Goodwin–Staton integral
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In mathematics, the Goodwin–Staton integral is defined as :[1]
The integral satisfies the following third-order nonlinear differential equation：
Properties
Symmetry:
Expansion for small z:
![{\displaystyle {\begin{aligned}G(z)={}&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}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/6a215371dcbfc53d98d098e5584abed0d791fba6)