习题 19, § 1

One knows (A, IX, §10, exerc. 2) that in the complex plane $\mathbf{C}^2$ the group $A$ of angles of directed lines is isomorphic to the orthogonal group $\mathbf{O}_2(\mathbf{C})$, the canonical isomorphism associating the angle $\theta$ with rotation through the angle $\theta$; one transports the topology of $\mathbf{O}_2(\mathbf{C})$ (considered as a subspace of the space $\mathbf{M}_2(\mathbf{C})$ of matrices of order 2 over $\mathbf{C}$) to the group $A$ by this isomorphism, so making $A$ a locally compact topological group;

此外，映射$\theta \mapsto \cos \theta + i \sin \theta$是群A在复数$\neq 0$的乘法群$\mathbf{C}^*$上的*同构的*，逆同构由公式$\cos \theta = \frac{1}{2}(z + 1/z)$定义， $\sin \theta = \frac{1}{2i}(z - 1/z)$。从这些关系推导出加法群 $\mathbf{C}$ 的每个连续同态 $z \mapsto \varphi(z)$ 在 A 上，使得复数函数 $\cos \varphi(z)$, $\sin \varphi(z)$) 在 $\mathbf{C}$ 上可微，由关系定义$\cos \varphi(z) = \cos az$、$\sin \varphi(z) = \sin az$（$a$复数）。