I was reading a research article where I encountered a complex integral $$\theta(t)=\frac{1}{2\pi i}\int_{-i\infty}^{-i\infty}z^{-1/2}\frac{I_1(z^{1/2})}{I_0(z^{1/2})}e^{zt}dz$$ Here $z$ is a purely imaginary number and $I_1$ and $I_0$ are Modified Bessel functions of first kind of first and zeroth order respectively. It is said that the integral can be evaluated numerically using double exponential formula. I don't fully understand double exponential formula.
Can anyone shed light on how to tackle this integral?