Goodwin–Staton integral

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

G ( z ) = ∫ 0 ∞ e − t 2 t + z d t {\displaystyle G(z)=\int _{0}^{\infty }{\frac {e^{-t^{2}}}{t+z}}\,dt}

It satisfies the following third-order nonlinear differential equation：

4 w ( z ) + 8 z d d z w ( z ) + ( 2 + 2 z 2 ) d 2 d z 2 w ( z ) + z d 3 d z 3 w ( z ) = 0 {\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}

Properties

Symmetry:

G ( − z ) = − G ( z ) {\displaystyle G(-z)=-G(z)}

Expansion for small z:

G ( z ) = 1 − γ − ln ⁡ ( z 2 ) − i csgn ⁡ ( i z 2 ) π + 2 i π z + ( − 2 + γ + ln ⁡ ( z 2 ) + i csgn ⁡ ( i z 2 ) π ) z 2 − 4 i 3 π z 3 + ( 5 4 − 1 2 γ − 1 2 ln ⁡ ( z 2 ) − 1 2 i csgn ⁡ ( i z 2 ) π ) z 4 + O ( z 5 ) {\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}}}

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
  • 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