page_title: Mathieu function
section_title: Qualitative behavior
hierarchical_section_title: Mathieu function / Properties / Qualitative behavior
caption_attribution_description: English: Sample plots of Mathieu functions of the first kind
context_page_description: In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation
  where and are parameters. They were first introduced by Émile Léonard Mathieu, who encountered them while studying vibrating elliptical drumheads. They have applications in many fields of the physical sciences, such as optics, quantum mechanics, and general relativity. They tend to occur in problems involving periodic motion, or in the analysis of partial differential equation boundary value problems possessing elliptic symmetry.
context_section_description: For small ,  and  behave similarly to  and . For arbitrary , they may deviate significantly from their trigonometric counterparts; however, they remain periodic in general. Moreover, for any real ,  and  have exactly  simple zeros in , and as  the zeros cluster about .
  For  and as  the modified Mathieu functions tend to behave as damped periodic functions.
  In the following, the  and  factors from the Fourier expansions for  and  may be referenced (see Explicit representation and computation). They depend on  and  but are independent of .
caption: Sample plots of Mathieu functions of the first kind
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