Lambert function and Wolfram or "±infinity+i×K=±infinity" ?

[Gil]

Yes, W0(-inf) = inf+i×pi is the limit that appears when x tends to very large negative real numbers.

But, again,
(inf+i×pi)*EXP(inf+i×pi) (1)
=(inf+i×pi)*EXP(inf) *EXP(i×pi)
=(inf+i×pi)*inf*(cos pi +i× sin pi)
=(inf+i×pi)*inf*(-1)
=(inf+i×inf)*(-1)
=-inf-i*inf ≠ -inf

if I am not mistaken, unless you force and say that first member in (1), inf+i×PI, = inf, with no more imaginary part (but that sounds strange : infinity of bananas [real part of number] + 1 orange×[imaginary part] =? infinity of bananas; on the other hand, infinity of bananas means that there are everywhere bananas in the universe, and consequently no place for a single orange).

And then annoying again, mixing up the definitions.

By the way, what does give your software for W0(infinity +10i). By what I understood, the output should be infinity + 10i, vs inf with Wolfram.