Exponentiation

Exponential Group Homomorphism - The \( \pi \)-Tuning

The morphism of groups:

\[\exp : \mathbf{C} \rightarrow \mathbf{C}^{\mathbf{x}}, \quad \boldsymbol{z} \mapsto \exp z \]

is a homomorphism of the additive group \( \mathrm{C} \) into the multiplicative group \( \mathrm{C}^{\times}. \) As in the case of a general group homomorphism \( \sigma: G \rightarrow H \) we should consider the image group \( \sigma(G):=\exp (\mathrm{C}), \) and the kernel:

\[\text { Ker } \sigma:=\{g \in G: \sigma(g)=\text { neutral element of } H\}\]

For the exponential group homomorphism we obtain:

\[\exp (\mathrm{C})=\mathrm{C}^{\times}, \quad \operatorname{Ker}(\mathrm{exp})=2 \pi i \mathbb{Z}.\]

We conclude that there is a uniquely defined real number \( \pi>0 \), such that the numbers \( 2 n \pi i, n \in \mathbb{Z}, \) constitute the set of numbers mapped on to 1 by the exponential mapping exp \( z \); Equivalently there is a unique tuning real number number \( \pi \) with the property that:

\[\{w \in \mathbf{C}: \exp w=1\}=2 \pi i \mathbb{Z}.\]

The above property characterizes \( \pi \) uniquely, thus it amounts to the definition of \( \pi \).