Goodwin–Staton integral explained

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

G(z) = \int ^{inf ty} _{0} (e ^{- t ^{2}})/(t + z) dt

It satisfies the following third-order nonlinear differential equation：

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

Properties

Symmetry:

G( - z) =- G(z)

Expansion for small z:

\begin{align} G(z) = {} & 1- \gamma -ln (z ^{2)} - i\operatorname{csgn} ( iz ^{2)} \pi + (2 i)/(\sqrt \pi) z \\[ 5 pt] & {} + ( -2 + \gamma + ln (z ^{2)} + i \operatorname{csgn} (iz ^{2)} \pi ) z ^{2} - (4 i)/(3 \sqrt\pi) z ^{3} \\[ 5 pt] & {} + \left( (5)/(4) - (1)/(2) \gamma - (1)/(2) ln (z ^{2)} - (1)/(2) i \operatorname{csgn} ( iz ^{2)} \pi \right) z ^{4} + O (z ^{5) \end{align}}

References

1. [Frank William John Olver]

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