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 (◇) gives

(8)

which can be separated into

			(9)
			(10)

so

(11)

(12)

Now use

(13)

(14)

to obtain

{c+1/2m^2[cosh(2u)-1]}U=0 " class="numberedequation" src="http://mathworld.wolfram.com/images/equations/HelmholtzDifferentialEquationEllipticCylindricalCoordinates/NumberedEquation13.gif" style="border:0px; height:40px; width:233px" />

(15)

{c-1/2m^2[1-cos(2v)]}V=0. " class="numberedequation" src="http://mathworld.wolfram.com/images/equations/HelmholtzDifferentialEquationEllipticCylindricalCoordinates/NumberedEquation14.gif" style="border:0px; height:40px; width:227px" />

(16)

Regrouping gives

(17)

(18)

Let  and , then these become

(19)

(20)

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.