derivation of integral representations of Jacobi functions
By rearranging the Fourier series of , one obtains the series
This equation which is valid for all real values of such that and all non-integral complex values of . By comparison with the convergent series , it follows that this series is absolutely convergent. Note that this series may be viewed as a Mittag-Leffler partial fraction expansion.
Let be a positive real number. Multiply both by and integrate.
Because of the exponential, the integrand decays rapidly as provided that , and hence the integral converges absolutely. Make a change of variables
Since the squares closest in absolute value to are and , it follows that for all . Hence, we have
When , we have . Hence, and
Finally since when . Also, . From these observations, we conclude that
Note that this quantity approaches 0 in the limit .
Taking the limit we obtain
Going back to the beginning of the proof, where the integral on the left hand side was expressed as an integral with respect to , we obtain
Making a change of variables and tidying up some, we obtain
Because of the initial assumption about the Fourier series, we only know that this formula is valid when is purely imaginary with strictly positive imaginary part and is real and . However, we can use analytic continuation to extend the domain of its validity. On the one hand, the theta function on the right-hand side is analytic for all and all such that .
On the other hand, I claim that the integral on the left hand side is also an analytic function of and whenever . To validate this claim, we need to examine the behaviour of the integrand as . The contribution of the denominator is bounded;
for some constant whenever . The absolute value of the cosine in the numerator is easy to bound:
To bound the remaining term, let us examine the argument of the exponential carefully:
Therefore, if , it will be the case that , and so
Taken together, the estimates of the last paragraph imply that
when . If we impose the further conditions
it will be the case that
Likewise, under the same restriction on ,
will be an analytic function of and . Suppose that and are restricted to bounded regions of the complex plane and that, furthermore, is positive and bounded away from zero. Then the inequalities of the last paragraph imply that the integral converges uniformly as , and hence
is an analytic function of and in the domain .
Thus, by the fundamental theorem of analytic continuation, we may conclude that
throughout this domain.
Finally, integral representations of the remaining three theta functions may be easily obtained from this one by adding the appropriate half-quasiperiods to .
|Title||derivation of integral representations of Jacobi functions|
|Date of creation||2013-03-22 14:39:54|
|Last modified on||2013-03-22 14:39:54|
|Last modified by||rspuzio (6075)|