$∫_0^xe^{x^2-t^2}$渐近式

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WWch8 page 152, 证明$$\int_{0}^{x} e^{x ^2-t^2} d t \sim \pm \frac{1}{2} e^{x^2} \sqrt{\pi}-\left(\frac{1}{2 x}-\frac{1}{2^{2} x^{3}}+\frac{1·3}{2^{3} x^5}-\frac{1·3·5}{2^{4} x^{7}}+\ldots\right)$$