古德温 - 斯塔顿积分

古德温 - 斯塔顿积分（Template:Lang-en）定义如下^[1]

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

它是下列三阶非线性常微分方程的一个解： $4 w (z) + 8 z \frac{d}{d z} w (z) + (2 + 2 z^{2}) \frac{d^{2}}{d z^{2}} w (z) + z \frac{d^{3}}{d z^{3}} w (z) = 0$

对称关系

$G (- z) = G (z)$

$G (z) = \frac{1}{2} \frac{G_{2, 3}^{3, 2} (z^{2} |_{1 / 2, 0, 0}^{0, 1 / 2})}{π}$ MeijerG 函数

$G (z) = e^{- z^{2}} + U (1, 1, - z^{2}) e^{z^{2}} e^{- z^{2}} + \frac{2 i e^{- z^{2}} z M (1 / 2, 3 / 2, z^{2})}{\sqrt{π}}$
$G (z) = e^{- z^{2}} + 𝐸 𝑖 (1, - z^{2}) e^{- z^{2}} + \frac{2 i e^{- z^{2}} z 𝐻 𝑒 𝑢 𝑛 𝐵 (1, 0, 1, 0, \sqrt{z^{2}})}{\sqrt{π}}$
$G (z) = e^{- z^{2}} + 𝐸 𝑖 (1, - z^{2}) e^{- z^{2}} + \frac{z e^{- z^{2}} (- i 𝑒 𝑟 𝑓 𝑐 (\sqrt{- z^{2}}) + i)}{\sqrt{- z^{2}}}$

$G (z) = e^{- z^{2}} + 𝐸 𝑖 (1, - z^{2}) e^{- z^{2}} + i e^{- z^{2}} \sqrt{π} z 𝐿 𝑎 𝑔 𝑢 𝑒 𝑟 𝑟 𝑒 𝐿 (- 1 / 2, 1 / 2, z^{2})$

$G (z) = 10 z^{- 1} - 50 z^{- 2} - \frac{1000}{3} \frac{z^{2} - 1}{z^{3}} + 2500 \frac{z^{2} - 1}{z^{4}} + 10000 \frac{2 - 2 z^{2} + z^{4}}{z^{5}} - \frac{250000}{3} \frac{2 - 2 z^{2} + z^{4}}{z^{6}} - \frac{5000000}{21} \frac{- 6 + 6 z^{2} - 3 z^{4} + z^{6}}{z^{7}} + \frac{6250000}{3} \frac{- 6 + 6 z^{2} - 3 z^{4} + z^{6}}{z^{8}} + \frac{125000000}{27} \frac{24 - 24 z^{2} + 12 z^{4} - 4 z^{6} + z^{8}}{z^{9}} - \frac{125000000}{3} \frac{24 - 24 z^{2} + 12 z^{4} - 4 z^{6} + z^{8}}{z^{10}}$
$G (z) = (1 - γ - \ln (z^{2}) - i 𝑐 𝑠 𝑔 𝑛 (i z^{2}) π + \frac{2 i}{\sqrt{π}} z + (- 2 + γ + \ln (z^{2}) + i 𝑐 𝑠 𝑔 𝑛 (i z^{2}) π) z^{2} + \frac{- 4 / 3 i}{\sqrt{π}} z^{3} + (\frac{5}{4} - 1 / 2 γ - 1 / 2 \ln (z^{2}) - 1 / 2 i 𝑐 𝑠 𝑔 𝑛 (i z^{2}) π) z^{4} + O (z^{5}))$

↑ Frank Oliver, NIST Handbook of Mathematical Functions, p160,Cambridge University Press 2010Template:En