De la Viquipèdia, l'enciclopèdia lliure
Gràfic 2D de la integral de Goodwin-Staton
Gràfic 3D de la integral de Goodwin-Staton
En matemàtiques, la integral de Goodwin-Staton es defineix com:[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)
que satisfà la següent equació diferencial no lineal de tercer ordre:
![{\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)
Relació amb altres funcions[modifica]
La funció G de Meijer[modifica]
![{\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)
La funció exponencial i la funció d'error[modifica]
![{\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)
![{\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)
La funció de Laguerre[modifica]
![{\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 {L}}\left(-1/2,1/2,{z}^{2}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2e0b198823e2a4dce6ec331e50a5768f3e76f39)
Desenvolupament de la sèrie[modifica]
![{\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)
Enllaços externs[modifica]