…
► where the sum is
over
1
≤
g
<
k
≤
n
and
n
≥
h
>
ℓ
≥
1
.
…
► For
(
j
,
k
)
∈
B
,
B
∖
[
j
,
k
]
denotes
B
after removal of all elements of the form
(
j
,
t
)
or
(
t
,
k
)
,
t
=
1
,
2
,
…
,
n
.
B
∖
(
j
,
k
)
denotes
B
with the element
(
j
,
k
)
removed.
►
26.15.5
R
(
x
,
B
)
=
x
R
(
x
,
B
∖
[
j
,
k
]
)
+
R
(
x
,
B
∖
(
j
,
k
)
)
.
…
…
►
33.8.2
H
ℓ
±
′
H
ℓ
±
=
c
±
i
ρ
a
b
2
(
ρ
−
η
±
i
)
+
(
a
+
1
)
(
b
+
1
)
2
(
ρ
−
η
±
2
i
)
+
⋯
,
…
►
a
=
1
+
ℓ
±
i
η
,
►
b
=
−
ℓ
±
i
η
,
►
c
=
±
i
(
1
−
(
η
/
ρ
)
)
.
…
►
F
ℓ
=
±
(
q
−
1
(
u
−
p
)
2
+
q
)
−
1
/
2
,
…
…
► The set
{
n
≥
1
|
n
≡
±
j
(
mod
k
)
}
is denoted by
A
j
,
k
.
…
► where the last right-hand side is the sum
over
m
≥
0
of the generating functions for partitions into distinct parts with largest part equal to
m
.
…
► where the sum is
over nonnegative integer values of
k
for which
n
−
1
2
(
3
k
2
±
k
)
≥
0
.
…
► where the sum is
over nonnegative integer values of
k
for which
n
−
(
3
k
2
±
k
)
≥
0
.
…
► where the sum is
over nonnegative integer values of
m
for which
n
−
1
2
k
m
2
−
m
+
1
2
k
m
≥
0
.
…
…
► If in addition
f
is periodic,
f
∈
C
k
(
ℝ
)
, and the integral is taken
over a period, then
…
►
Table 3.5.1: Nodes and weights for the 5-point Gauss–Legendre formula.
►
►
►
Table 3.5.2: Nodes and weights for the 10-point Gauss–Legendre formula.
►
►
►
Table 3.5.3: Nodes and weights for the 20-point Gauss–Legendre formula.
►
►
►
Table 3.5.4: Nodes and weights for the 40-point Gauss–Legendre formula.
►
►
…
…
►
∫
e
±
z
z
ν
𝒵
ν
(
z
)
d
z
=
e
±
z
z
ν
+
1
2
ν
+
1
(
𝒵
ν
(
z
)
∓
𝒵
ν
+
1
(
z
)
)
,
ν
≠
−
1
2
,
►
∫
e
±
z
z
−
ν
𝒵
ν
(
z
)
d
z
=
e
±
z
z
−
ν
+
1
1
−
2
ν
(
𝒵
ν
(
z
)
∓
𝒵
ν
−
1
(
z
)
)
,
ν
≠
1
2
.
►
§10.43(ii) Integrals over the Intervals
(
0
,
x
)
and
(
x
,
∞
)
…
►
§10.43(iv) Integrals over the Interval (
0
,
∞
)
…
►
…
…
►
§33.2(ii) Regular Solution
F
ℓ
(
η
,
ρ
)
…
►
§33.2(iii) Irregular Solutions
G
ℓ
(
η
,
ρ
)
,
H
ℓ
±
(
η
,
ρ
)
► The functions
H
ℓ
±
(
η
,
ρ
)
are defined by
…
► As in the case of
F
ℓ
(
η
,
ρ
)
, the solutions
H
ℓ
±
(
η
,
ρ
)
and
G
ℓ
(
η
,
ρ
)
are analytic functions of
ρ
when
0
<
ρ
<
∞
.
Also,
e
∓
i
σ
ℓ
(
η
)
H
ℓ
±
(
η
,
ρ
)
are analytic functions of
η
when
−
∞
<
η
<
∞
.
…