Continued fraction for 1/sqrt(pi) integral from -infinity to infinity of e^(-t^2)/(z-t) dt

Theorem

The following formula holds: $$\dfrac{1}{\sqrt{\pi}} \displaystyle\int_{-\infty}^{\infty} \dfrac{e^{-t^2}}{z-t} \mathrm{d}t = \dfrac{1}{z-\dfrac{\frac{1}{2}}{z-\dfrac{1}{z-\dfrac{\frac{3}{2}}{z-\dfrac{2}{z-\ldots}}}}}=\dfrac{1}{\sqrt{\pi}} \displaystyle\lim_{n \rightarrow \infty} \displaystyle\sum_{k=1}^n \dfrac{H_k^{(n)}}{z-x_k^{(n)}},$$ where $\pi$ denotes pi, $e^{-t^2}$ denotes the exponential function, $x_k^{(n)}$ denotes the zeros of the Hermite polynomials, and $H_k^{(n)}$ denote the weight function for the Hermite polynomials.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 7.1.15