…
►If
and
, then set
and
, without loss of generality.
…Similar results hold for
with
and
.
…
►Let
and
be solutions of
, where
…and assume
.
…
►for
, where
…
…
►where
and
.
►For the values of
and
used in the formulas below
…
►For corresponding formulas for second, third, and fourth derivatives, with
, see
Collatz (1960, Table III, pp. 538–539).
…
►
,
.
…
►For partial derivatives we use the notation
.
…
…
►This reference gives
,
, and their logarithmic
-derivatives to 4D for
,
, where
is the modular angle given by
►
20.15.1
►Spenceley and Spenceley (1947) tabulates
,
,
,
to 12D for
,
, where
and
is defined by (
20.15.1), together with the corresponding values of
and
.
►Lawden (1989, pp. 270–279) tabulates
,
, to 5D for
,
, and also
to 5D for
.
►Tables of Neville’s theta functions
,
,
,
(see §
20.1) and their logarithmic
-derivatives are given in
Abramowitz and Stegun (1964, pp. 582–585) to 9D for
, where (in radian measure)
, and
is defined by (
20.15.1).
…
…
►These expansions are uniform with respect to
, including the turning point
and its neighborhood, and the region of validity often includes cut neighborhoods (§
1.10(vi)) of other singularities of the differential equation, especially irregular singularities.
…
►The number
can also be replaced by any real constant
in the sense that
is analytic and nonvanishing at
; moreover,
is permitted to have a single or double pole at
.
…
►In regions in which the function
has a simple pole at
and
is analytic at
(the case
in §
10.72(i)), asymptotic expansions of the solutions
of (
10.72.1) for large
can be constructed in terms of Bessel functions and modified Bessel functions of order
, where
is the limiting value of
as
.
…
►In (
10.72.1) assume
and
depend continuously on a real parameter
,
has a simple zero
and a double pole
, except for a critical value
, where
.
…These approximations are uniform with respect to both
and
, including
, the cut neighborhood of
, and
.
…