Addition formula for $\mathrm{\wp }$

(Setup for generic $w$.) Let $w\in \mathbb{C}$ with $2w\mathrm{\notin }\mathrm{\Lambda }$; then $w\mathrm{\notin }\mathrm{\Lambda }$ and ${\mathrm{\wp }}^{\mathrm{\prime }}(w)\mathrm{\ne }0$ by [F3]. Define, in the field of meromorphic functions on $\mathbb{C}$, $Q_w(z):=\frac{{\mathrm{\wp }}^{\mathrm{\prime }}(z)-{\mathrm{\wp }}^{\mathrm{\prime }}(w)}{\mathrm{\wp }(z)-\mathrm{\wp }(w)},{\mathrm{\Phi }}_w(z):=\mathrm{\wp }(z+w)+\mathrm{\wp }(z)+\mathrm{\wp }(w)-\frac{1}{4}{Q}_w(z{)}^2\mathrm{.}$ The denominator $\mathrm{\wp }(z)-\mathrm{\wp }(w)$ is not the zero function of $z$ because $\mathrm{\wp }$ is nonconstant by [F2], so $Q_w$ and ${\mathrm{\Phi }}_w$ are meromorphic by [F7]; moreover $Q_w$ and ${\mathrm{\Phi }}_w$ are $\mathrm{\Lambda }$-periodic in $z$, since $\mathrm{\wp }(z)$, ${\mathrm{\wp }}^{\mathrm{\prime }}(z)$ and $\mathrm{\wp }(z+w)$ are $\mathrm{\Lambda }$-periodic in $z$ by [F2] and the formula uses only these. Proving ${\mathrm{\Phi }}_w\equiv 0$ is exactly the identity for this $w$, so it suffices to prove that.

(Expansion of $\mathrm{\wp }$ and ${\mathrm{\wp }}^{\mathrm{\prime }}$ at $0$.) By [F2] the function $\mathrm{\wp }(z)-z^{-2}$ is holomorphic near $0$ and even, so $\mathrm{\wp }(z)=z^{-2}+a_0+a_2{z}^2+a_4{z}^4+O(z^6)$ and, differentiating the series of [F2], ${\mathrm{\wp }}^{\mathrm{\prime }}(z)=-2z^{-3}+2a_{2}z+4a_4{z}^3+O(z^5)$ for $\mathrm{\mid }z\mathrm{\mid }$ small, with Laurent/Taylor coefficients unique by [F5]. Substituting these expansions into [F4] on a punctured disc and comparing the coefficient of $z^{-4}$ gives $0=12a_0$, since $({\mathrm{\wp }}^{\mathrm{\prime }}{)}^2=4z^{-6}-8a_2{z}^{-2}+O(1)$ has no $z^{-4}$ term while $4{\mathrm{\wp }}^3-g_2\mathrm{\wp }-g_3=4z^{-6}+12a_0{z}^{-4}+O(z^{-2})$ has coefficient $12a_0$ there; hence $a_0=0$, that is $\mathrm{\wp }(z)-z^{-2}\to 0$ and $\mathrm{\wp }(z)=z^{-2}+a_2{z}^2+O(z^4)$, ${\mathrm{\wp }}^{\mathrm{\prime }}(z)=-2z^{-3}+2a_{2}z+O(z^3)$ near $0$.

(${\mathrm{\Phi }}_w$ is entire.) Away from $\mathrm{\Lambda }$, $Q_w$ can have poles only where $\mathrm{\wp }(z)=\mathrm{\wp }(w)$, i.e. at $z\equiv \pm w$ modulo $\mathrm{\Lambda }$ by [F3], and $\mathrm{\wp }(z)$, $\mathrm{\wp }(z+w)$ can have poles only at $\mathrm{\Lambda }$ and at $-w+\mathrm{\Lambda }$, every point of $\mathbb{C}$ outside the three discrete sets $\mathrm{\Lambda }$, $w+\mathrm{\Lambda }$, $-w+\mathrm{\Lambda }$ is a point where ${\mathrm{\Phi }}_w$ is holomorphic; we check the three exceptional loci. (i) At $z_0\in \mathrm{\Lambda }$, write $u=z-z_0$; by [F2] $\mathrm{\wp }(z_0+u)=\mathrm{\wp }(u)=u^{-2}+a_2{u}^2+O(u^4)$ and ${\mathrm{\wp }}^{\mathrm{\prime }}(z_0+u)={\mathrm{\wp }}^{\mathrm{\prime }}(u)=-2u^{-3}+2a_{2}u+O(u^3)$ by step 1.2, so $Q_w(z_0+u)=(-2u^{-3}+O(u)-{\mathrm{\wp }}^{\mathrm{\prime }}(w))/(u^{-2}+O(1)-\mathrm{\wp }(w))=-2u^{-1}-2\mathrm{\wp }(w)u-{\mathrm{\wp }}^{\mathrm{\prime }}(w)u^2+O(u^3)$ and $\frac{1}{4}{Q}_w^2=u^{-2}+2\mathrm{\wp }(w)+{\mathrm{\wp }}^{\mathrm{\prime }}(w)u+O(u^2)$; hence $\mathrm{\wp }(z)-\frac{1}{4}{Q}_w(z{)}^2=(u^{-2}+O(u^2))-(u^{-2}+2\mathrm{\wp }(w)+{\mathrm{\wp }}^{\mathrm{\prime }}(w)u+O(u^2))=O(1)$ is bounded near $z_0$ and, being holomorphic on a punctured neighbourhood, extends holomorphically across $z_0$ by [F5], while $\mathrm{\wp }(z+w)$ and $\mathrm{\wp }(w)$ are holomorphic near $z_0$ because $z_0+w\mathrm{\notin }\mathrm{\Lambda }$ as $w\mathrm{\notin }\mathrm{\Lambda }$. (ii) At $z_0\equiv -w$ modulo $\mathrm{\Lambda }$, write $z=z_0+u=-w+u$; using $\mathrm{\wp }(-w+u)=\mathrm{\wp }(w-u)$ and ${\mathrm{\wp }}^{\mathrm{\prime }}(-w+u)=-{\mathrm{\wp }}^{\mathrm{\prime }}(w-u)$ by parity [F2], and the Taylor expansions $\mathrm{\wp }(w-u)=\mathrm{\wp }(w)-{\mathrm{\wp }}^{\mathrm{\prime }}(w)u+O(u^2)$, ${\mathrm{\wp }}^{\mathrm{\prime }}(w-u)={\mathrm{\wp }}^{\mathrm{\prime }}(w)-{\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}(w)u+O(u^2)$ from [F5], the numerator of $Q_w$ is $-{\mathrm{\wp }}^{\mathrm{\prime }}(w-u)-{\mathrm{\wp }}^{\mathrm{\prime }}(w)=-2{\mathrm{\wp }}^{\mathrm{\prime }}(w)+{\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}(w)u+O(u^2)$ and its denominator is $\mathrm{\wp }(w-u)-\mathrm{\wp }(w)=-{\mathrm{\wp }}^{\mathrm{\prime }}(w)u+\frac{1}{2}{\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}(w)u^2+O(u^3)=-u{\mathrm{\wp }}^{\mathrm{\prime }}(w)(1-\frac{{\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}(w)}{2{\mathrm{\wp }}^{\mathrm{\prime }}(w)}u+O(u^2))$, with ${\mathrm{\wp }}^{\mathrm{\prime }}(w)\mathrm{\ne }0$; hence $Q_w(z_0+u)=\frac{2}{u}(1+O(u^2))=2u^{-1}+O(u)$ and $\frac{1}{4}{Q}_w(z{)}^2=u^{-2}+O(1)$, while $\mathrm{\wp }(z+w)=\mathrm{\wp }(u)=u^{-2}+O(u^2)$; thus $\mathrm{\wp }(z+w)-\frac{1}{4}{Q}_w(z{)}^2=O(1)$ extends holomorphically across $z_0$ by [F5], and $\mathrm{\wp }(z)+\mathrm{\wp }(w)$ is holomorphic near $z_0\mathrm{\notin }\mathrm{\Lambda }$. (iii) At $z_0\equiv w$, write $z=w+u$; then ${\mathrm{\wp }}^{\mathrm{\prime }}(w+u)-{\mathrm{\wp }}^{\mathrm{\prime }}(w)={\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}(w)u+O(u^2)$ and $\mathrm{\wp }(w+u)-\mathrm{\wp }(w)={\mathrm{\wp }}^{\mathrm{\prime }}(w)u+O(u^2)$ by [F5], so $Q_w(w+u)={\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}(w)\mathrm{/}{\mathrm{\wp }}^{\mathrm{\prime }}(w)+O(u)$ is holomorphic at $u=0$ because ${\mathrm{\wp }}^{\mathrm{\prime }}(w)\mathrm{\ne }0$, and $\mathrm{\wp }(z)$, $\mathrm{\wp }(z+w)$ are holomorphic near $z_0=w$ because $w\mathrm{\notin }\mathrm{\Lambda }$ and $2w\mathrm{\notin }\mathrm{\Lambda }$. Therefore ${\mathrm{\Phi }}_w$ is holomorphic at every point of $\mathbb{C}$.

(${\mathrm{\Phi }}_w$ is zero.) By step 2.1, ${\mathrm{\Phi }}_w$ is entire and by step 1.1 it is $\mathrm{\Lambda }$-periodic; the closed parallelogram $P$ is compact by [F6] and every $z$ differs from a point of $P$ by a lattice element by [F1], so $\mathrm{\mid }{\mathrm{\Phi }}_w\mathrm{\mid }$ is bounded on $\mathbb{C}$ by the boundedness of the continuous function $\mathrm{\mid }{\mathrm{\Phi }}_w\mathrm{\mid }$ on the compact set $P$ [F6]; hence ${\mathrm{\Phi }}_w$ is constant by Liouville [F6]. Its value is ${\lim }_{z\to 0}{\mathrm{\Phi }}_w(z)$, which step 2.1 shows is finite; expanding with step 1.2, $\mathrm{\wp }(z)-\mathrm{\wp }(w)=z^{-2}(1-\mathrm{\wp }(w)z^2+O(z^4)),{\mathrm{\wp }}^{\mathrm{\prime }}(z)-{\mathrm{\wp }}^{\mathrm{\prime }}(w)=-2z^{-3}+2a_{2}z+O(z^3)-{\mathrm{\wp }}^{\mathrm{\prime }}(w),$ so $Q_w(z)=-2z^{-1}-2\mathrm{\wp }(w)z-{\mathrm{\wp }}^{\mathrm{\prime }}(w)z^2+O(z^3)$, $\frac{1}{4}{Q}_w(z{)}^2=z^{-2}+2\mathrm{\wp }(w)+{\mathrm{\wp }}^{\mathrm{\prime }}(w)z+O(z^2)$, and $\mathrm{\wp }(z)-\frac{1}{4}{Q}_w(z{)}^2=-2\mathrm{\wp }(w)-{\mathrm{\wp }}^{\mathrm{\prime }}(w)z+O(z^2)\to -2\mathrm{\wp }(w)$ as $z\to 0$; therefore ${\mathrm{\Phi }}_w(z)\to \mathrm{\wp }(w)-2\mathrm{\wp }(w)+\mathrm{\wp }(w)=0$. Hence ${\mathrm{\Phi }}_w\equiv 0$, that is, $\mathrm{\wp }(z+w)=-\mathrm{\wp }(z)-\mathrm{\wp }(w)+\frac{1}{4}{Q}_w(z{)}^2$ as meromorphic functions of $z$ for every $w$ with $2w\mathrm{\notin }\mathrm{\Lambda }$.

(Meromorphic continuation in the second variable.) Fix $z\mathrm{\notin }\mathrm{\Lambda }$ and put $W_z(w):=\mathrm{\wp }(z+w)+\mathrm{\wp }(z)+\mathrm{\wp }(w)-\frac{1}{4}{\left(\frac{{\mathrm{\wp }}^{\mathrm{\prime }}(z)-{\mathrm{\wp }}^{\mathrm{\prime }}(w)}{\mathrm{\wp }(z)-\mathrm{\wp }(w)}\right)}^2\mathrm{.}$ As a function of $w$, each of $\mathrm{\wp }(z+w)$, $\mathrm{\wp }(w)$, ${\mathrm{\wp }}^{\mathrm{\prime }}(w)$ is meromorphic on $\mathbb{C}$ by [F2], the quantities $\mathrm{\wp }(z),{\mathrm{\wp }}^{\mathrm{\prime }}(z)$ are constants, and the denominator $\mathrm{\wp }(z)-\mathrm{\wp }(w)$ is not the zero function of $w$ because $\mathrm{\wp }$ is nonconstant [F2]; hence $W_z$ is meromorphic on $\mathbb{C}$ by [F7]. Let $V:=\{w\in \mathbb{C}:2w\mathrm{\notin }\mathrm{\Lambda },w\mathrm{\notin }z+\mathrm{\Lambda },w\mathrm{\notin }-z+\mathrm{\Lambda }\}$: the sets $\frac{1}{2}\mathrm{\Lambda }$, $z+\mathrm{\Lambda }$, $-z+\mathrm{\Lambda }$ are discrete and closed, hence have empty interior, so $V$ is a nonempty open subset of $\mathbb{C}$. For $w\in V$ the point $z$ is outside $\mathrm{\Lambda }$, $w+\mathrm{\Lambda }$ and $-w+\mathrm{\Lambda }$, so step 3.1 applied to $w$ gives the identity at the point $z$, i.e. $W_z(w)=0$; since the meromorphic function $W_z$ vanishes on the nonempty open set $V$, it is identically zero by [F7]. Thus for every $z\mathrm{\notin }\mathrm{\Lambda }$ and every $w\in \mathbb{C}$, the identity holds in the meromorphic sense in $w$.

(Exceptional parameters and poles.) For fixed $w\mathrm{\notin }\mathrm{\Lambda }$, the expression in step 1.1 is meromorphic in $z$. Step 4.1 gives its vanishing at all ordinary pairs $z\mathrm{\notin }\mathrm{\Lambda }$, $\mathrm{\wp }(z)\mathrm{\ne }\mathrm{\wp }(w)$, so [F7] gives the identity for this fixed $w$, including nonzero half-periods. At $z\equiv w$, the quotient is removable when $2w\mathrm{\notin }\mathrm{\Lambda }$, as in step 2.1(iii). If instead $w=h$ is a nonzero half-period, [F3] gives ${\mathrm{\wp }}^{\mathrm{\prime }}(h)=0$ and ${\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}(h)\mathrm{\ne }0$; Taylor expansion yields $Q_h(h+u)=2\mathrm{/}u+O(1)$, so $Q_h^2\mathrm{/}4$ and $\mathrm{\wp }(2h+u)=\mathrm{\wp }(u)$ both have genuine double poles. For a lattice parameter $w=\lambda +u$, fix $z\mathrm{\notin }\mathrm{\Lambda }$. Periodicity and the expansions of steps 1.2 and 2.1(i), with the variables interchanged, give $\frac{{\mathrm{\wp }}^{\mathrm{\prime }}(z)-{\mathrm{\wp }}^{\mathrm{\prime }}(\lambda +u)}{\mathrm{\wp }(z)-\mathrm{\wp }(\lambda +u)}=-2u^{-1}-2\mathrm{\wp }(z)u-{\mathrm{\wp }}^{\mathrm{\prime }}(z)u^2+O(u^3),$ and therefore $-\mathrm{\wp }(z)-\mathrm{\wp }(\lambda +u)+\frac{1}{4}{\left(\frac{{\mathrm{\wp }}^{\mathrm{\prime }}(z)-{\mathrm{\wp }}^{\mathrm{\prime }}(\lambda +u)}{\mathrm{\wp }(z)-\mathrm{\wp }(\lambda +u)}\right)}^2=\mathrm{\wp }(z)+{\mathrm{\wp }}^{\mathrm{\prime }}(z)u+O(u^2)\mathrm{.}$ The combined right side thus extends in $w$ at $\lambda$ with value $\mathrm{\wp }(z)$; its restriction to $w=\lambda$ extends meromorphically in $z$ as $\mathrm{\wp }(z)=\mathrm{\wp }(z+\lambda )$. The same reasoning applies with the variables interchanged, since the formula is symmetric. Away from the exceptional loci the combined right side equals $\mathrm{\wp }(z+w)$, so its meromorphic continuation across them is this same meromorphic function; no individual infinite term is evaluated as a complex constant. In particular no finite value is asserted at a genuine pole.

(Assembly.) Step 3.1 proves the identity as meromorphic functions of $z$ for every $w$ with $2w\mathrm{\notin }\mathrm{\Lambda }$; step 4.1 extends the identity, in the second variable, to all $w$ for $z\mathrm{\notin }\mathrm{\Lambda }$; and step 5.1 treats half-period poles and the lattice-parameter restriction by explicit continuation, yielding the symmetric meromorphic identity in $(z,w)$ together with the interpretation of the apparent exceptional cases. This is the assertion of the theorem. ∎