I experienced a bit with the problem of finding x in equations like \( y = \;^2 x \) and \( y = \;^3 x \) and \( y = \;^n x \) where y is given.
This was mainly motivated by frequent questions in MSE and/or MO for solutions where \( y=- 1 \) or \( y= i \)
What I mainly found was the requirement for a generalization of the Lambert-W (but a nicely straightforward one!) and some insight into the occuring power series.
Although having now an accessible entry-point into the general problem, I did not yet find explicite, simple closed form expressions for the occuring coefficients except when \( n=2 \) (but those are already well known...), so it's an open field for pattern-detection and research on radii of convergence.
It is too much to write it here in this limited box, so I made a pdf-file. I upload it as attachment but put it also on my webspace, see http://go.helms-net.de/math/tetdocs/Wexz...erroot.pdf
Collaboration is appreciated...
Gottfried
This was mainly motivated by frequent questions in MSE and/or MO for solutions where \( y=- 1 \) or \( y= i \)
What I mainly found was the requirement for a generalization of the Lambert-W (but a nicely straightforward one!) and some insight into the occuring power series.
Although having now an accessible entry-point into the general problem, I did not yet find explicite, simple closed form expressions for the occuring coefficients except when \( n=2 \) (but those are already well known...), so it's an open field for pattern-detection and research on radii of convergence.
It is too much to write it here in this limited box, so I made a pdf-file. I upload it as attachment but put it also on my webspace, see http://go.helms-net.de/math/tetdocs/Wexz...erroot.pdf
Collaboration is appreciated...
Gottfried
Gottfried Helms, Kassel