Bounds for Modulus of e^z on Circle x^2 + y^2 - 2x - 2y - 2 = 0

Theorem
Consider the circle $C$ embedded in the complex plane defined by the equation:
 * $x^2 + y^2 - 2 x - 2 y - 2 = 0$

Let $z = x + i y \in \C$ be a point lying on $C$.

Then:
 * $e^{-1} \le \cmod {e^z} \le e^3$

Proof
This defines a circle whose center is at $1 + i$ and whose radius is $2$.

From Modulus of Exponential is Exponential of Real Part:
 * $\cmod {e^z} = e^x$

If $z \in C$ then from the geometry of the circle $C$:
 * $-1 \le x \le 3$

Then from Exponential is Strictly Increasing:
 * $e^{-1} \le e^x \le e^3$

Hence the result.