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

[Gil]

Now Wk(inf +ib)

W-3(INF +i20) = inf-i(6pi)
W-3(INF - i20) = inf-i(6pi)

W-3(-INF - i20) = inf-i(5pi)
W-3(-INF + i20) = inf-i(7pi)
W-3(-INF - i0) = inf-i(5pi)

W-2(INF +i20) = inf-i(4pi)
W-2(INF - i20) = inf-i(4pi)

W-1(INF +i20) = inf-i(2pi)
W-1(INF - i20) = inf-i(2pi)

W0(INF +i20) = inf-i(0pi)
W0(INF - i20) = inf-i(0pi)

W1(INF +i20) = inf+i(2pi)
W1(INF - i20) = inf+i(2pi)

W2(INF +i20) = inf+i(4pi)
W2(INF - i20) = inf+i(4pi)

W2(INF +i0) = inf+i(4pi)

Conclusion & summary for Wk(inf+ib)
Wk(INF +ib) = inf+2ik


z=a+ib, a=±infinity, b≠±infinity
Wk(±infinity+ib) =+inf +
i × (2k, if a>0; (else 2k+1, if b>=0; 2k-1, else))