Intresting functions not ?

[Gottfried]

Not sure how to define " intresting " though.

...

exp(a+bi) = a+bi
tommy1729

I don't know what makes a function interesting either or to what extent any of these functions have been studied. However, I found the answer to exp(a+bi) = a+bi, albeit not properly at all since I used a calculator to graph a certain part ((b/sin(b))-exp(b/tan(b)) = 0) to solve for the b portion. I feel like that one must have been done before since a calculator can be used to find a and b. I could be very wrong on this.
Since there is an answer in the set of complex numbers it doesn't seem as interesting to me as it would if another set had to be used.

Surely this has been done may times; I think I've seen this for instance in Corless et al. profound article on the LambertW-function; but also D. Shell / (?) Thron (after who the region of convergence of the infinite powertower in the comple numbers has been named) should have needed the solution for this. For me it was in the beginning of my encounter with tetration interesting to find complex fixpoints for real bases (of a continuous interval), so I made this picture (it seems to be a bit overcomplicated, but well...)
http://go.helms-net.de/math/tetdocs/real...uction.png
and
http://go.helms-net.de/math/tetdocs/real...tion_2.png


I don't understand your last remark: "if another set would be used" - what do you mean?

Gottfried