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

[Gil]

Input {k X},
k for the branch
& X, real or a complex number,

with real part of X = ± inf
and/or imaginary part of X = ±inf.

Here is a program or rather a subroutine summarising the above discussed cases, with output in form of a string like "oo ± iPI#",
with oo standing for "limit tending to +inf"
& with iPI together, to remember the Euler's formula when EXP(iPI#):

Taking the real part of the given result, we get
"something—> inf" × EXP( "something—>inf")
And that latter expression in bold is "something—> inf".

This last limit is still to be multiplied by a "fixed number", ie, EXP(iPI#), here always a real number of the form (±1+i0) or a complex number of the form (0±i).

Consequently, the final result of the above mentioned multiplication will always be either
+inf, or -inf, or 0+i×inf, or 0-i×inf.

In other words...
Having for instance x=a+i×inf, a≠0 & a≠±inf
W0(x) = (inf + iPI/2)
But (that last result=inf + iPI/2) × EXP(that last result = inf + iPI/2) = "something tending to inf" × (0+i×inf) =(0+i×inf),
and never the initial value of x=a+i×inf.

Code:

\<< OBJ\-> DROP \-> k x
  \<< x RE ABS \->NUM \oo \->NUM == x IM ABS \->NUM \oo \->NUM == OR
    IF
    THEN "\oo "
      CASE x RE \->NUM \oo \->NUM ==
        THEN 2 k * DUP 0 \>=
          IF
          THEN "+ i\pi\183"
          ELSE "- i\pi\183"
          END SWAP ABS R\->I
        END x RE \->NUM \oo \->NUM NEG ==
        THEN 2 k * 1 x IM \->NUM 0 >
          IF
          THEN +
          ELSE -
          END DUP 0 \>=
          IF
          THEN "+ i\pi\183"
          ELSE "- i\pi\183"
          END SWAP ABS R\->I
        END 2 k * .5 x IM \->NUM 0 >
        IF
        THEN +
        ELSE -
        END DUP 0 \>=
        IF
        THEN "+ i\pi\183"
        ELSE "- i\pi\183"
        END SWAP ABS \->Q \->STR "'" "" SREPL DROP
      END + + "\1831/" "/" SREPL DROP "<U>" + "i" "<U>i" SREPL DROP "<U>" 19 CHR 3 CHR OVER + + SREPL DROP
    ELSE
    END
  \>>
\>>