In Elliptic Cylindrical Coordinates, the Scale Factors are
, , and the separation functions are
, giving a
Stäckel Determinant of
. The Helmholtz differential
equation is

(1) 
Attempt Separation of Variables by writing

(2) 
then the Helmholtz Differential Equation becomes

(3) 
Now divide by to give

(4) 
Separating the part,

(5) 
so

(6) 
which has the solution

(7) 
Rewriting (5) gives

(8) 
which can be separated into
so

(11) 

(12) 
Now use

(13) 

(14) 
to obtain

(15) 

(16) 
Regrouping gives

(17) 

(18) 
Let
and
, then these become

(19) 

(20) 
Here, (20) is the Mathieu Differential Equation and (19) is the modified Mathieu Differential
Equation. These solutions are known as Mathieu Functions.
See also Elliptic Cylindrical Coordinates, Mathieu Differential Equation, Mathieu Function
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York:
McGrawHill, pp. 514 and 657, 1953.
© 19969 Eric W. Weisstein
19990525