Asymptotic Expansions
Contents
- Poincaré-Type Expansions
- Error Bounds and Exponential Improvement
- Ratios
Poincaré-Type Expansions
As
z
→
∞
in the sector
|
ph
z
|
≤
π
-
δ
(
<
π
)
,
ln
Γ
(
z
)
∼
(
z
-
1
2
)
ln
z
-
z
+
1
2
ln
(
2
π
)
+
∑
k
=
1
∞
B
2
k
2
k
(
2
k
-
1
)
z
2
k
-
1
ψ
(
z
)
∼
ln
z
-
1
2
z
-
∑
k
=
1
∞
B
2
k
2
k
z
2
k
For the Bernoulli numbers
B
2
k
,
Also,
Γ
(
z
)
∼
ⅇ
-
z
z
z
(
2
π
z
)
1
2
(
∑
k
=
0
∞
g
k
z
k
)
g
0
=
1
,
g
1
=
1
12
,
g
2
=
1
288
,
g
3
=
-
139
51840
,
g
4
=
-
571
24 88320
,
g
5
=
1 63879
2090 18880
,
g
6
=
52 46819
7 52467 96800
g
k
=
2
(
1
2
)
k
a
2
k
,
where
a
0
=
1
2
2
, and
a
0
a
k
+
1
2
a
1
a
k
-
1
+
1
3
a
2
a
k
-
2
+
…
+
1
k
+
1
a
k
a
0
=
1
k
a
k
-
1
k
≥
1
.
Wrench(1968)
gives exact values of
g
k
up to
g
20
.
Spira(1971)
corrects errors in Wrench's results and also supplies exact and 45D values of
g
k
for
k
=
21
,
22
,
…
,
30
. For an asymptotic expansion of
g
k
as
k
→
∞
see
Boyd(1994)
.
With the same conditions
Γ
(
a
z
+
b
)
∼
2
π
ⅇ
-
a
z
(
a
z
)
a
z
+
b
-
(
1
2
)
where
a
(
>
0
)
and
b
(
∈
ℂ
)
are both fixed, and
ln
Γ
(
z
+
h
)
∼
(
z
+
h
-
1
2
)
ln
z
-
z
+
1
2
ln
(
2
π
)
+
∑
k
=
2
∞
(
-
1
)
k
B
k
(
h
)
k
(
k
-
1
)
z
k
-
1
where
h
(
∈
[
0
,
1
]
)
is fixed.
Also as
y
→
±
∞
,
|
Γ
(
x
+
ⅈ
y
)
|
∼
2
π
|
y
|
x
-
(
1
2
)
ⅇ
-
π
|
y
|
2
uniformly for bounded real values of
x
.
Error Bounds and Exponential Improvement
If the sums in the expansions
(Equation 1) and
(Equation 2) are terminated at
k
=
n
-
1
(
k
≥
0
) and
z
is real and positive, then the remainder terms are bounded in magnitude by
the first neglected terms and have the same sign. If
z
is complex, then the remainder terms are bounded in magnitude by
sec
2
n
(
1
2
ph
z
)
for
(Equation 1), and
sec
2
n
+
1
(
1
2
ph
z
)
for
(Equation 2), times the first neglected terms.
For the remainder term in
(Equation 3) write
Γ
(
z
)
=
ⅇ
-
z
z
z
(
2
π
z
)
1
2
(
∑
k
=
0
K
-
1
g
k
z
k
+
R
K
(
z
)
)
K
=
1
,
2
,
3
,
…
.
Then
|
R
K
(
z
)
|
≤
(
1
+
ζ
(
K
)
)
Γ
(
K
)
2
(
2
π
)
K
+
1
|
z
|
K
(
1
+
min
(
sec
(
ph
z
)
,
2
K
1
2
)
)
|
ph
z
|
≤
1
2
π
Ratios
If
a
(
∈
ℂ
)
and
b
(
∈
ℂ
)
are fixed as
z
→
∞
in
|
ph
z
|
≤
π
-
δ
(
<
π
)
, then
Γ
(
z
+
a
)
Γ
(
z
+
b
)
∼
z
a
-
b
Γ
(
z
+
a
)
Γ
(
z
+
b
)
∼
z
a
-
b
∑
k
=
0
∞
G
k
(
a
,
b
)
z
k
Also, with the added condition
ℜ
(
b
-
a
)
>
0
,
Γ
(
z
+
a
)
Γ
(
z
+
b
)
∼
(
z
+
a
+
b
-
1
2
)
a
-
b
∑
k
=
0
∞
H
k
(
a
,
b
)
(
z
+
1
2
(
a
+
b
-
1
)
)
2
k
Here
G
0
(
a
,
b
)
=
1
,
G
1
(
a
,
b
)
=
1
2
(
a
-
b
)
(
a
+
b
-
1
)
,
G
2
(
a
,
b
)
=
1
12
(
a
-
b
2
)
(
3
(
a
+
b
-
1
)
2
-
(
a
-
b
+
1
)
)
H
0
(
a
,
b
)
=
1
,
H
1
(
a
,
b
)
=
-
1
12
(
a
-
b
2
)
(
a
-
b
+
1
)
,
H
2
(
a
,
b
)
=
1
240
(
a
-
b
4
)
(
2
(
a
-
b
+
1
)
+
5
(
a
-
b
+
1
)
2
)
In terms of generalized Bernoulli polynomials we have for
k
=
0
,
1
,
…
G
k
(
a
,
b
)
=
(
a
-
b
k
)
B
k
(
a
-
b
+
1
)
(
a
)
H
k
(
a
,
b
)
=
(
a
-
b
2
k
)
B
2
k
(
a
-
b
+
1
)
(
a
-
b
+
1
2
)
Γ
(
z
+
a
)
Γ
(
z
+
b
)
Γ
(
z
+
c
)
∼
∑
k
=
0
∞
(
-
1
)
k
(
c
-
a
)
k
(
c
-
b
)
k
k
!
Γ
(
a
+
b
-
c
+
z
-
k
)