Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Connection formulas between the solutions of Mathieu's equation


Author: Gregory H. Wannier
Journal: Quart. Appl. Math. 11 (1953), 33-59
MSC: Primary 33.0X
DOI: https://doi.org/10.1090/qam/57390
MathSciNet review: 57390
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Abstract: The problem of connecting the various types of solutions of Mathieu's equation is solved by the introduction of a new parameter $ \Phi $ which is a function of the the two equation parameters $ a$ and $ q$. This quantity $ \Phi $ is introduced and enclosed between two very close analytic limits in section 2. In sections 3, 4, 5 precise definitions are given and information is collected for the three main types of functions which are to be connected. Section 6 contains the connection formulas. Section 7 reviews the status of knowledge achieved. Section 8 is an appendix on integral equations which are more general than those developed earlier in the text, but which appear to be of no use for the main purpose of this paper.


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DOI: https://doi.org/10.1090/qam/57390
Article copyright: © Copyright 1953 American Mathematical Society


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