= Reciprocal chart at infinity
{title2=$w=1/z,\quad w(\infty)=0$}
The reciprocal chart makes infinity an ordinary coordinate origin. Its domain is the <Riemann sphere> minus the point $z=0$; $w(\infty)=0$ and $w(z)=1/z$ elsewhere. A <neighbourhood> $U_R$ of infinity becomes the <open disc> $|w|<1/R$. In sphere coordinates this is exactly the cap
$$
Z>\frac{R^2-1}{R^2+1}.
$$
For $R=2$, the exterior $|z|>2$ together with infinity becomes $Z>3/5$ and $|w|<1/2$. The boundary itself is excluded from each <open set>.
\Image[/riemann-sphere-reciprocal-cap.png]
{title=The same neighbourhood of infinity as a plane exterior, a north-pole cap, and a disc in the reciprocal chart}
The point $z=2+i$ lies in this <neighbourhood> because $|z|=\sqrt5>2$. Its reciprocal coordinate is $w=(2-i)/5$, with $|w|=1/\sqrt5<1/2$, and its sphere height is $Z=2/3>3/5$. The negative <imaginary part> of $w$ is the axis reversal required by the <holomorphic stereographic atlas of the sphere>.
For a complex-valued <holomorphic function> $f$ defined outside a large disc, having a <removable singularity at infinity> means that $g(w)=f(1/w)$ extends to a <holomorphic function> at $w=0$. This definition reduces questions at infinity to familiar questions at a finite point.<Complex Analysis Lecture Notes by Dan Romik>[Complex Analysis Lecture Notes, section 9, items 81–82, printed p. 32, reciprocal definition at infinity]{ref} For example,
$$
f(z)=\frac{z-2}{z+3},\qquad g(w)=\frac{1-2w}{1+3w}=1-5w+15w^2-45w^3+\cdots,\qquad |w|<1/3.
$$
The <geometric series> gives a <Taylor series> at the new origin, so $f(\infty)=1$ and $g'(0)=-5$. The coefficient $-5$ is the <derivative> in the specified $w$ coordinate. In contrast, $f(z)=z^2$ becomes $g(w)=w^{-2}$, displaying a <pole> of order two at infinity when the target is the <complex plane>. As a <holomorphic map> to the <Riemann sphere>, it uses the target reciprocal coordinate $1/f(1/w)=w^2$, which is regular at zero.
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