r}(q)} have the following notations: ce and se are abbreviations for cosine-elliptic and sine-elliptic, respectively.
Even periodic solution: c e r ( ω , q ) = ∑ m A r , m cos m ω for a = a r ( q ) {\displaystyle ce_{r}(\omega ,q)=\sum _{m}A_{r,m}\cos {m\omega }{\text{ for }}a=a_{r}(q)}
Odd periodic solution: s e r ( ω , q ) = ∑ m A r , m sin m ω for a = b r ( q ) {\displaystyle se_{r}(\omega ,q)=\sum _{m}A_{r,m}\sin {m\omega }{\text{ for }}a=b_{r}(q)}
where the sums are taken over even (respectively odd) values of m if the period of y is π {\displaystyle \pi } (respectively 2 π {\displaystyle 2\pi } ). Given r, we denote henceforth A r , m {\displaystyle A_{r,m}} by A m {\displaystyle A_{m}} , for short. Interesting relationships are found when q → 0 {\displaystyle q\to 0} , r ≠ 0 {\displaystyle r\neq 0} :
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