…
►Here
denotes an arbitrary
small positive constant and
…Lastly, for
we define
…For large
,
…
►Numerical values of
are given in Table
9.7.1 for
to 2D.
…
►As special cases, when
…
…
►When the differences are moderately
small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (
19.19.7) is evaluated.
…
►Remedies for cancellation when
is real and near 0 are supplied in
Midy (1975).
…
►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).
…
►When the values of complete integrals are known, addition theorems with
(§
19.11(ii)) ease the computation of functions such as
when
is
small and positive.
Similarly, §
19.26(ii) eases the computation of functions such as
when
(
) is
small compared with
.
…
…
►
is the number of lattice paths from
to
that stay on or above the line
and are composed of directed line segments of the form
,
, or
.
…
►
is the number of lattice paths from
to
that stay on or above the line
, are composed of directed line segments of the form
or
, and for which there are exactly
occurrences at which a segment of the form
is followed by a segment of the form
.
…
►
is the number of paths from
to
that stay on or above the diagonal
and are composed of directed line segments of the form
,
, or
.
…
►For sufficiently
small
and
,
…
►
26.6.8
…
…
►Stokes sets are surfaces (codimension one) in
space, across which
or
acquires an exponentially-
small asymptotic contribution (in
), associated with a complex critical point of
or
.
…
►The Stokes set consists of the rays
in the complex
-plane.
…
►where
are the two smallest positive roots of the equation
…
►When
the Stokes set
is given by
…
►Alternatively, when
…
…
►Bessel functions first appear in the investigation of a physical problem in Daniel Bernoulli’s analysis of the
small oscillations of a uniform heavy flexible chain.
…
►See
Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2),
Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and
Slater (1942, Chapter 4, §§20, 25).
…
►On separation of variables into cylindrical coordinates, the Bessel functions
, and modified Bessel functions
and
, all appear.
…
►The analysis of the current distribution in circular conductors leads to the Kelvin functions
,
,
, and
.
See
Relton (1965, Chapter X, §§10.2, 10.3),
Bowman (1958, Chapter III, §§51–53),
McLachlan (1961, Chapters VIII and IX), and
Russell (1909).
…
…
►where
is an arbitrary
small positive constant.
…
►
11.6.3
,
►
11.6.4
,
…
►
11.6.5
.
…
►
…