In elliptic cylindrical coordinates, the scale factors are , , and the separation functions are , giving a Stäckel determinant of . The Helmholtz differential equation is
(1)
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Attempt separation of variables by writing
(2)
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then the Helmholtz differential equation becomes
(3)
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Now divide by to give
(4)
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Separating the part,
(5)
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so
(6)
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which has the solution
(7)
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Rewriting (◇) gives
(8)
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which can be separated into
(9)
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(10)
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so
(11)
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(12)
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Now use
(13)
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(14)
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to obtain
(15)
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(16)
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Regrouping gives
(17)
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(18)
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Let and , then these become
(19)
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(20)
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Here, (19) is the mathieu differential equation and (20) is the modified mathieu differential equation. These solutions are known as mathieu functions.