…
►
§3.7(ii) Taylor-Series Method: Initial-Value Problems
…
►
§3.7(iii) Taylor-Series Method: Boundary-Value Problems
…
►Let
be a finite or
infinite interval and
be a real-valued continuous (or piecewise continuous) function on the closure of
.
…with limits taken in (
3.7.16) when
or
, or both, are
infinite.
…
►The method consists of a set of rules each of which is equivalent to a truncated Taylor-
series expansion, but the rules avoid the need for analytic differentiations of the differential equation.
…
…
►where the
series terminates when the product of the first
primes exceeds
.
…
►
27.12.4
…
►
changes sign
infinitely often as
; see
Littlewood (1914),
Bays and Hudson (2000).
…
►If
is relatively prime to the modulus
, then there are
infinitely many primes congruent to
.
…
►There are
infinitely many Carmichael numbers.
…
…
►The Fourier
series of a Floquet solution
►
28.2.18
…
►For given
and
, equation (
28.2.16) determines an
infinite discrete set of values of
, the
eigenvalues or
characteristic
values, of Mathieu’s equation.
…
►Near
,
and
can be expanded in power
series in
(see §
28.6(i)); elsewhere they are determined by analytic continuation (see §
28.7).
…
…
►In the region
, called the
critical strip,
has
infinitely many zeros, distributed symmetrically about the real axis and about the
critical
line
.
…
►Because
changes sign
infinitely often,
has
infinitely many zeros with
real.
…
►The error term
can be expressed as an asymptotic
series that begins
…
►For further information on the Riemann–Siegel
expansion see
Berry (1995).
…
►
§1.3(iii) Infinite Determinants
…
►
►Of importance for special functions are
infinite determinants of
Hill’s
type.
These have the property that the double
series
…
►
Orthonormal Expansions
…
…
►converges for all sufficiently large
, and
is
infinitely differentiable in a neighborhood of the origin.
…
►Then the
series obtained by substituting (
2.3.7) into (
2.3.1) and integrating formally term by term yields an asymptotic
expansion:
…
►Assume that
again has the
expansion (
2.3.7) and this
expansion is
infinitely differentiable,
is
infinitely differentiable on
, and each of the integrals
,
, converges at
, uniformly for all sufficiently large
.
Then
…
►
(b)
As the asymptotic expansions (2.3.14) apply,
and each is infinitely differentiable. Again , , and are
positive.
…
…
►This especially included updated information on matrix analysis, measure theory, spectral analysis, and a new section on linear second order differential operators and eigenfunction
expansions.
…
►
Expansion
§4.13 has been enlarged. The Lambert -function
is multi-valued and we use the notation , , for the
branches. The original two solutions are identified via
and .
Other changes are the introduction of the Wright -function and tree
-function in (4.13.1_2) and (4.13.1_3), simplification formulas
(4.13.3_1) and (4.13.3_2), explicit representation (4.13.4_1) for
, additional Maclaurin series (4.13.5_1) and
(4.13.5_2), an explicit expansion about the branch point at in
(4.13.9_1), extending the number of terms in asymptotic expansions (4.13.10)
and (4.13.11), and including several integrals and integral representations for
Lambert -functions in the end of the section.
…
►
Subsection 25.2(ii) Other Infinite Series
It is now mentioned that
(25.2.5), defines the Stieltjes constants .
Consequently, in (25.2.4), (25.6.12)
are now identified as the Stieltjes constants.
…
►
Equation (2.11.4)
Because (2.11.4) is not an asymptotic expansion, the symbol
that was used originally is incorrect and has been replaced with ,
together with a slight change of wording.
►
Equation (13.9.16)
Originally was expressed in term of asymptotic symbol . As a consequence of the use of
the order symbol on the right-hand side, was replaced by .
…