Hill Determinant

A determinant which arises in the solution of the second-order ordinary differential equation

(1)

Writing the solution as a power series

(2)

gives a recurrence relation

(3)

The value of  can be computed using the Hill determinant

(4)

where

			(5)
			(6)
			(7)

and  is the variable to solve for. The determinant can be given explicitly by the amazing formula

(8)

where

(9)

leading to the implicit equation for ,

(10)

REFERENCES:

Hill, G. W. "On the Part of the Motion of Lunar Perigee Which is a Function of the Mean Motions of the Sum and Moon." Acta Math. 8, 1-36, 1886.

Magnus, W. and Winkler, S. Hill's Equation. New York: Dover, 1979.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 555-562, 1953.