Goodwin–Staton integral

From HandWiki

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

[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

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



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