a curious limit

[bo198214]

let's take the limit from positive (keep it simple first)

so:
\( \lim_{h\to\0^+}\, (1-e^{vi})ln(h)\, =\, f(v) \)

Is there any way of re-expressing this limit?

But then its not difficult, since \( \ln(h)\to -\infty \) on the reals, the whole limit goes to (complex) \( \infty \) except for \( 1-e^{vi}=0 \), for which the whole limit is 0.