\(\therefore d\xi={-\frac{1}{2}t}^{-3/2}\text{\ dt}\) |
(8) |
\(\therefore dt={-2\ t}^{3/2}\ d\xi=-2\ \xi^{-3}\text{\ dξ}\) |
(9) |
\(\therefore\int{\text{erfc}\left(\frac{G}{\sqrt{t}}\right)\text{\ dt}}=-2\int{\text{erfc}\left(\text{G\ ξ}\right)\ \xi^{-3}\text{\ dξ}}\) |
(10) |
\(\therefore-2\int{\text{erfc}\left(\text{G\ ξ}\right)\ \xi^{-3}\text{\ dξ}}=-2\left[-\frac{\text{erfc\ }\left(\text{G\ ξ}\right)}{\left(3-1\right)\ \xi^{n-1}}-\frac{2\ G}{\left(3-1\right)\ \sqrt{\pi}}\int\frac{e^{-G^{2}\text{\ ξ}^{2}}}{\xi^{3-1}}\text{dξ}\right]\) |
|
\(=-2\left[-\frac{\text{erfc\ }\left(\text{G\ ξ}\right)}{2\ \xi^{2}}-\frac{2\ G}{2\ \sqrt{\pi}}\int\frac{e^{-G^{2}\text{\ ξ}^{2}}}{\xi^{2}}\text{dξ}\right]\) |
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\(=t\text{\ erfc\ }\left(\frac{G}{\sqrt{t}}\right)+\frac{2\ G}{\sqrt{\pi}}\int\frac{e^{-G^{2}\text{\ ξ}^{2}}}{\xi^{2}}\text{dξ}\) |
(11) |