Mathieu function

They were first introduced by Émile Léonard Mathieu, who encountered them while studying vibrating elliptical drumheads.[1][2][3] They have applications in many fields of the physical sciences, such as optics, quantum mechanics, and general relativity., called characteristic numbers, conventionally indexed as two separate sequences, besides serving to arrange the characteristic numbers in ascending order, is convenient in that, Mathieu's equation never possesses two (independent) basically periodic solutions for the same values ofAn equivalent statement of Floquet's theorem is that Mathieu's equation admits a complex-valued solution of form whereFor almost all choices of these parameters, Floquet theory says that any solution either converges to zero or diverges to infinity.non-integral) Mathieu functions of the first kind can be represented as Fourier series:[5] The expansion coefficientsBy substitution into the Mathieu equation, they can be shown to obey three-term recurrence relations in the lower index.Moreover, in this particular case, an asymptotic analysis[17] shows that one possible choice of fundamental solutions has the property In particular,To overcome these issues, more sophisticated semi-analytical/numerical approaches are required, for instance using a continued fraction expansion,[18][5] casting the recurrence as a matrix eigenvalue problem,[19] or implementing a backwards recurrence algorithm.[17] The complexity of the three-term recurrence relation is one of the reasons there are few simple formulas and identities involving Mathieu functions.[20] In practice, Mathieu functions and the corresponding characteristic numbers can be calculated using pre-packaged software, such as Mathematica, Maple, MATLAB, and SciPy., the form of the series must be chosen carefully to avoid subtraction errors.[4] Solutions of Mathieu's equation satisfy a class of integral identities with respect to kernels[32] There also exist integral relations between functions of the first and second kind, for instance:[23] valid for any complex:[33] Thus, the modified Mathieu functions decay exponentially for large real argument., tunneling through the barriers becomes possible (in physical language), leading to a splitting of the characteristic numbers(in quantum mechanics called eigenvalues) corresponding to even and odd periodic Mathieu functions.one obtains The corresponding characteristic numbers or eigenvalues then follow by expansion, i.e. Insertion of the appropriate expressions above yields the result ForMathieu's differential equations appear in a wide range of contexts in engineering, physics, and applied mathematics.Many of these applications fall into one of two general categories: 1) the analysis of partial differential equations in elliptic geometries, and 2) dynamical problems which involve forces that are periodic in either space or time.[37] In general relativity, an exact plane wave solution to the Einstein field equation can be given in terms of Mathieu functions.More recently, Mathieu functions have been used to solve a special case of the Smoluchowski equation, describing the steady-state statistics of self-propelled particles.; hence, these coordinates are convenient for solving the Helmholtz equation on domains with elliptic boundaries.As a specific physical example, the Helmholtz equation can be interpreted as describing normal modes of an elastic membrane under uniform tension.then arise due to imposition of physical conditions on some bounding surface, such as an elliptic boundary defined by, which in turn requires These conditions define the normal modes of the system.In such cases, knowledge of the general properties of Mathieu's equation— particularly with regard to stability of the solutions—can be essential for understanding qualitative features of the physical dynamics.The modified Mathieu equation also arises when describing the quantum mechanics of singular potentials.