lambertw, all branches

[Gil]

Again, W(k=-1,a=+PI/2 )
seems impossible to give — at least with my HP50G program —the nice answer (0-i×PI),
returning "tiny real - i×PI"

And also W(k=-1, a=-4×PI/2 )
HP output is "tiny - i×2PI"

And also W(k=-2,a=i×6PI/2)
HP output is "tiny - i×3PI"

LambertW (k=-3,0+10*i*pi/2)
HP output is "tiny - i×5PI"

LambertW (k=-1000,i*3998 *pi/2)
Output Wolfram Alpha
i ×
6280.04371452599668368682412317572626551014162935083653612891424=

Output HP:
:W-1000
(
'i*3998*(pi/2)'=(0.,6280.04371455)
)
: (3.6616915941E-12,-6280.04371452)

And LambertW (-5, -20*i*pi/2)
=-i*10*pi/2 =-i*20*pi/2 = a


Pattern
W(k=n, n<0, a=i×(-n×4-2)) = a —> complex with no real part
W(k=n, n>0, a=i×(n×4)) = a —> complex with no real part

All the results are quite OK, though unsatisfactory, as already said.

Rule of thumb:
IF {imaginary (result) = integer≠0 × PI
and abs (real (result) < 1E-10}
THEN result = i × imaginary (result)
END

And more generally
With W0(-pi/2)=0+i×Pi/2
With W-1(-pi/2)=0-i×Pi/2
IF {imaginary (result) = integer≠0 × PI/2
and abs (real (result) < 1E-10}
THEN result = i × imaginary (result)
END

Is that deduction or assumption correct?

Of course, because of the roundings and the fact that we never have the real value of pi, we might never — or rather will never — have a real integer multiple of pi/2.

Final program steps/test
M0=abs(Imaginary part of the final result) /(pi/2)
M1= round (M0, 0) —> to get the nearest integer to M0
M2= real part of the final result
IF {abs (M0-M1)<0.0001 and abs (M2) < 1E-11}
THEN
M0 is considered as a multiple of (pi/2)
& real part of the result, found to be "tiny",
is to be set to exactly = zero:
M2=0
final result = M2+i×(Imaginary part of the final result)
(ELSE
The found result is OK and is not to be "tampered":
final result = final result)
END

Check a=real, multiple of pi/2, with no imaginary part
Tests with Wolfram
Conclusion
W(k, k<0 or k>0, a= real =(-4k-1)×pi/2=-i×a (and zero for the real part)

Overall conclusion
Too many tests about the input on k & a=±i × multiple of pi/2.
Easier to test the final result and correct the real part of that final result (tiny into exactly zero), as described above in Final program steps/test, if necessary or wished, ie if & only if when both conditions abs (M0-M1)<0.0001 and abs (M2) < 1e-11 are fulfilled.