A math professor of mine once said of this (paraphrased): yes it is cool, but really it is just a trivial consequence of how we defined our extension of the rea

A math professor of mine once said of this (paraphrased): yes it is cool, but really it is just a trivial consequence of how we defined our extension of the real exponential map to complex numbers. The really cool part, is that this extension is the _only_ possible holomorphic (differentiable for complex numbers) extension. All that to say that the really nifty part of this identity is the uniqueness of analytic continuations (on the whole complex plane).