古德温 - 斯塔顿积分 （英語：Goodwin-Staton Integral ）定义如下[ 1]
Goodwin-Staton Integral Maple 2D plot Goodwin-Station integral Maple complex 3D plot
G
(
z
)
=
∫
0
∞
e
−
t
2
t
+
z
d
t
{\displaystyle G(z)=\int _{0}^{\infty }\!{\frac {{\rm {e}}^{-{t}^{2}}}{t+z}}{dt}}
它是下列三阶非线性常微分方程的一个解：
4
w
(
z
)
+
8
z
d
d
z
w
(
z
)
+
(
2
+
2
z
2
)
d
2
d
z
2
w
(
z
)
+
z
d
3
d
z
3
w
(
z
)
=
0
{\displaystyle 4\,w\left(z\right)+8\,z{\frac {d}{dz}}w\left(z\right)+\left(2+2\,{z}^{2}\right){\frac {d^{2}}{d{z}^{2}}}w\left(z\right)+z{\frac {d^{3}}{d{z}^{3}}}w\left(z\right)=0}
G
(
−
z
)
=
G
(
z
)
{\displaystyle G(-z)=G(z)}
Meijer G-函数
G
(
z
)
=
1
2
G
2
,
3
3
,
2
(
z
2
|
1
/
2
,
0
,
0
0
,
1
/
2
)
π
{\displaystyle G(z)={\frac {1}{2}}\,{\frac {G_{2,3}^{3,2}\left({z}^{2}\,{\Big \vert }\,_{1/2,0,0}^{0,1/2}\right)}{\pi }}}
MeijerG 函数 指数函数 与误差函数
G
(
z
)
=
e
−
z
2
+
E
i
(
1
,
−
z
2
)
e
−
z
2
+
e
−
z
2
e
r
f
(
i
z
)
{\displaystyle G\left(z\right)={{\rm {e}}^{-{z}^{2}}}+{\it {Ei}}\left(1,-{z}^{2}\right){{\rm {e}}^{-{z}^{2}}}+{{\rm {e}}^{-{z}^{2}}}{{\rm {erf}}\left(iz\right)}}
G
(
z
)
=
e
−
z
2
+
U
(
1
,
1
,
−
z
2
)
e
z
2
e
−
z
2
+
2
i
e
−
z
2
z
M
(
1
/
2
,
3
/
2
,
z
2
)
π
{\displaystyle G\left(z\right)={{\rm {e}}^{-{z}^{2}}}+{{\rm {U}}\left(1,\,1,\,-{z}^{2}\right)}{{\rm {e}}^{{z}^{2}}}{{\rm {e}}^{-{z}^{2}}}+{\frac {2\,i{{\rm {e}}^{-{z}^{2}}}z{{\rm {M}}\left(1/2,\,3/2,\,{z}^{2}\right)}}{\sqrt {\pi }}}}
G
(
z
)
=
e
−
z
2
+
E
i
(
1
,
−
z
2
)
e
−
z
2
+
2
i
e
−
z
2
z
H
e
u
n
B
(
1
,
0
,
1
,
0
,
z
2
)
π
{\displaystyle G\left(z\right)={{\rm {e}}^{-{z}^{2}}}+{\it {Ei}}\left(1,-{z}^{2}\right){{\rm {e}}^{-{z}^{2}}}+{\frac {2\,i{{\rm {e}}^{-{z}^{2}}}z{\it {HeunB}}\left(1,0,1,0,{\sqrt {{z}^{2}}}\right)}{\sqrt {\pi }}}}
G
(
z
)
=
e
−
z
2
+
E
i
(
1
,
−
z
2
)
e
−
z
2
+
z
e
−
z
2
(
−
i
e
r
f
c
(
−
z
2
)
+
i
)
−
z
2
{\displaystyle G\left(z\right)={{\rm {e}}^{-{z}^{2}}}+{\it {Ei}}\left(1,-{z}^{2}\right){{\rm {e}}^{-{z}^{2}}}+{\frac {z{{\rm {e}}^{-{z}^{2}}}\left(-i{\it {erfc}}\left({\sqrt {-{z}^{2}}}\right)+i\right)}{\sqrt {-{z}^{2}}}}}
拉盖尔函数
G
(
z
)
=
e
−
z
2
+
E
i
(
1
,
−
z
2
)
e
−
z
2
+
i
e
−
z
2
π
z
L
a
g
u
e
r
r
e
L
(
−
1
/
2
,
1
/
2
,
z
2
)
{\displaystyle G\left(z\right)={{\rm {e}}^{-{z}^{2}}}+{\it {Ei}}\left(1,-{z}^{2}\right){{\rm {e}}^{-{z}^{2}}}+i{{\rm {e}}^{-{z}^{2}}}{\sqrt {\pi }}z{\it {LaguerreL}}\left(-1/2,1/2,{z}^{2}\right)}
{\displaystyle }
G
(
z
)
=
10
z
−
1
−
50
z
−
2
−
1000
3
z
2
−
1
z
3
+
2500
z
2
−
1
z
4
+
10000
2
−
2
z
2
+
z
4
z
5
−
250000
3
2
−
2
z
2
+
z
4
z
6
−
5000000
21
−
6
+
6
z
2
−
3
z
4
+
z
6
z
7
+
6250000
3
−
6
+
6
z
2
−
3
z
4
+
z
6
z
8
+
125000000
27
24
−
24
z
2
+
12
z
4
−
4
z
6
+
z
8
z
9
−
125000000
3
24
−
24
z
2
+
12
z
4
−
4
z
6
+
z
8
z
10
{\displaystyle G(z)=10\,{z}^{-1}-50\,{z}^{-2}-{\frac {1000}{3}}\,{\frac {{z}^{2}-1}{{z}^{3}}}+2500\,{\frac {{z}^{2}-1}{{z}^{4}}}+10000\,{\frac {2-2\,{z}^{2}+{z}^{4}}{{z}^{5}}}-{\frac {250000}{3}}\,{\frac {2-2\,{z}^{2}+{z}^{4}}{{z}^{6}}}-{\frac {5000000}{21}}\,{\frac {-6+6\,{z}^{2}-3\,{z}^{4}+{z}^{6}}{{z}^{7}}}+{\frac {6250000}{3}}\,{\frac {-6+6\,{z}^{2}-3\,{z}^{4}+{z}^{6}}{{z}^{8}}}+{\frac {125000000}{27}}\,{\frac {24-24\,{z}^{2}+12\,{z}^{4}-4\,{z}^{6}+{z}^{8}}{{z}^{9}}}-{\frac {125000000}{3}}\,{\frac {24-24\,{z}^{2}+12\,{z}^{4}-4\,{z}^{6}+{z}^{8}}{{z}^{10}}}}
G
(
z
)
=
(
1
−
γ
−
ln
(
z
2
)
−
i
c
s
g
n
(
i
z
2
)
π
+
2
i
π
z
+
(
−
2
+
γ
+
ln
(
z
2
)
+
i
c
s
g
n
(
i
z
2
)
π
)
z
2
+
−
4
/
3
i
π
z
3
+
(
5
4
−
1
/
2
γ
−
1
/
2
ln
(
z
2
)
−
1
/
2
i
c
s
g
n
(
i
z
2
)
π
)
z
4
+
O
(
z
5
)
)
{\displaystyle G(z)=(1-\gamma -\ln \left({z}^{2}\right)-i{\it {csgn}}\left(i{z}^{2}\right)\pi +{\frac {2\,i}{\sqrt {\pi }}}z+\left(-2+\gamma +\ln \left({z}^{2}\right)+i{\it {csgn}}\left(i{z}^{2}\right)\pi \right){z}^{2}+{\frac {-4/3\,i}{\sqrt {\pi }}}{z}^{3}+\left({\frac {5}{4}}-1/2\,\gamma -1/2\,\ln \left({z}^{2}\right)-1/2\,i{\it {csgn}}\left(i{z}^{2}\right)\pi \right){z}^{4}+O\left({z}^{5}\right))}
^ Frank Oliver, NIST Handbook of Mathematical Functions, p160,Cambridge University Press 2010（英文）
