En matemàtiques, la integral de Goodwin-Staton es defineix com:[1] G ( z ) = ∫ 0 ∞ e − t 2 t + z d t {\displaystyle G(z)=\int _{0}^{\infty }{\frac {e^{-t^{2}}}{t+z}}\,dt} Gràfic 2D de la integral de Goodwin-Staton Gràfic 3D de la integral de Goodwin-Staton que satisfà la següent equació diferencial no lineal de tercer ordre: 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 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} G ( − z ) = − G ( z ) {\displaystyle G(-z)=-G(z)} G ( z ) = ( 1 − γ − ln ( z 2 ) − i csgn ( i z 2 ) π + 2 i π z + ( − 2 + γ + ln ( z 2 ) + i csgn ( i z 2 ) π ) z 2 − 4 i 3 π z 3 + ( 5 4 − 1 2 γ − 1 2 ln ( z 2 ) − 1 2 i csgn ( i z 2 ) π ) z 4 + O ( z 5 ) ) {\displaystyle {\begin{aligned}G(z)={}&{\Big (}1-\gamma -\ln(z^{2})-i\operatorname {csgn} (iz^{2})\pi +{\frac {2i}{\sqrt {\pi }}}z\\[5pt]&\qquad {}+(-2+\gamma +\ln(z^{2})+i\operatorname {csgn} (iz^{2})\pi {\Big )}z^{2}-{\frac {4i}{3{\sqrt {\pi }}}}z^{3}\\[5pt]&\qquad {}+\left({\frac {5}{4}}-{\frac {1}{2}}\gamma -{\frac {1}{2}}\ln(z^{2})-{\frac {1}{2}}i\operatorname {csgn} (iz^{2})\pi \right)z^{4}+O(z^{5}))\end{aligned}}} La funció G de Meijer 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 }}} La funció exponencial i la funció d'error Vegeu també: Funció exponencial 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}}}}} La funció de Laguerre G ( z ) = e − z 2 + E i ( 1 , − z 2 ) e − z 2 + i e − z 2 π z ; 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 {L}}\left(-1/2,1/2,{z}^{2}\right)} 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))} [1]Frank William John Olver (ed.), N. M. Temme (Chapter contr.), NIST Handbook of Mathematical Functions, Chapter 7, p160, Cambridge University Press 2010 «Evaluation of the generalized Goodwin–Staton integral using binomial expansion theorem». Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 105, pàg. 8–11. DOI: 10.1016/j.jqsrt.2006.09.018. F. W. J. Olver, Werner Rheinbolt, Academic Press, 2014, Mathematics,Asymptotics and Special Functions, 588 pages, ISBN 9781483267449 gbook The Generalized Goodwin-Station Integral (anglès) Error Functions, Dawson's and Fresnel Integrals (anglès) The generalized Goodwin-Staton integral (anglès) Analytical and Numerical Aspects of a Generalization of the Complementary Error Function (anglès)Wikiwand in your browser!Seamless Wikipedia browsing. 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