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Imaginary zeros of f(z)= z^(1/z) (real valued solutions f(z)>e^(1/e))
It is well known that the (lower real) solution of is
where is the Lambert W function which is the solution of where .
Short derivation:

Now the series development of the Lambert W function is well known to be:

with your formula is confirmed Wink

However the radius of convergence of the series is and hence the radius of convergence of is but Gottfried was just interested in a solution for .

The radius of convergence is limited because the LambertW function has a singularity at . It can however analytically continued past that singularity to values below which are then complex.

Such an analytic continuation is somewhat cumbersome.
You start with the development point .
Then you compute the development at a point with then you compute the development at a point with and so on until you can reach an .
See the attachment for a depiction.

A development at point turns into the development at point via

For our task it should suffice to have 4 steps to reach a development at .

Taking 3 steps (each 60 degrees) would result in which is not allowed.

So the four step solution would result in

But nobody wants to use this *lol*

Attached Files
.pdf   analytic continuation.pdf (Size: 23.44 KB / Downloads: 416)

Messages In This Thread
RE: Tetration below 1 - by Gottfried - 09/09/2007, 07:04 AM
RE: The Complex Lambert-W - by Gottfried - 09/09/2007, 04:54 PM
RE: The Complex Lambert-W - by andydude - 09/10/2007, 06:58 AM

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