Does there exist a function such that $\int_{\mathbb{R}_+^{\star} } t^nf(t)dt=0$?
Solution 1:
Start with your example, take the imaginary part, just user every fourth step with variable $t=s^{1/4}$. We get: $$ F(s) = \exp(-s^{1/4})\sin(s^{1/4}), $$ for which $$ \int_0^\infty s^n F(s)\;ds = 0 $$ for all nonnegative integers $n$.