Say that I have an entire function $f$ on $\mathbb{C}_\infty$. I was thinking the following. If say the integral $$\int_{-\infty}^{+\infty} f dz=k$$ for some number $k$, does this mean that all integrals through infinity exist? I thought this is obvious by Cauchy's theorem applied on the Riemann sphere, however, I am not quite sure. The issue is that if I would go by this analogy, since $f$ is entire, I should have $\int f =0$ because of analiticty of the "interior" (?). I am not sure quite sure what is going on. This is motivated from thinking about asymptotic expansions through contour integrals and how, at least my course, assumes the above fact.
Question: If an entire function $f$ on $\mathbb{C}_\infty$has $$\int_{-\infty}^{+\infty} f dz=k$$ for some number $k$, what can we say about all integrals along paths going through infinity (as viwed on the riemann sphere)?
No. $\int_{-\infty}^{+\infty} e^{-z^{2}} dz$ exists but the integral over the imaginary axis of this function does not.