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In mathematics, the Mathieu functions are certain special functions useful for treating a variety of problems in applied mathematics, including: *vibrating elliptical drumheads, *quadrupole mass analyzers and quadrupole ion traps for mass spectrometry *wave motion in periodic media, such as ultracold atoms in an optical lattice *the phenomenon of parametric resonance in forced oscillators, *exact plane wave solutions in general relativity, *the Stark effect for a rotating electric dipole, *in general, the solution of differential equations that are separable in elliptic cylindrical coordinates. They were introduced by in the context of the first problem. ==Mathieu equation== The canonical form for Mathieu's differential equation is : The Mathieu equation is a Hill equation with only 1 harmonic mode. Closely related is Mathieu's modified differential equation : which follows on substitution . The two above equations can be obtained from the Helmholtz equation in two dimensions, by expressing it in elliptical coordinates and then separating the two variables.() This is why they are also known as angular and radial Mathieu equation, respectively. The substitution transforms Mathieu's equation to the ''algebraic form'' : This has two regular singularities at and one irregular singularity at infinity, which implies that in general (unlike many other special functions), the solutions of Mathieu's equation ''cannot'' be expressed in terms of hypergeometric functions. Mathieu's differential equations arise as models in many contexts, including the stability of railroad rails as trains drive over them, seasonally forced population dynamics, the four-dimensional wave equation, and the Floquet theory of the stability of limit cycles. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Mathieu function」の詳細全文を読む スポンサード リンク
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