e^x=x revisted

[bishnu]

The recent thread e^x=x has gotten me intressted in what it roots are. Through my look at it i think there is a infintude of roots...let me explain

 

one root is very easy to get take the root of x on both sides and you get e=x^1/x which you can easily solve with newtons method setting x0->i

 

but on further anaylsis e^(a+bi)=a+bi

a+bi=ln(a+bi) take the polar a+bi=ln((a^2+b^2)^(1/2)cis(arctan(b/a))

set it to exponetial a+bi=ln((a^2+b^2)^(1/2)e^(arctan(b/a)*i)) reduce down and you get a+bi=ln(a^2+b^2)/2+arctan(b/a)*i now because of the periodicness of arctan there should be an infinte # of solutions i belive but i dont know how to solve for them....wait i just relized that there might not be infinity but possibly more than one...does anyone know how to solve.