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I posed this problem to my calculus teacher a few days ago and we have not been able to come close to solving it thus far. The problem is to find the intersection of the solids (complex functions require use of the 4d space, so I assume that the function would be a solid) f(z)=e^z and g(z)=z (where z is a complex number).

What I managed to show was that this system is equivalent to the complex intersection between ln(z) and e^z. This is quite simple:

e^z=z

ln(e^z)=ln(z)

z=ln(z)

by basic substitution, a new system with equivalent solutions is born: e^z=ln(z).

Furthermore, raising both sides of this equation to a power of e results in z=e^(e^z). Thus, this new function is equivalent to ln(z), and one can continue this process infinitely by replacing every z with e^z.

Any ideas? Maybe there is a theorem regarding the intersections of inverse equations