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[[File:Goodwin-Staton Integral.png|thumb|300px|Goodwin-Staton Integral Maple 2D plot]] |
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[[File:Goodwin-Staton Integral.png|thumb|300px|Goodwin-Staton Integral Maple 2D plot]] |
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[[File:Goodwin-Station integral Maple complex 3D plot.png|thumb|300px|Goodwin-Station integral Maple complex 3D plot]] |
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[[File:Goodwin-Station integral Maple complex 3D plot.png|thumb|300px|Goodwin-Station integral Maple complex 3D plot]] |
Revision as of 16:32, 18 March 2015
Goodwin-Staton Integral Maple 2D plot
Goodwin-Station integral Maple complex 3D plot
In mathematicsGoodwin-Staton integralis defined as :[1]
It satisfy the following differential equation:
Symmmetry
In terms of othe special functions
![{\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)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6affab30d66ed0115a65a90af9ef1667ca78ca4)
in terms of MeijerG function
![{\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 }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ec47e28886e55ef67106540a9e022df32f60346)
![{\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 }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a595f443292b0dbb7b85d1b2228fe3b913070aca)
![{\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}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2b16a7214ed346ca25eef8b08bf46c15f39ceaf)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1902cbe1fb018603207bc2d10f926d8e70c4a30)
Series expansion
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13a72ede354b50e2a6e31d07faae64b1f0cdc6c7)
![{\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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba22fd9fa7a161fdd0081f287c10f9540f2f9ad6)
References
- ^ Frank Oliver, NIST Handbook of Mathematical Functions, p160,Cambridge University Press 2010