Goodwin–Staton integral

In mathematics the Goodwin–Staton integral is defined as :

[math]\displaystyle{ G(z)=\int_0^\infty \frac {e^{-t^2}}{t+z} \, dt }[/math]

It satisfies the following third-order nonlinear differential equation：

[math]\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 }[/math]

Properties

Symmetry:

[math]\displaystyle{ G(-z)=-G(z) }[/math]

Expansion for small z:

[math]\displaystyle{ \begin{align} 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{align} }[/math]

References

• http://journals.cambridge.org/article_S0013091504001087
• Mamedov, B.A. (2007). "Evaluation of the generalized Goodwin–Staton integral using binomial expansion theorem". Journal of Quantitative Spectroscopy and Radiative Transfer 105: 8–11. doi:10.1016/j.jqsrt.2006.09.018. 
• 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, ISBN:9781483267449 gbook