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We write the Mathieu equation as
And consider the case when
and write
and have solutions
. We then
suppose that the case when q is nonzero can be taken into
account as a series based on this initial solution.
Let
![$\displaystyle a= 1 + \sum_{i=1}^{\infty}\alpha_i q^i$](img146.png) |
(5.1) |
Then we suppose that
![$\displaystyle y=\cos z + \sum_{i=1}^{\infty}q^{i}c_i(z)$](img147.png) |
(5.2) |
We determine the nature of the
functions as follows. Plugging
our solution into the Mathieu equation, we get
Using the identity
Coefficents are matched:
The particular solution corresponding to
is
which is not bounded, thus we require
that
such that
The arguments presented before imply that
Following [4], we write
and so we find the c functions can be represted by ``cosine-elliptic'' function,
![$\displaystyle a=1 + q -\frac{q^2}{8} -\frac{q^3}{64} + O(q^4)$](img175.png) |
(5.4) |
Next: Fractional order
Up: Technical details
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tim jones
2008-07-07