\(\therefore t\text{\ erfc\ }\left(\frac{G}{\sqrt{t}}\right)+\frac{2\ G}{\sqrt{\pi}}\int\frac{e^{-G^{2}\text{\ ξ}^{2}}}{\xi^{2}}d\xi=t\text{\ erfc\ }\left(\frac{G}{\sqrt{t}}\right)+\frac{2\ G}{\sqrt{\pi}}\left[-\frac{e^{-G^{2}\text{\ ξ}^{2}}}{\xi}-\sqrt{G^{2}\text{\ π}}\operatorname{erf\ }\left(\text{G\ ξ}\right)\right]\) |
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\(=t\text{\ erfc\ }\left(\frac{G}{\sqrt{t}}\right)-\frac{2\ G\ \sqrt{t}\ e^{\frac{-G^{2}}{t}}}{\sqrt{\pi}}-2\ G^{2}\operatorname{erf\ }\left(\frac{G}{\sqrt{t}}\right)\) |
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\(=t\text{\ erfc\ }\left(\frac{G}{\sqrt{t}}\right)-\frac{2\ G\ \sqrt{t}\ e^{\frac{-G^{2}}{t}}}{\sqrt{\pi}}-2\ G^{2}\left[\operatorname{1-erfc}\left(\frac{G}{\sqrt{t}}\right)\right]\) |
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\(=t\text{\ erfc\ }\left(\frac{G}{\sqrt{t}}\right)-\frac{2\ G\ \sqrt{t}\ e^{-\frac{G^{2}}{t}}}{\sqrt{\pi}}\operatorname{-2+2\ erfc\ }\left(\frac{G}{\sqrt{t}}\right)\) |
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\(=2\ G^{2}\text{\ erfc\ }\left(\frac{G}{\sqrt{t}}\right)+t\text{\ erfc\ }\left(\frac{G}{\sqrt{t}}\right)-\frac{2\ G\ \sqrt{t}\ e^{\frac{-G^{2}}{t}}}{\sqrt{\pi}}-2\ G^{2}\) |
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\(=\left(2\ G^{2}+t\right)\text{\ erfc\ }\left(\frac{G}{\sqrt{t}}\right)-\frac{2\ G\ \sqrt{t}\ e^{\frac{-G^{2}}{t}}}{\sqrt{\pi}}-2\ G^{2}\) |
(13) |