Let $F(z)$ be defined by

$$F(z)=\int_{0}^{\infty} \frac{e^{-z t}}{1+t^{2}} d t, \quad|\arg z|<\frac{\pi}{2}$$

Let $\tilde{F}(z)$ be defined by

$$\tilde{F}(z)=\mathcal{P} \int_{0}^{\infty e^{-\frac{i \pi}{2}}} \frac{e^{-z \zeta}}{1+\zeta^{2}} d \zeta, \quad 0<\arg z<\pi$$

where $\mathcal{P}$ denotes principal value integral and the contour is the negative imaginary axis.

By computing $F(z)-\tilde{F}(z)$, obtain a formula for the analytic continuation of $F(z)$ for $\frac{\pi}{2} \leqslant \arg z<\pi$.