Making sense of the complex exponential function I read in a book that if we want to make sense of the $e^z$ where $z=x+iy$, we already know how to interpret $e^x $, so the only thing we have to make sense of is $e^{iy}$. For $e^{iy}$ to make sense it has to obey the calculus rule. So $e^{iy}$ should be defined as the solution to the initial value problem: $$\frac{d}{dx}e^{iy}=ze^{zt}; e^{z*0}=1$$
I don't understand why for the $e^{iy}$ to make sense should be defined as the solution to the initial value problem? 
 A: The "should" is indeed problematic here.
So you (the author/reader of that text) want a nice complex function, which means holomorphic/analytical, that on the real axis is equal to the real exponential function. But that function already has a nice and nicely convergent power series expansion. By the series identification problem the complex function can not help but to have the same expansion (that is, the complex power series expansion with the same coefficients).
Now you want perhaps to avoid to invoke power series and their properties. Now what uniquely identifying properties of the real exponential function are there that could be seamlessly expanded to the complex domain?


*

*The classical definition $e^x=\lim_{n\to\infty}\left(1+\frac xn\right)^n$. One would need to prove convergence over the convergence of the single terms. The order structure of the reals can obviously not be used.

*Cauchy's functional equation $f(x+y)=f(x)f(y)$ with $f$ differentiable in $x=0$ and $f'(0)=1$. The solution is obtained by identifying $f(0)=1$ and $f(x)=f(\frac xn)^n=\left(1+\frac xn+o(\frac xn)\right)^n$ leads then back to the first variant.

*Taking the derivative in $y=0$ gives the differential equation $f'(x)=f(x)$, $f(0)=f'(0)=1$. Defining what a solution over the complex plane is leads back to Euler steps and again to the first definition.
These are all connected, their solution invariably leads back to the binomial power and the exponential series. So the latter two variants are just more compressed definitions, there is no inherent necessity in them.
