Let be analytic within an ellipse with foci , and
18.18.1 | |||
Then
18.18.2 | |||
when lies in the interior of . Moreover, the series (18.18.2) converges uniformly on any compact domain within .
Alternatively, assume is real and continuous and is piecewise continuous on . Assume also the integrals and converge. Then (18.18.2), with replaced by , applies when ; moreover, the convergence is uniform on any compact interval within .
This is the case of Jacobi. Equation (18.18.1) becomes
18.18.3 | |||
Assume is real and continuous and is piecewise continuous on . Assume also converges. Then
18.18.4 | |||
, | |||
where
18.18.5 | |||
The convergence of the series (18.18.4) is uniform on any compact interval in .
Assume is real and continuous and is piecewise continuous on . Assume also converges. Then
18.18.6 | |||
, | |||
where
18.18.7 | |||
The convergence of the series (18.18.6) is uniform on any compact interval in .
18.18.10 | |||
18.18.11 | |||
18.18.12 | |||
18.18.13 | |||
18.18.14 | ||||
18.18.15 | ||||
and a similar pair of equations by symmetry; compare the second row in Table 18.6.1.
18.18.16 | ||||
18.18.17 | ||||
18.18.18 | ||||
18.18.19 | ||||
18.18.20 | |||
18.18.21 | |||
18.18.22 | |||
18.18.23 | |||
With
18.18.24 | |||
18.18.25 | |||
18.18.26 | |||
18.18.27 | |||
. | |||
For the modified Bessel function see §10.25(ii).
18.18.28 | |||
. | |||
These Poisson kernels are positive, provided that are real, , and in the case of (18.18.27) .
In this subsection the variables and are not confined to the closures of the intervals of orthogonality; compare §18.2(i).
18.18.29 | |||
18.18.30 | |||
18.18.31 | ||||
18.18.32 | ||||
18.18.33 | ||||
18.18.34 | ||||
18.18.35 | ||||
18.18.36 | |||
18.18.37 | |||
18.18.38 | |||
18.18.39 | |||
18.18.40 | |||