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Previous: Basics and Flouqent's Theorem
With Floquent's theorem we assume a series solution, due to G. W. Hill,
![$\displaystyle w=e^{\mu \eta}\phi(\eta)=e^{\mu \eta}\sum^{\infty}_{r=-\infty}c_{2r}e^{2ri\eta}=\sum^{\infty}_{r=-\infty}c_{2r}e^{(\mu + 2ri)\eta}$](img37.png) |
(2.5) |
When we put this into Mathieu's equation,
matching terms in power of r, we get the equation
![$\displaystyle -qc_{2r-2} + \left((\mu + 2ir)^2 + a\right)c_{2r} - qc_{2r+2}=0$](img39.png) |
(2.6) |
Multiplying through by
, and then dividing by the middle term,
![$\displaystyle \frac{q}{(2r-\mu i)^2 - a}c_{2r-2} + c_{2r} + \frac{q}{\left(2r-\mu i\right)^2-a}c_{2r+2}=0$](img41.png) |
(2.7) |
We now define
That these coefficents,
have non-trivial solutions
requires the infinite determinant
to vanish for noninfinite r:
![$\displaystyle \Delta(i\mu)=\left\vert\begin{array}{ccccc}\cdots&&&&\ \gamma_{-...
...}&1&\gamma_{0}&\\
&&\gamma_2&1&\gamma_2\\
&&&&\cdots \end{array}\right\vert=0$](img45.png) |
(2.8) |
But of course, this is not a simple object to understand and solve. We can
approach this problem from a rather clever angle introduced by E. T. Whittaker.
Consider the function
Like our determinant,
has a simple pole at
,
so that the function
has no singularities if
is
chosen properly and is bound at infinity, where
since
the
functions all vanish and the diagonal term is all that remains,
and
since
limits to zero as x tends towards
infinity.
By Liouville's theorem (of complex calculus), since this
limits to a constant, it is a constant always, so we have
Next we consider the
case and find,
Next we suppose that
is chosen to satisfy our requirement that the determinant vanish.
We thus have
Recall that our solution took the form,
This solution will be unbounded unless
, in which case we have
![$\displaystyle \mu=\frac{1}{\pi}\cos^{-1}\left(1-\Delta(0)(1-\cosh \pi \sqrt{a})\right)$](img62.png) |
(2.9) |
We can easily encode this result, say,
if(a>=0){ mu=acos( 1 - (d[100])*(1-cos(pi*sqrt(a)))) / (pi);}
if(a<0){ mu=acos( 1 - (d[100])*(1-cosh(pi*sqrt(fabs(a))))) / (pi);}
if (mu != mu){mu=0.000000;} //If mu=nan then make it zero
But first we must calculate
. This task has been made
exceedingly simple by the recent work of J. E. Sträng [5]
who has found an efficient recursion formula.
Next: Sträng's recursion formula for
Up: Mathieu's Equation, solution, and
Previous: Basics and Flouqent's Theorem
tim jones
2008-07-07