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Date: 18-7-2018
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Date: 13-7-2018
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Date: 21-7-2018
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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.
REFERENCES:
Abramowitz, M. and Stegun, I. A. (Eds.). "Mathieu Functions." Ch. 20 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 721-746, 1972.
Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed.New York: Springer-Verlag, pp. 17-19, 1988.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 514 and 657, 1953.
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