Let $F(z)$ be defined by

$$F(z)=\int_{0}^{\infty} \frac{e^{-2 z t}}{1+t^{3}} d t, \quad|\arg z|<\frac{\pi}{2}$$

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

$$\tilde{F}(z)=\int_{0}^{-i \infty} \frac{e^{-2 z \zeta}}{1+\zeta^{3}} d \zeta, \quad \alpha<\arg z<\beta$$

where the above integral is along the negative imaginary axis of the complex $\zeta$-plane and the real constants $\alpha$ and $\beta$ are to be determined.

Using Cauchy's theorem, or otherwise, compute $F(z)-\tilde{F}(z)$ and hence find a formula for the analytic continuation of $F(z)$ for $\frac{\pi}{2} \leqslant \arg z<\pi$.