Goodwin–Staton integral

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

G (z) = \int_{0}^{\infty} \frac{e^{- t^{2}}}{t + z} d t

It satisfies the following third-order nonlinear differential equation：

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

Properties

Symmetry:

G (- z) = - G (z)

Expansion for small z:

\begin{aligned} G (z) = & 1 - γ - \ln (z^{2}) - i csgn (i z^{2}) π + \frac{2 i}{\sqrt{π}} z \\ + (- 2 + γ + \ln (z^{2}) + i csgn (i z^{2}) π) z^{2} - \frac{4 i}{3 \sqrt{π}} z^{3} \\ + (\frac{5}{4} - \frac{1}{2} γ - \frac{1}{2} \ln (z^{2}) - \frac{1}{2} i csgn (i z^{2}) π) z^{4} + O (z^{5}) \end{aligned}

↑ 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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