Goodwin–Staton Integral

In mathematics, the Goodwin–Staton integral is defined as : Frank William John Olver (ed.), N. M. Temme (Chapter contr.), NIST Handbook of Mathematical Functions, Chapter 7, p160,Cambridge University Press 2010

: $G(z)=\int_0^\infty \frac \, dt$

The integral satisfies the following third-order nonlinear differential equation：

: $4w(z) +8\,z \frac w (z) + (2+2\,z^2) \frac w (z) +z \frac w \left( z \right) =0$

Properties

Symmetry:
: $G(-z)=-G(z)$

Expansion for small ''z'':
:
\begin
G(z) = & 1-\gamma-\ln(z^2) -i\operatorname ( iz^2) \pi +\frac z \\ pt& \qquad + (-2 + \gamma + \ln(z^2) +i \operatorname (iz^2) \pi \Big) z^2 - \frac z^3 \\ pt& \qquad + \left( \frac 5 4 - \frac 1 2 \gamma - \frac 1 2 \ln (z^2) - \frac 1 2 i \operatorname ( iz^2) \pi \right) z^4 + O (z^5)
\end

References

* http://journals.cambridge.org/article_S0013091504001087
*
* http://dlmf.nist.gov/7.2
* https://web.archive.org/web/20150225035306/http://discovery.dundee.ac.uk/portal/en/research/the-generalized-goodwinstaton-integral(3db9f429-7d7f-488c-a1d7-c8efffd01158).html
* https://web.archive.org/web/20150225105452/http://discovery.dundee.ac.uk/portal/en/research/the-generalized-goodwinstaton-integral(3db9f429-7d7f-488c-a1d7-c8efffd01158)/export.html
* http://www.damtp.cam.ac.uk/user/na/NA_papers/NA2009_02.pdf
* F. W. J. Olver, Werner Rheinbolt, Academic Press, 2014, Mathematics,''Asymptotics and Special Functions'', 588 pages,
gbook

{{DEFAULTSORT:Goodwin-Staton integral
Special functions