…
βΊ 1942
in Dorchester, U.
…He began his academic career
in 1964 at the University of Lancaster, U.
…
βΊWalker’s books are
An Introduction to Complex Analysis, published by Hilger
in 1974,
The Theory of Fourier Series and Integrals, published by Wiley
in 1986,
Elliptic Functions. A Constructive Approach, published by Wiley
in 1996, and
Examples and Theorems in Analysis, published by Springer
in 2004.
…
βΊWalker is now retired and living
in Cheltenham, UK.
…
βΊ
…
…
βΊThey lie
in the sectors
and
, and are denoted by
,
, respectively,
in the former sector, and by
,
,
in the conjugate sector, again arranged
in ascending order of absolute value (modulus) for
See §
9.3(ii) for visualizations.
βΊFor the distribution
in
of the zeros of
, where
is an arbitrary complex constant, see
MuraveΔ (1976) and
Gil and Segura (2014).
…
βΊ
§9.9(iv) Asymptotic Expansions
…
βΊFor error bounds for the asymptotic
expansions of
,
,
, and
see
Pittaluga and Sacripante (1991), and a conjecture given
in Fabijonas and Olver (1999).
…
βΊTables
9.9.3 and
9.9.4 give the corresponding results for the first ten complex zeros of
and
in the upper half plane.
…
…
βΊHe currently resides
in Santa Fe, NM.
βΊReinhardt is a theoretical chemist and atomic physicist, who has always been interested
in orthogonal polynomials and
in the analyticity properties of the functions of mathematical physics.
…
βΊThis is closely connected with his interests
in classical dynamical “chaos,” an area where he coauthored a book,
Chaos in atomic physics with Reinhold Blümel.
…
βΊ
…
βΊIn November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters
20,
22, and
23.
…
βΊLastly, when
,
(Hermite polynomial case)
has
zeros and they lie
in the interval
.
…
βΊ
§12.11(ii) Asymptotic Expansions of Large Zeros
…
βΊNumerical calculations
in this case show that
corresponds to the
th zero on the string; compare §
7.13(ii).
βΊ
§12.11(iii) Asymptotic Expansions for Large Parameter
βΊFor large negative values of
the real zeros of
,
,
, and
can be approximated by reversion of the Airy-type asymptotic
expansions of §§
12.10(vii) and
12.10(viii).
…
§5.11 Asymptotic Expansions
βΊ
§5.11(i) Poincaré-Type Expansions
…
βΊWrench (1968) gives exact values of
up to
.
…
βΊFor re-
expansions of the remainder terms
in (
5.11.1) and (
5.11.3)
in series of incomplete gamma functions with exponential improvement (§
2.11(iii))
in the asymptotic
expansions, see
Berry (1991),
Boyd (1994), and
Paris and Kaminski (2001, §6.4).
…
βΊ