In this blackpenredpen video for $i!$ he obtains
$$i! = \int_0^\infty e^{-t} \cos(\ln t) \rm{dt} + i\int_0^\infty e^{-t} sin(\ln t) \rm{dt}$$
At 4:10 claims that these converge at $t=0$ but then he sketches a graph of $e^{-t} \cos(\ln t)$ which looks like it does not converge at $0$.
Can you give a better explanation please?