Goodwin–Staton integral

In mathematics the Goodwin–Staton integral is defined as :^[1]

$G(z)=\int_0^\infty \frac {e^{-t^2}}{t+z} \, dt$

It satisfies the following third-order nonlinear differential equation：

$4w(z) +8\,z \frac {d}{dz} w (z) + (2+2\,z^2) \frac {d^{2}}{dz^2} w (z) +z \frac {d^3}{dz^3} w \left( z \right) =0$

Symmetry

$G(-z)=-G(z)$

• $G(z)= \left(1-\gamma-\ln(z^2) -i\operatorname{csgn} ( iz^2) \pi +\frac {2\,i}{\sqrt {\pi }} z + (-2 + \gamma + \ln(z^2) +i \operatorname{csgn} (iz^2) \pi\right) z^2 + \frac {-4/3\,i}{\sqrt {\pi }} z^3+ \left( \frac {5}{4} -1/2\,\gamma-1/2\,\ln (z^2) -1/2\,i \operatorname{csgn} ( iz^2) \pi \right) z^4+O (z^5))$

References

1. Frank William John Olver (ed.), N. M. Temme (Chapter contr.), NIST Handbook of Mathematical Functions, Chapter 7, p160,Cambridge University Press 2010

• http://journals.cambridge.org/article_S0013091504001087
• http://www.sciencedirect.com/science/article/pii/S0022407306002500
• http://dlmf.nist.gov/7.2
• http://discovery.dundee.ac.uk/portal/en/research/the-generalized-goodwinstaton-integral%283db9f429-7d7f-488c-a1d7-c8efffd01158%29.html
• http://discovery.dundee.ac.uk/portal/en/research/the-generalized-goodwinstaton-integral%283db9f429-7d7f-488c-a1d7-c8efffd01158%29/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, ISBN 9781483267449 gbook