Complex exponential

For \(z=x+iy\) we define \(e^{z}=e^{x}(\cos y+i\sin y)\). When \(y=0\) we recover the real exponential, and when \(x=0\) we get Euler's formula \(e^{iy}=\cos y+i\sin y\), which gives for instance \(e^{i\pi}=-1\). With this notation the polar form \(r(\cos\theta+i\sin\theta)\) is written compactly as \(re^{i\theta}\).

The basic property is \(e^{z+w}=e^{z}e^{w}\), which amounts to multiplying moduli and adding arguments, and it implies \(e^{-z}=1/e^{z}\). Also \(|e^{z}|=e^{x}>0\), so the exponential is never zero; on the other hand, adding \(2\pi\) to \(y\) does not change the angle, which is why \(e^{z+2\pi i}=e^{z}\).

This periodicity is the big novelty compared with the real case: \(e^{z}\) is not injective, and \(e^{z}=e^{w}\) holds exactly when \(z-w=2\pi i k\) with \(k\) an integer. Geometrically, the vertical line \(x=c\) is sent to the circle of radius \(e^{c}\), traversed once every \(2\pi\), and a horizontal strip of height \(2\pi\) covers all of \(\mathbb{C}\setminus\{0\}\).

It is continuous on all of \(\mathbb{C}\) (since \(e^{x}\), \(\cos y\) and \(\sin y\) are). A common mistake is to assume \(e^{z}=1\) only for \(z=0\); in fact it holds at every \(2\pi i k\). This loss of injectivity is exactly what forces the logarithm to be defined as a multivalued function.