Approximations
Contents
- Rational Approximations
- Expansions in Chebyshev Series
- Approximations in the Complex Plane
Rational Approximations
Cody and Hillstrom(1967)
gives minimax rational approximations for
ln
Γ
(
x
)
for the ranges
0.5
≤
x
≤
1.5
,
1.5
≤
x
≤
4
,
4
≤
x
≤
12
; precision is variable.
Hart et.al.(1968)
gives minimax polynomial and rational approximations to
Γ
(
x
)
and
ln
Γ
(
x
)
in the intervals
0
≤
x
≤
1
,
8
≤
x
≤
1000
,
12
≤
x
≤
1000
; precision is variable.
Cody et.al.(1973)
gives minimax rational approximations for
ψ
(
x
)
for the ranges
0.5
≤
x
≤
3
and
3
≤
x
<
∞
; precision is variable.
For additional approximations see
Hart et.al.(1968)
(Appendix B),
Luke(1975)
(pp. 22–23), and
Weniger(2003)
.
Expansions in Chebyshev Series
Luke(1969)
gives the coefficients to 20D for the Chebyshev-series expansions of
Γ
(
1
+
x
)
,
1
Γ
(
1
+
x
)
,
Γ
(
x
+
3
)
,
ln
Γ
(
x
+
3
)
,
ψ
(
x
+
3
)
, and the first six derivatives of
ψ
(
x
+
3
)
for
0
≤
x
≤
1
. These coefficients are reproduced in
Luke(1975)
.
Clenshaw(1962)
also gives 20D Chebyshev-series coefficients for
Γ
(
1
+
x
)
and its reciprocal for
0
≤
x
≤
1
. See
Luke(1975)
(pp. 22–23) for additional expansions.
Approximations in the Complex Plane
Rational approximations for
Γ
(
z
+
1
)
A
(
z
)
, where
A
(
z
)
=
(
2
π
)
1
2
(
z
+
c
+
1
2
)
z
+
1
exp
(
-
(
z
+
c
+
1
2
)
)
, and approximations for
Γ
(
z
+
1
)
based on the Padé approximants for two forms of the incomplete
gamma function are in
Luke(1969)
.
Luke(1975)
(pp. 13–16) provides explicit rational approximations for
ψ
(
z
)
+
γ
