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*<math>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) </math> |
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*<math>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) </math> |
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Revision as of 23:50, 20 March 2015
Goodwin–Staton integral Maple 2D plot
Goodwin–Station integral Maple complex 3D plot
In mathematics the Goodwin–Staton integral is defined as :[1]
![{\displaystyle G(z)=\int _{0}^{\infty }{\frac {e^{-t^{2}}}{t+z}}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6dff5cf213b9639a478edd55ddb94ce1fb85c4d7)
It satisfies the following third-order nonlinear differential equation:
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cd7e6a32d0ae22684c336efdee10f6b3e859978)
Symmmetry
In terms of othe special functions
- Exponential integral and Error function
![{\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}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df88dcadf6045d69dfaa7406f0d21116ff0d8142)
- Meijer G function
![{\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 }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c7c7e46bda37593d2e0405f76266b8312a3c713)
- Kummer function
![{\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 }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/587f71a5f915de275fe93e4ecdbd0f2f5b753c97)
- Heun function
![{\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 }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51e89d6addcae40ec14bb8fb9849602074e0852a)
- Laguerrel polynomials
![{\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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba3f63097cad1ade44acba87e41861e3a9e7b98)
Series expansion
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/040c4f60cd4556a81d6664ff7cba898c5eb05d28)
![{\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}))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8929675cd5782a17e419ffc1d198ea9e3beed024)
References