HP49-50G,VER16.2 Lambert Funktion Wk(x), k=k= 0, ±1, ±2, ±3, ±4..., x real or complex

[Gil]

New version 16.2 of W(x).

For x real between -1/e and 0, two results: one for the upper branch, the default branch W0(x); and the other, for the lower branch branch, W-1.

For W-1, x—>0 is a special case: if x—>0- (x "slightly negative") —> W-1 = -infinity; and if x—>0+ (x "slightly positive"), W-1—> "- infinity - iPI“, but simplified in Wolfram to -infinity.

To get the answer for W (k=100, z=i×infinity), procede as follows:
1) 100 ENTER
2) 0 (for the real part)
3) LS +" number zero", ie white arrow + "key right to On-key" (to get the infinity character or corresponding the infinity number 9.99999999999E499)
4) —>NUM (to convert infinity character to the real number 9.99999999999E499)
5) command R—>C (to convert the two numbers in the stack, 0 and 9.99999999999E499, into a complex (0, 9.99999999999E499))
6) 2 —>LIST (to get a list with two objects: 100, =k the branch number, as first object of the list ; and (0, 9.99999999999E499), the complex z, as second object of the list)
7) Having the appropriate list {100 (0, 9.99999999999E499)} in stack level 1, launch the program.

You should get, as a result, "infinity + i × 401×pi/2“.

Approximate check with Wolfram
LambertW (100, i*1E500000)
gives1.1512785901131729534744838666712253701249035476598766913241 × 10^6 +629.88877992373306983756183247568673861816336497433660193071 i

And 629.8887799/(pi/2)
gives 400.99965169 —> 401

Note that, if you enter in Wolfram
LambertW (100, i*infinity),
you get infinity, and not "infinity + i × 401×pi/2“.

For more details about the LambertW function and its subtilities, look at the recent special thread in this forum on LambertW function by Albert Chan.