Interesting value for W, h involving phi,Omega?

[andydude]

W(-pi-I*2*ln2) = ln2 - I*(pi/2)

True (numerically verified).

W(-3pi/2-I*3*ln3) = ln3 - I*(pi/2)

True (numerically verified).

W(-2pi-I*4*ln4) = ln4 + I*(pi/2)

True (numerically verified).

H((I/2)^(I/2)^(-1)) = I/2

True (numerically verified), plus this is practically the definition of H.

H((I/3)^(I/3)^(-1)) = I/3

(not numerically verified), this is practically the definition of H, so it should be true, but for some reason my CAS gives a different value. These are the values I'm getting:
\(
\begin{tabular}{rl}
H((i/3)^{1/(i/3)}) \approx -0.008442966 & +\ 0.343894471 i \\
i/3 = 0 & +\ 0.333333333 i
\end{tabular}
\)
although, this is bothering me, since it is the definition of H that this should be true. I'm guessing that that large value \( |(i/3)^{(3/i)}|\approx 111.318 \) might be why. Maybe my CAS never tested the LambertW for this large a value, and maybe it really does give the wrong value...

About your claim for all n, since you have found 3 instances of it being true, it is probably true, but that does not constitute a proof. Since your logic doesn't require n to be constant, you could probably re-work the math to start with n rather than 2 or 3. Then it would be a proof.

Andrew Robbins