古德温 - 斯塔顿积分

古德温 - 斯塔顿积分（英語：Goodwin-Staton Integral）定义如下 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） http://journals.cambridge.org/article_S0013091504001087 http://www.sciencedirect.com/science/article/pii/S0022407306002500 （页面存档备份，存于互联网档案馆） http://dlmf.nist.gov/7.2 （页面存档备份，存于互联网档案馆） 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 （页面存档备份，存于互联网档案馆） F. W. J. Olver, Werner Rheinbolt, Academic Press, 2014 , Mathematics,Asymptotics and Special Functions, 588 pages , ISBN 9781483267449 gbook （页面存档备份，存于互联网档案馆）