We define $f(z)=e^z-3$ for all complex numbers $z$.
I am trying to represent the modulus $|e^z-3|$ in terms of the real part of $e^z-3$? Then maybe I can tell whether $|f^n(z)|\to\infty$ if and only if the real part of $f^n(z)$ goes to infinity. Is that true? The $f^n$ means the function $f$ composed with itself $n$ times.