Helmholtz Differential Equation--Circular Cylindrical Coordinates

In Cylindrical Coordinates, the Scale Factors are , , and the separation functions are , , , so the Stäckel Determinant is 1. Attempt Separation of Variables by writing

(1)

(2)

(3)

(4)

(5)

(6)

(7)

The solution to the second part of (7) must not be sinusoidal at for a physical solution, so thedifferential equation has a Positive separation constant

(8)

(9)

(10)

(11)

(12)

(13)

	(14)

Actually, the Helmholtz Differential Equation is separable for general of the form

(15)

References

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 514 and 656-657, 1953.

• Liouville's Boundedness Theorem
• Liouville's Conformality Theorem
• Liouville's Conic Theorem
• Liouville's Constant
• Liouville's Elliptic Function Theorem
• Liouville Space
• Liouville's Phase Space Theorem
• Liouville's Rational Approximation Theorem
• Liouville's Sphere-Preserving Theorem
• Lipschitz Condition
• Lipschitz Function
• Lipschitz's Integral
• Lissajous Curve
• Lissajous Figure
• List
• Lituus
• Lituus Inverse Curve
• LLL Algorithm
• Ln
• Lobachevsky-Bolyai-Gauss Geometry
• Lobachevsky's Formula
• Lobatto Quadrature
• Lobster
• Local Cell
• Local Degree