…
βΊThe power-series expansions of §§
33.6 and
33.19 converge for all finite values of the radii
and
,
respectively, and may be used
to compute the regular and irregular solutions.
…
βΊ
§33.23(iii) Integration of Defining Differential Equations
βΊWhen numerical values of the Coulomb functions are available for some radii, their values for other radii may be obtained by direct numerical
integration of equations (
33.2.1) or (
33.14.1), provided that the
integration is carried out in a stable direction (§
3.7).
…On the other hand, the irregular solutions of §§
33.2(iii) and
33.14(iii) need
to be
integrated in the direction of decreasing radii beginning, for example, with values obtained from asymptotic expansions (§§
33.11 and
33.21).
…
βΊCurtis (1964a, §10) describes the use of series, radial
integration, and other methods
to generate the tables listed in §
33.24.
…
…
βΊThe
integration path begins at
, encircles
once in the positive sense, followed by
once in the positive sense, and so on, returning finally
to
.
The
integration path is called a
Pochhammer double-loop
contour (compare Figure
5.12.3).
…
βΊ
31.9.3
…
βΊ
31.9.5
,
…
βΊand the
integration paths
,
are Pochhammer double-loop contours encircling distinct pairs of singularities
,
,
.
…
…
βΊ
A numerical approach to the recursion coefficients and quadrature abscissas and weights
…
βΊHaving now directly connected computation of the quadrature abscissas and weights
to the moments, what follows uses these for a Stieltjes–Perron
inversion to regain
.
…
βΊResults of low (
to
decimal digits) precision for
are easily obtained for
to
.
…
βΊInterpolation of the midpoints of the jumps followed by differentiation with
respect to
yields a Stieltjes–Perron inversion
to obtain
to a precision of
decimal digits for
.
…
βΊFurther,
exponential convergence in
, via the Derivative Rule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros
to approximate
for these OP systems on
and
respectively,
Reinhardt (2018), and
Reinhardt (2021b),
Reinhardt (2021a).
…
…
βΊFurthermore,
and
are entire functions of
, and
and
are meromorphic functions of
with simple poles at
and
,
respectively.
…
βΊ(obtained from (
5.2.1) by rotation of the
integration path) is also needed.
…
βΊIn these representations the
integration paths do not cross the negative real axis, and in the case of (
8.21.4) and (
8.21.5) the paths also exclude the origin.
…
βΊ
§8.21(v) Special Values
…
βΊWhen
with
(
),
…
…
βΊWhile non-normalizable continuum, or scattering, states are mentioned, with appropriate references in what follows, focus is on the
eigenfunctions corresponding
to the point, or discrete, spectrum, and representing
bound rather than
scattering states, these former being expressed in terms of OP’s or EOP’s.
…
βΊKuijlaars and Milson (2015, §1) refer
to these, in this case complex zeros, as
exceptional, as opposed
to regular, zeros of the EOP’s, these latter belonging
to the (real) orthogonality
integration range.
…
βΊOrthogonality and normalization of eigenfunctions of this form is
respect to the measure
.
…
βΊwith an infinite set of orthonormal
eigenfunctions
…
βΊFor either sign of
, and
chosen such that
,
, truncation of the basis
to
terms, with
, the discrete eigenvectors are the orthonormal
functions
…
…
βΊIf
is absolutely
integrable on
, then
is continuous,
as
, and
…
βΊIn many applications
is absolutely
integrable and
is continuous on
.
…
βΊThe
Fourier cosine transform and
Fourier sine transform are defined
respectively by
…
βΊ
Differentiation and Integration
…
βΊNote: If
is continuous and
and
are real numbers such that
as
and
as
, then
is
integrable on
for all
.
…
…
βΊSufficient conditions for the limit
to exist are that
is continuous, or piecewise continuous, on
.
…
βΊ
Change of Order of Integration
…
βΊIn the cases (
1.5.30) and (
1.5.33) they are defined by taking limits in the repeated integrals (
1.5.32) and (
1.5.34) in an analogous manner
to (
1.4.22)–(
1.4.23).
…
βΊA more general concept of
integrability (both finite and infinite) for functions on domains in
is
Lebesgue integrability.
…
βΊAgain the mapping is one-
to-one except perhaps for a set of points of volume zero.
…