A nonsingular affine conic is a punctured-plane Riemann surface

(The conic is nonsingular.) The curve is cut out by the single polynomial $f(x,y)=x^2+y^2-1$ with gradient $\mathrm{\nabla }f=(2x,2y)$; if $\mathrm{\nabla }f$ vanished at a point of $C_0$ then $x=y=0$ and $x^2+y^2=0\mathrm{\ne }1$, so $\mathrm{\nabla }f$ is nowhere zero on $C_0$ and the Jacobian-rank condition of [F1] holds at every point.

(The two factors are reciprocal.) For $(x,y)\in C_0$ one has $(x+iy)(x-iy)=x^2+y^2=1$, so $u=x+iy$ satisfies $u\mathrm{\ne }0$ and $x-iy=u^{-1}$; adding and subtracting the two equations gives $x=(u+u^{-1})\mathrm{/}2$ and $y=(u-u^{-1})\mathrm{/}(2i)$.

($u$ is a bijection onto the punctured plane.) If $u(x,y)=u(x^{\mathrm{\prime }},y^{\mathrm{\prime }})$ then $x+iy=x^{\mathrm{\prime }}+i{y}^{\mathrm{\prime }}$ and, by step 1.2, also $x-iy=u^{-1}=x^{\mathrm{\prime }}-i{y}^{\mathrm{\prime }}$, so $(x,y)=(x^{\mathrm{\prime }},y^{\mathrm{\prime }})$ and $u$ is injective; conversely, for $u\in {\mathbb{C}}^{\times }$ the point $v(u)=((u+u^{-1})\mathrm{/}2,(u-u^{-1})\mathrm{/}(2i))$ satisfies $v(u)\in C_0$ because its coordinates give $x+iy=u$ and $x-iy=u^{-1}$, whose product is $1$, and $u(v(u))=u$, so $u$ is surjective with the displayed inverse.

($u$ and its inverse are continuous.) The map $u$ is the restriction of the polynomial map $(x,y)\mapsto x+iy$, which is complex differentiable and hence continuous by [F6]; the inverse $v$ has components that are rational functions of $u$ and $u^{-1}$, complex differentiable on ${\mathbb{C}}^{\times }$ by [F6] and so continuous; hence $u:C_0\to {\mathbb{C}}^{\times }$ is a continuous bijection with continuous inverse, that is, a homeomorphism.

(Transport of the complex structure.) Give $C_0$ the one-chart atlas $\{u\}$, the chart $u$ being the homeomorphism of step 3.1 onto the plane domain ${\mathbb{C}}^{\times }$; its only transition with itself is the identity, which is holomorphic, and $C_0$ is nonempty, connected as a continuous image of the connected ${\mathbb{C}}^{\times }$ under $v$ by [F5], Hausdorff and second countable by [F4], so $C_0$ is a Riemann surface by [F3]; with these charts the chart expression of $u$ is the identity and the chart expression of its inverse $v$ is also the identity, so $u$ is a biholomorphism $C_0\to {\mathbb{C}}^{\times }$ by [F3].

(Conclusion.) The curve $x^2+y^2=1$ is nonsingular at every point by step 1.1, and step 4.1 exhibits it as a Riemann surface biholomorphic to ${\mathbb{C}}^{\times }$ through $u=x+iy$, with inverse $x=(u+u^{-1})\mathrm{/}2$, $y=(u-u^{-1})\mathrm{/}(2i)$, exactly as claimed.