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[[File:Goodwin-Staton Integral.png|thumb|300px|Goodwin–Staton integral [[Maple]] 2D plot]]
[[File:Goodwin-Staton Integral.png|thumb|300px|Goodwin–Staton integral [[Maple]] 2D plot]]
[[File:Goodwin-Station integral Maple complex 3D plot.png|thumb|300px|Goodwin–Station integral Maple complex 3D plot]]
[[File:Goodwin-Station integral Maple complex 3D plot.png|thumb|300px|Goodwin–Station integral Maple complex 3D plot]]
[[File:Goodwin-staton integral derivative Re 3D plot.png|thumb|300px|Goodwin-staton integral derivative Re 3D plot]]
In mathematics the '''Goodwin–Staton integral''' is defined as :<ref>Frank Oliver, NIST Handbook of Mathematical Functions, p160,[[Cambridge University Press]] 2010</ref>
In mathematics the '''Goodwin–Staton integral''' is defined as :<ref>Frank Oliver, NIST Handbook of Mathematical Functions, p160,[[Cambridge University Press]] 2010</ref>
Revision as of 15:31, 22 March 2015
Goodwin–Staton integral Maple 2D plot Goodwin–Station integral Maple complex 3D plot Goodwin-staton integral derivative Re 3D plot In mathematics the Goodwin–Staton integral is defined as :[ 1]
G
(
z
)
=
∫
0
∞
e
−
t
2
t
+
z
d
t
{\displaystyle G(z)=\int _{0}^{\infty }{\frac {e^{-t^{2}}}{t+z}}\,dt}
It satisfies the following third-order nonlinear differential equation ：
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 4w(z)+8\,z{\frac {d}{dz}}w(z)+(2+2\,z^{2}){\frac {d^{2}}{dz^{2}}}w(z)+z{\frac {d^{3}}{dz^{3}}}w\left(z\right)=0}
Symmmetry
G
(
−
z
)
=
−
G
(
z
)
{\displaystyle G(-z)=-G(z)}
In terms of othe special functions
Exponential integral and Error function
G
(
z
)
=
e
−
z
2
+
Ei
(
1
,
−
z
2
)
e
−
z
2
+
e
−
z
2
erf
(
i
z
)
{\displaystyle G(z)=e^{-z^{2}}+\operatorname {Ei} (1,-z^{2})e^{-z^{2}}+e^{-z^{2}}\operatorname {erf} (iz)}
G
(
z
)
=
e
−
z
2
+
Ei
(
1
,
−
z
2
)
e
−
z
2
+
z
e
−
z
2
(
−
i
erfc
(
−
z
2
)
+
i
)
−
z
2
{\displaystyle G(z)=e^{-z^{2}}+\operatorname {Ei} (1,-z^{2})e^{-z^{2}}+{\frac {ze^{-z^{2}}\left(-i\operatorname {erfc} \left({\sqrt {-z^{2}}}\right)+i\right)}{\sqrt {-z^{2}}}}}
Meijer G function
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 }}}
Kummer function
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(z)=e^{-z^{2}}+{\rm {U}}(1,\,1,\,-z^{2})e^{z^{2}}e^{-z^{2}}+{\frac {2ie^{-z^{2}}z{\rm {M}}(1/2,3/2,z^{2})}{\sqrt {\pi }}}}
Heun function
G
(
z
)
=
e
−
z
2
+
Ei
(
1
,
−
z
2
)
e
−
z
2
+
2
i
e
−
z
2
z
HeunB
(
1
,
0
,
1
,
0
,
z
2
)
π
{\displaystyle G(z)=e^{-z^{2}}+\operatorname {Ei} (1,-z^{2})e^{-z^{2}}+{\frac {2ie^{-z^{2}}z\operatorname {HeunB} \left(1,0,1,0,{\sqrt {z^{2}}}\right)}{\sqrt {\pi }}}}
Laguerrel polynomials
G
(
z
)
=
e
−
z
2
+
E
i
(
1
,
−
z
2
)
e
−
z
2
+
i
e
−
z
2
π
z
LaguerreL
(
−
1
/
2
,
1
/
2
,
z
2
)
{\displaystyle G(z)=e^{-z^{2}}+{\it {Ei}}(1,-z^{2})e^{-z^{2}}+i{e^{-z^{2}}}{\sqrt {\pi }}z\operatorname {LaguerreL} (-1/2,1/2,z^{2})}
Series expansion
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)=10z^{-1}-50z^{-2}-{\frac {1000}{3}}\,{\frac {z^{2}-1}{z^{3}}}+2500\,{\frac {z^{2}-1}{z^{4}}}+10000\,{\frac {2-2z^{2}+z^{4}}{z^{5}}}-{\frac {250000}{3}}\,{\frac {2-2z^{2}+z^{4}}{z^{6}}}-{\frac {5000000}{21}}\,{\frac {-6+6z^{2}-3z^{4}+z^{6}}{z^{7}}}+{\frac {6250000}{3}}\,{\frac {-6+6z^{2}-3z^{4}+z^{6}}{z^{8}}}+{\frac {125000000}{27}}\,{\frac {24-24z^{2}+12z^{4}-4z^{6}+z^{8}}{z^{9}}}-{\frac {125000000}{3}}\,{\frac {24-24z^{2}+12z^{4}-4z^{6}+z^{8}}{z^{10}}}}
G
(
z
)
=
(
1
−
γ
−
ln
(
z
2
)
−
i
csgn
(
i
z
2
)
π
+
2
i
π
z
+
(
−
2
+
γ
+
ln
(
z
2
)
+
i
csgn
(
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
csgn
(
i
z
2
)
π
)
z
4
+
O
(
z
5
)
)
{\displaystyle G(z)=\left(1-\gamma -\ln(z^{2})-i\operatorname {csgn} (iz^{2})\pi +{\frac {2\,i}{\sqrt {\pi }}}z+(-2+\gamma +\ln(z^{2})+i\operatorname {csgn} (iz^{2})\pi \right)z^{2}+{\frac {-4/3\,i}{\sqrt {\pi }}}z^{3}+\left({\frac {5}{4}}-1/2\,\gamma -1/2\,\ln(z^{2})-1/2\,i\operatorname {csgn} (iz^{2})\pi \right)z^{4}+O(z^{5}))}
Derivative
d
d
z
G
(
z
)
=
∫
0
∞
−
e
−
t
2
(
t
+
z
)
2
d
t
=
−
2
e
−
z
2
(
z
2
−
z
2
π
+
e
z
2
−
z
2
π
+
E
i
(
1
,
−
z
2
)
z
2
−
z
2
π
+
i
z
3
e
r
f
(
−
z
2
)
π
−
i
e
z
2
z
−
z
2
)
z
−
z
2
π
{\displaystyle {\frac {d}{dz}}G\left(z\right)=\int _{0}^{\infty }\!-{\frac {{\rm {e}}^{-{t}^{2}}}{\left(t+z\right)^{2}}}{dt}=-2\,{\frac {{{\rm {e}}^{-{z}^{2}}}\left({z}^{2}{\sqrt {-{z}^{2}}}{\sqrt {\pi }}+{{\rm {e}}^{{z}^{2}}}{\sqrt {-{z}^{2}}}{\sqrt {\pi }}+{\it {Ei}}\left(1,-{z}^{2}\right){z}^{2}{\sqrt {-{z}^{2}}}{\sqrt {\pi }}+i{z}^{3}{{\rm {erf}}\left({\sqrt {-{z}^{2}}}\right)}{\sqrt {\pi }}-i{{\rm {e}}^{{z}^{2}}}z{\sqrt {-{z}^{2}}}\right)}{z{\sqrt {-{z}^{2}}}{\sqrt {\pi }}}}}
Gallery
Goodwin-staton integral Im complex 3D plot Goodwin-staton integral derivative 2D plot Goodwin-Staton integral Im density plot
References
