…
►He has recently carried out research on non-linear dynamics of Bose–Einstein condensates that served
to motivate his interest in elliptic
functions.
Older work on the scattering theory of the atomic Coulomb problem led
to the discovery of new classes of orthogonal polynomials
relating to the spectral theory of Schrödinger operators, and new uses of old ones: this work was strongly motivated by his original ownership of a 1964 hard copy printing of the original
AMS 55 NBS Handbook of Mathematical Functions.
…
►
►
…
►In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters
20,
22, and
23.
…
►When
,
, (
25.12.1) becomes
…
►The special case
is the Riemann zeta
function:
.
…
►Further properties include
…and
…
►In terms of polylogarithms
…
§20.11 Generalizations and Analogs
…
►However, in this case
is no longer regarded as an independent complex variable within the unit circle, because
is
related to the variable
of the
theta functions via (
20.9.2).
…
…
►
§20.11(iv) Theta Functions with Characteristics
…
►Such sets of twelve equations include derivatives, differential equations, bisection
relations, duplication
relations, addition formulas (including new ones for
theta functions), and pseudo-addition formulas.
…
…
►‘✓’ indicates that a software package implements the
functions in a section; ‘
a’ indicates available
functionality through optional or add-on packages; an empty space indicates no known support.
…
►In the list below we identify four main sources of software for computing special
functions.
…
►
Open Source Collections and Systems.
These are collections of software (e.g. libraries) or interactive systems of
a somewhat broad scope. Contents may be adapted from research software or may
be contributed by project participants who donate their services to the project.
The software is made freely available to the public, typically in source code form.
While formal support of the collection may not be provided by its developers,
within active projects there is often a core group who
donate time to consider bug reports and make updates to the collection.
…
►
Commercial Software.
Such software ranges from a collection of reusable software
parts (e.g., a library) to fully functional interactive computing environments
with an associated computing language. Such software is usually professionally
developed, tested, and maintained to high standards. It is available for purchase,
often with accompanying updates and consulting support.
…
►
Guide to Available Mathematical Software
A cross index of mathematical software in use at NIST.
…
►
§9.9(ii) Relation to Modulus and Phase
…
►
§9.9(iii) Derivatives With Respect to
…
►
§9.9(iv) Asymptotic Expansions
…
►
§9.9(v) Tables
►Tables
9.9.1 and
9.9.2 give 10D values of the first ten real zeros of
,
,
,
, together with the associated values of the derivative or the
function.
…
…
►Legendre’s integrals can be computed from symmetric integrals by using the
relations in §
19.25(i).
…
►If
and
, so that
, then this procedure reduces
to the AGM method for the complete integral.
►The step from
to
is an ascending Landen transformation if
(leading ultimately
to a hyperbolic case of
) or a descending Gauss transformation if
(leading
to a circular case of
).
…
►
§19.36(iii) Via Theta Functions
…
►For computation of Legendre’s integral of the third kind, see
Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20).
…