…
►In the following subsections, only Stokes sets involving
at least one real saddle are included unless stated otherwise.
…
►
. Swallowtail
►The Stokes set takes different forms for
,
, and
.
►For
, the set consists of the two curves
…
►This consists of three separate cusp-edged sheets connected to the cusp-edged sheets of the bifurcation set, and related by rotation about the
-axis by
.
…
…
►For the
values of
and
used in the formulas below
…where
is defined by (
3.3.12), with numerical
values as in §
3.3(ii).
…
►Taking
to be a circle of radius
centered
at
, we obtain
…
►With the choice
(which is crucial when
is large because of numerical cancellation) the integrand equals
at the dominant points
, and in combination with the factor
in front of the integral sign this gives a rough approximation to
.
…
►For additional formulas involving
values of
and
on square, triangular, and cubic grids, see
Collatz (1960, Table VI, pp. 542–546).
…
…
►
–
possess hierarchies of rational solutions for special
values of the parameters which are generated from “seed solutions” using the Bäcklund transformations and often can be expressed in the form of determinants.
…
►where the
are monic polynomials (coefficient of highest power of
is
) satisfying
…with
,
.
…
►Next, let
be the polynomials defined by
for
, and
…
►where
,
,
,
, and
, with
,
, independently, and
at least one of
,
,
or
is an integer.
…
…
►Abramowitz and Stegun (1964, Chapter 23) includes exact
values of
,
,
;
,
,
,
, 20D;
,
, 18D.
►Wagstaff (1978) gives complete prime factorizations of
and
for
and
, respectively.
…
…
► 1976 in Leidschendam, the Netherlands) is an Associate Professor
at the Delft University of Technology in Delft, The Netherlands.
…
► in mathematics
at the Delft University of Technology in 2004.
…
►As of September
20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter
18 Orthogonal Polynomials.
…
…
►Also, in further development along the lines of the notations of Neville (§
20.1) and of Glaisher (§
22.2), the identities (
20.7.6)–(
20.7.9) have been recast in a more symmetric manner with respect to suffices
.
…
►See
Lawden (1989, pp. 19–20).
…
►See also
Carlson (2011, §3).
…
►In the following equations
, and all square roots assume their principal
values.
…
►
20.7.34
…
…
►There are many ways to implement these first two steps, noting that the expressions for
and
of equation (
18.2.30) are of little practical numerical
value, see
Gautschi (2004) and
Golub and Meurant (2010).
…
►The question is then: how is this possible given only
, rather than
itself?
often converges to smooth results for
off the real axis for
at a distance greater than the pole spacing of the
, this may then be followed by
approximate numerical analytic continuation via fitting to lower order continued fractions (either Padé, see §
3.11(iv), or pointwise continued fraction approximants, see
Schlessinger (1968, Appendix)), to
and evaluating these on the real axis in regions of higher pole density that those of the approximating function.
Results of low (
to
decimal digits) precision for
are easily obtained for
to
.
…
►This is a challenging case as the desired
on
has an essential singularity
at
.
…
►Achieving precisions
at this level shown above requires higher than normal computational precision, see
Gautschi (2009).
…