…
►
►
►
►An often used alternative to the
symbol is the Clebsch–Gordan coefficient
►
34.1.1
…
…
►Leading terms of the power series for
and
for
are:
…
►The coefficients of the power series of
,
and also
,
are the same until the terms in
and
, respectively.
…
►Numerical values of the radii of convergence
of the power series (
28.6.1)–(
28.6.14) for
are given in Table
28.6.1.
Here
for
,
for
, and
for
and
.
…
►
§28.6(ii) Functions and
…
…
►
26.12.9
…
►
26.12.10
…
►
26.12.11
…
►The notation
denotes the sum over all plane partitions contained in
, and
denotes the number of elements in
.
…
►where
is the sum of the squares of the divisors of
.
…
…
►Given numerical values of
and
, the solution
of the equation
…These errors have the effect of perturbing the solution by unwanted small multiples of
and of an independent solution
, say.
…
►The unwanted multiples of
now decay in comparison with
, hence are of little consequence.
…
►The latter method is usually superior when the true value of
is zero or pathologically small.
…
►beginning with
.
…
…
►
•
Blanch and Rhodes (1955) includes , ,
, ; 8D.
The range of is 0 to 0.1, with step sizes ranging from 0.002
down to 0.00025. Notation:
,
.
►
•
Ince (1932) includes eigenvalues , , and Fourier coefficients
for or , ; 7D. Also
, for ,
, corresponding to the eigenvalues in the tables; 5D. Notation:
, .
…
►
•
Stratton et al. (1941) includes , , and the corresponding Fourier
coefficients for and for
or , . Precision is mostly 5S. Notation:
, , , and for
, see §28.1.
…
►
•
Ince (1932) includes the first zero for ,
for or , ; 4D. This reference
also gives zeros of the first derivatives, together with expansions for small
.
…
►For other tables prior to 1961 see
Fletcher et al. (1962, §2.2) and
Lebedev and Fedorova (1960, Chapter 11).
…
►A transformation of a convergent sequence
with limit
into a sequence
is called
limit-preserving if
converges to the same limit
.
…
►This transformation is accelerating if
is a
linearly convergent
sequence, i.
…
►Then the transformation of the sequence
into a sequence
is given by
…
►Then
.
…
►We give a special form of
Levin’s transformation in which the sequence
of partial sums
is transformed into:
…
…
►The path is partitioned at
points labeled successively
, with
,
.
…
►Write
,
, expand
and
in Taylor series (§
1.10(i)) centered at
, and apply (
3.7.2).
…
►If, for example,
, then on moving the contributions of
and
to the right-hand side of (
3.7.13) the resulting system of equations is not tridiagonal, but can readily be made tridiagonal by annihilating the elements of
that lie below the main diagonal and its two adjacent diagonals.
…
►The values
are the
eigenvalues and the corresponding solutions
of the differential equation are the
eigenfunctions.
…
►where
and
…