Addition and duplication for $\mathrm{\wp }$

(The second-derivative identity on $\mathbb{C}\setminus \mathrm{\Lambda }$.) Differentiating [F3] with the rules of [F6] gives $2{\mathrm{\wp }}^{\mathrm{\prime }}{\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}=(12{\mathrm{\wp }}^2-g_2){\mathrm{\wp }}^{\mathrm{\prime }}$ on $\mathbb{C}\setminus \mathrm{\Lambda }$; at every point with ${\mathrm{\wp }}^{\mathrm{\prime }}\mathrm{\ne }0$ this gives ${\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}=6{\mathrm{\wp }}^2-\frac{1}{2}{g}_2$. Let $h\mathrm{\notin }\mathrm{\Lambda }$ with ${\mathrm{\wp }}^{\mathrm{\prime }}(h)=0$: by [F2] the zero of ${\mathrm{\wp }}^{\mathrm{\prime }}$ at $h$ is of order one, so [F5] gives ${\mathrm{\wp }}^{\mathrm{\prime }}(w)=(w-h)g(w)$ with $g(h)\mathrm{\ne }0$, and hence ${\mathrm{\wp }}^{\mathrm{\prime }}(w)\mathrm{\ne }0$ for $0<\mathrm{\mid }w-h\mathrm{\mid }<\rho$ with some $\rho >0$; shrinking $\rho$ if necessary, the pole set of the meromorphic function $\mathrm{\wp }$ is discrete by [F7], so the disc $D(h,\rho )$ contains no lattice point. Both ${\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}$ and $6{\mathrm{\wp }}^2-\frac{1}{2}{g}_2$ are holomorphic on $D(h,\rho )$ and agree on the punctured disc $D(h,\rho )\setminus \{h\}$, which is a nonempty connected open set, so [F5] makes them agree on all of $D(h,\rho )$, in particular at $h$. Every point of $\mathbb{C}\setminus \mathrm{\Lambda }$ either has ${\mathrm{\wp }}^{\mathrm{\prime }}\mathrm{\ne }0$ or is such a zero $h$ by [F2], so ${\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}=6{\mathrm{\wp }}^2-\frac{1}{2}{g}_2$ on all of $\mathbb{C}\setminus \mathrm{\Lambda }$.

(The duplication identity where ${\mathrm{\wp }}^{\mathrm{\prime }}(z)\mathrm{\ne }0$.) Fix $z\in \mathbb{C}\setminus \mathrm{\Lambda }$ with ${\mathrm{\wp }}^{\mathrm{\prime }}(z)\mathrm{\ne }0$; then $2z\mathrm{\notin }\mathrm{\Lambda }$ by [F2], and $\mathrm{\wp }$ is holomorphic near $2z$. Choose $\rho >0$ so small that $D(z,\rho )$ avoids $\mathrm{\Lambda }$, $(z+\mathrm{\Lambda })\setminus \{z\}$ and $-z+\mathrm{\Lambda }$, and that $z+w\mathrm{\notin }\mathrm{\Lambda }$ for $w\in D(z,\rho )$. For $0<\mathrm{\mid }w-z\mathrm{\mid }<\rho$ the addition formula [F4] applies and gives $\mathrm{\wp }(z+w)=-\mathrm{\wp }(z)-\mathrm{\wp }(w)+\frac{1}{4}Q(w{)}^2$ with $Q(w)=({\mathrm{\wp }}^{\mathrm{\prime }}(z)-{\mathrm{\wp }}^{\mathrm{\prime }}(w))\mathrm{/}(\mathrm{\wp }(z)-\mathrm{\wp }(w))$. Both numerator and denominator vanish at $w=z$, and the denominator has a simple zero there because its derivative is $-{\mathrm{\wp }}^{\mathrm{\prime }}(z)\mathrm{\ne }0$; the numerator has a zero of at least order one. Thus $Q$ extends holomorphically with $Q(z)={\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}(z)\mathrm{/}{\mathrm{\wp }}^{\mathrm{\prime }}(z)$, whether or not ${\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}(z)$ vanishes. Letting $w\to z$ gives $\mathrm{\wp }(2z)=-2\mathrm{\wp }(z)+\frac{1}{4}({\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}(z)\mathrm{/}{\mathrm{\wp }}^{\mathrm{\prime }}(z){)}^2$.

(Rational expression in $\mathrm{\wp }$ and ${\mathrm{\wp }}^{\mathrm{\prime }}$.) Substituting the identity of step 1.1 into the duplication identity of step 1.2 gives, for every $z\in \mathbb{C}\setminus \mathrm{\Lambda }$ with ${\mathrm{\wp }}^{\mathrm{\prime }}(z)\mathrm{\ne }0$, $\mathrm{\wp }(2z)=-2\mathrm{\wp }(z)+\frac{1}{4}{\left(\frac{6\mathrm{\wp }(z{)}^2-\frac{1}{2}{g}_2}{{\mathrm{\wp }}^{\mathrm{\prime }}(z)}\right)}^2=-2\mathrm{\wp }(z)+\frac{(6\mathrm{\wp }(z{)}^2-\frac{1}{2}{g}_2{)}^2}{4{\mathrm{\wp }}^{\mathrm{\prime }}(z{)}^2},$ a value of the rational function $R(x,y)=-2x+{\displaystyle \frac{(6x^2-\frac{1}{2}{g}_2{)}^2}{4y^2}}$ of the two variables $x,y$ with coefficients in the field generated by $g_2$ over $\mathbb{C}$ (here the denominator $4y^2$ does not vanish because ${\mathrm{\wp }}^{\mathrm{\prime }}(z)\mathrm{\ne }0$).

(Meromorphic extension and pole set.) The function $F(z):=\mathrm{\wp }(2z)$ is meromorphic on $\mathbb{C}$: it is holomorphic off $\frac{1}{2}\mathrm{\Lambda }=\{z:2z\in \mathrm{\Lambda }\}$, and if $z_0\in \frac{1}{2}\mathrm{\Lambda }$ with $\lambda :=2z_0\in \mathrm{\Lambda }$, then [F1] gives that $\mathrm{\wp }(w)-(w-\lambda {)}^{-2}$ is holomorphic near $\lambda$, so substituting $w=2z$ shows $F(z)-\frac{1}{4}(z-z_0{)}^{-2}$ holomorphic near $z_0$: each $z_0\in \frac{1}{2}\mathrm{\Lambda }$ is a double pole of $F$, and there are no others. The right-hand side $R(\mathrm{\wp }(z),{\mathrm{\wp }}^{\mathrm{\prime }}(z))$ is meromorphic on $\mathbb{C}$ by the field property [F7], because $\mathrm{\wp }$ and ${\mathrm{\wp }}^{\mathrm{\prime }}$ are meromorphic and ${\mathrm{\wp }}^{\mathrm{\prime }}$ is not the zero function by [F2]. By steps 1.2 and 2.1 the two meromorphic functions agree on $\{z\in \mathbb{C}\setminus \mathrm{\Lambda }:{\mathrm{\wp }}^{\mathrm{\prime }}(z)\mathrm{\ne }0\}=\mathbb{C}\setminus \frac{1}{2}\mathrm{\Lambda }$, a nonempty open subset of the connected domain $\mathbb{C}$ [F8]; hence their difference vanishes on a nonempty open set and is identically zero by [F7]. Therefore the duplication identity is an identity of meromorphic functions on $\mathbb{C}$: it holds wherever both sides are finite, and at the points of $\frac{1}{2}\mathrm{\Lambda }$, where $F$ has a double pole and the right-hand side likewise has a pole (at half-periods because ${\mathrm{\wp }}^{\mathrm{\prime }}$ has a zero of order one and ${\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}$ is nonzero there, and at lattice points by the equality of the two meromorphic functions), no finite value is asserted.

(Assembly.) Step 1.1 gives the identity ${\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}=6{\mathrm{\wp }}^2-\frac{1}{2}{g}_2$ on $\mathbb{C}\setminus \mathrm{\Lambda }$, extended holomorphically across the half-periods where the division by ${\mathrm{\wp }}^{\mathrm{\prime }}$ was only apparently problematic; step 1.2 gives the duplication identity for ${\mathrm{\wp }}^{\mathrm{\prime }}(z)\mathrm{\ne }0$; step 2.1 exhibits it as the rational expression in $\mathrm{\wp }(z)$ and ${\mathrm{\wp }}^{\mathrm{\prime }}(z)$; and step 3.1 upgrades the duplication identity to an identity of meromorphic functions on $\mathbb{C}$, with the genuine poles retained. These are exactly the assertions of the example. ∎