查看“︁古德温 - 斯塔顿积分”︁的源代码

[[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]]
'''古德温 - 斯塔顿[[积分]]'''（{{lang-en|Goodwin-Staton Integral}}）定义如下<ref>Frank Oliver, NIST Handbook of Mathematical Functions, p160,Cambridge University Press 2010{{en}}</ref>

<math>G(z)=\int _{0}^{\infty }\!{\frac {{{\rm e}^{-{t}^{2}}}}{t+z}}{dt}</math>

它是下列三阶非线性常微分方程的一个解：
<math>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
</math>

==对称关系==
<math>G(-z)=G(z)</math>

==与其他函数的关系==
;[[Meijer G-函数]]
*<math>G(z)=\frac{1}{2}\,{\frac {
G^{3, 2}_{2, 3}\left({z}^{2}\, \Big\vert\,^{0, 1/2}_{1/2, 0, 0}\right)
}{\pi }}</math> [[MeijerG 函数]]

;[[指数函数]]与[[误差函数]]

*<math>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)}</math>



;

*<math>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 }}}
  </math>
*<math> 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 }}}
 </math>
*<math>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}
}}}
  </math>

;[[拉盖尔函数]]
*<math>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) </math>

*<math>  </math>

==级数展开==
*<math>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}}}</math>
*<math>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) )
</math>

==参考文献==
<references/>
* http://journals.cambridge.org/article_S0013091504001087
* http://www.sciencedirect.com/science/article/pii/S0022407306002500 {{Wayback|url=http://www.sciencedirect.com/science/article/pii/S0022407306002500 |date=20150924154401 }}
* http://dlmf.nist.gov/7.2 {{Wayback|url=http://dlmf.nist.gov/7.2 |date=20201205140227 }}
* https://web.archive.org/web/20150225035306/http://discovery.dundee.ac.uk/portal/en/research/the-generalized-goodwinstaton-integral(3db9f429-7d7f-488c-a1d7-c8efffd01158).html
* https://web.archive.org/web/20150225105452/http://discovery.dundee.ac.uk/portal/en/research/the-generalized-goodwinstaton-integral(3db9f429-7d7f-488c-a1d7-c8efffd01158)/export.html
* http://www.damtp.cam.ac.uk/user/na/NA_papers/NA2009_02.pdf {{Wayback|url=http://www.damtp.cam.ac.uk/user/na/NA_papers/NA2009_02.pdf |date=20160304043431 }}
* F. W. J. Olver, Werner Rheinbolt, Academic Press, 2014 , Mathematics,''Asymptotics and Special Functions'', 588 pages , ISBN 9781483267449 [https://books.google.com/books?id=JLziBQAAQBAJ&pg=PA567&lpg=PA567&dq=Goodwin-Staton+Integral&source=bl&ots=hESMlEIped&sig=czrxemmIQvMD1rV2yzCO1NJ91r0&sa=X&ei=IDvsVL03iOTyBdnwgJgD&ved=0CDEQ6AEwAjgK#v=onepage&q=Goodwin-Staton%20Integral&f=false gbook] {{Wayback|url=https://books.google.com/books?id=JLziBQAAQBAJ&pg=PA567&lpg=PA567&dq=Goodwin-Staton+Integral&source=bl&ots=hESMlEIped&sig=czrxemmIQvMD1rV2yzCO1NJ91r0&sa=X&ei=IDvsVL03iOTyBdnwgJgD&ved=0CDEQ6AEwAjgK#v=onepage&q=Goodwin-Staton%20Integral&f=false |date=20190605192742 }}


[[Category:特殊函数]]