En matemàtiques, la integral de Goodwin-Staton es defineix com:[1]
que satisfà la següent equació diferencial no lineal de tercer ordre:
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}}}
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d32323d8f21f2417d90f110df6a1f712de17ef55)
