…
►
15.10.1
…
►
Singularity
…
►
Singularity
…
►
Singularity
…
►(b) If
equals
, and
, then fundamental solutions in the neighborhood of
are given by
and
…
…
►(For other notation see
Notation for the Special Functions.)
►
►The main
functions treated in this chapter are the Airy
functions
and
, and the Scorer
functions
and
(also known as inhomogeneous Airy
functions).
►Other notations that have been used are as follows:
and
for
and
(
Jeffreys (1928), later changed to
and
);
,
(
Fock (1945));
(
Szegő (1967, §1.81));
,
(
Tumarkin (1959)).
§16.13 Appell Functions
►The following four
functions of two real or complex variables
and
cannot be expressed as a product of two
functions, in general, but they satisfy partial differential equations that resemble the
hypergeometric differential equation (
15.10.1):
►
16.13.1
,
…
►
16.13.3
,
…
►
…
§9.12 Scorer Functions
…
►If
or
, and
is the modified Bessel
function (§
10.25(ii)), then
…
►where the integration contour separates the poles of
from those of
.
…
►
Functions and Derivatives
…
…
►
§20.2(i) Fourier Series
…
►Corresponding expansions for
,
, can be found by differentiating (
20.2.1)–(
20.2.4) with respect to
.
…
►For fixed
, each of
,
,
, and
is an analytic
function of
for
, with a natural boundary
, and correspondingly, an analytic
function of
for
with a natural boundary
.
…
►
§20.2(iii) Translation of the Argument by Half-Periods
…
►For
, the
-zeros of
,
, are
,
,
,
respectively.
§11.9 Lommel Functions
…
►Provided that
, (
11.9.1) has the general solution
…
►When
,
…
►
§11.9(iii) Asymptotic Expansion
…
►For further information on Lommel
functions see
Watson (1944, §§10.7–10.75) and
Babister (1967, Chapter 3).
…
§5.15 Polygamma Functions
►The
functions
,
, are called the
polygamma functions.
In particular,
is the
trigamma function;
,
,
are the
tetra-, penta-, and
hexagamma functions respectively.
Most properties of these
functions follow straightforwardly by differentiation of properties of the psi
function.
…
►In (
5.15.2)–(
5.15.7)
, and for
see §
25.6(i).
…
…
►
§5.2(i) Gamma and Psi Functions
►
Euler’s Integral
…
►It is a meromorphic
function with no zeros, and with simple poles of residue
at
.
…
►
5.2.2
.
…
►
5.2.3
…