09/10/2007, 01:33 PM

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:

.

Then

with your formula is confirmed

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*

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:

.

Then

with your formula is confirmed

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*