Why do we define the complex exponential as we do?

Question

Why do we define the complex exponential as we do? Defining $e^{z}$ as $e^{x}e^{iy}$ certainly seems to make sense, but I'm not sure the formal reason as to why it's defined like this. Was it from looking at Taylor Series, or was it just because we want the nice multiplicative structure? We can show that the real exponential function splits over addition, but the complex one has an addition that is different, so is it simply defined as I stated above, or is there an obvious thing I'm missing?

Answer 1

The short answer is: we don't always.

Many places will define the complex exponential by the power series:

$$\exp(z)=\sum_{n=0}^\infty {z^n\over n!}$$

and justify that by analyticity, this always matches the value on the real axis.

From here one can verify that it satisfies the functional equation

$$\exp(z+w)=\exp z\exp w$$

directly by treating $w$ as fixed, but arbitrary and performing the analysis on the variable $z$.

Certainly the definition you mention makes some sense, as wanting the complex exponential to behave the same way as the real one by demanding the functional equation hold can give rise to the equations

$${\partial\over \partial x}(e^xe^{iy})=e^xe^{iy},\; {\partial\over\partial y}e^xe^{iy}=ie^xe^{iy}$$

From there we use that

$${d\over dz}={1\over 2}\left({\partial\over\partial x}-i{\partial\over\partial y}\right)$$

With the given definition, we get:

$${d\over dz}e^z={1\over 2}\left(e^xe^{iy}-i\left(ie^xe^{iy}\right)\right)=e^xe^{iy}=e^z$$

so that the complex exponential satisfies the appropriate differential equation, one of it's best properties from real variables, so it supports the choice made there.