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
![h^2a_(n+1)+[2h^2-4b+16(n+1/2s)^2]a_n+h^2a_(n-1)=0.](http://mathworld.wolfram.com/images/equations/HillDeterminant/NumberedEquation3.gif) |
(3)
|
The value of 
can be computed using the Hill determinant
 |
(4)
|
where
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.
