Equivalence of Definitions of Complex Exponential Function/Power Series Expansion equivalent to Definition by Real Functions

Proof
We have the result:
 * Sum of Series equivalent to Solution of Differential Equation

which gives that the definition of $\exp z$ as the sum of the power series


 * $\exp z := \displaystyle \sum_{n \mathop = 0}^\infty \frac {z^n} {n!}$

is equivalent to the definition of $\exp z$ as the solution of the differential equation:


 * $\dfrac {\d y} {\d z} = y$

satisfying the initial condition $y \paren 0 = 1$.

Let:
 * $e: \R \to \R$ denote the real exponential function
 * $\sin: \R \to \R$ denote the real sine function
 * $\cos: \R \to \R$ denote the real cosine function.

Then: