Class of mathematical functions
In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions is also referred to as ℘-functions and they are usually denoted by the symbol ℘, a uniquely fancy script p. They play an important role in the theory of elliptic functions, i.e., meromorphic functions that are doubly periodic. A ℘-function together with its derivative can be used to parameterize elliptic curves and they generate the field of elliptic functions with respect to a given period lattice.
Symbol for Weierstrass ℘ {\displaystyle \wp } -function
Model of Weierstrass ℘ {\displaystyle \wp } -functionA cubic of the form C g 2 , g 3 C = { ( x , y ) ∈ C 2 : y 2 = 4 x 3 − g 2 x − g 3 } {\displaystyle C_{g_{2},g_{3}}^{\mathbb {C} }=\{(x,y)\in \mathbb {C} ^{2}:y^{2}=4x^{3}-g_{2}x-g_{3}\}} , where g 2 , g 3 ∈ C {\displaystyle g_{2},g_{3}\in \mathbb {C} } are complex numbers with g 2 3 − 27 g 3 2 ≠ 0 {\displaystyle g_{2}^{3}-27g_{3}^{2}\neq 0} , cannot be rationally parameterized.[1] Yet one still wants to find a way to parameterize it.
For the quadric K = { ( x , y ) ∈ R 2 : x 2 + y 2 = 1 } {\displaystyle K=\left\{(x,y)\in \mathbb {R} ^{2}:x^{2}+y^{2}=1\right\}} ; the unit circle, there exists a (non-rational) parameterization using the sine function and its derivative the cosine function: ψ : R / 2 π Z → K , t ↦ ( sin t , cos t ) . {\displaystyle \psi :\mathbb {R} /2\pi \mathbb {Z} \to K,\quad t\mapsto (\sin t,\cos t).} Because of the periodicity of the sine and cosine R / 2 π Z {\displaystyle \mathbb {R} /2\pi \mathbb {Z} } is chosen to be the domain, so the function is bijective.
In a similar way one can get a parameterization of C g 2 , g 3 C {\displaystyle C_{g_{2},g_{3}}^{\mathbb {C} }} by means of the doubly periodic ℘ {\displaystyle \wp } -function (see in the section "Relation to elliptic curves"). This parameterization has the domain C / Λ {\displaystyle \mathbb {C} /\Lambda } , which is topologically equivalent to a torus.[2]
There is another analogy to the trigonometric functions. Consider the integral function a ( x ) = ∫ 0 x d y 1 − y 2 . {\displaystyle a(x)=\int _{0}^{x}{\frac {dy}{\sqrt {1-y^{2}}}}.} It can be simplified by substituting y = sin t {\displaystyle y=\sin t} and s = arcsin x {\displaystyle s=\arcsin x} : a ( x ) = ∫ 0 s d t = s = arcsin x . {\displaystyle a(x)=\int _{0}^{s}dt=s=\arcsin x.} That means a − 1 ( x ) = sin x {\displaystyle a^{-1}(x)=\sin x} . So the sine function is an inverse function of an integral function.[3]
Elliptic functions are the inverse functions of elliptic integrals. In particular, let: u ( z ) = ∫ z ∞ d s 4 s 3 − g 2 s − g 3 . {\displaystyle u(z)=\int _{z}^{\infty }{\frac {ds}{\sqrt {4s^{3}-g_{2}s-g_{3}}}}.} Then the extension of u − 1 {\displaystyle u^{-1}} to the complex plane equals the ℘ {\displaystyle \wp } -function.[4] This invertibility is used in complex analysis to provide a solution to certain nonlinear differential equations satisfying the Painlevé property, i.e., those equations that admit poles as their only movable singularities.[5]
Visualization of the ℘ {\displaystyle \wp } -function with invariants g 2 = 1 + i {\displaystyle g_{2}=1+i} and g 3 = 2 − 3 i {\displaystyle g_{3}=2-3i} in which white corresponds to a pole, black to a zero.Let ω 1 , ω 2 ∈ C {\displaystyle \omega _{1},\omega _{2}\in \mathbb {C} } be two complex numbers that are linearly independent over R {\displaystyle \mathbb {R} } and let Λ := Z ω 1 + Z ω 2 := { m ω 1 + n ω 2 : m , n ∈ Z } {\displaystyle \Lambda :=\mathbb {Z} \omega _{1}+\mathbb {Z} \omega _{2}:=\{m\omega _{1}+n\omega _{2}:m,n\in \mathbb {Z} \}} be the period lattice generated by those numbers. Then the ℘ {\displaystyle \wp } -function is defined as follows:
This series converges locally uniformly absolutely in the complex torus C / Λ {\displaystyle \mathbb {C} /\Lambda } .
It is common to use 1 {\displaystyle 1} and τ {\displaystyle \tau } in the upper half-plane H := { z ∈ C : Im ( z ) > 0 } {\displaystyle \mathbb {H} :=\{z\in \mathbb {C} :\operatorname {Im} (z)>0\}} as generators of the lattice. Dividing by ω 1 {\textstyle \omega _{1}} maps the lattice Z ω 1 + Z ω 2 {\displaystyle \mathbb {Z} \omega _{1}+\mathbb {Z} \omega _{2}} isomorphically onto the lattice Z + Z τ {\displaystyle \mathbb {Z} +\mathbb {Z} \tau } with τ = ω 2 ω 1 {\textstyle \tau ={\tfrac {\omega _{2}}{\omega _{1}}}} . Because − τ {\displaystyle -\tau } can be substituted for τ {\displaystyle \tau } , without loss of generality we can assume τ ∈ H {\displaystyle \tau \in \mathbb {H} } , and then define ℘ ( z , τ ) := ℘ ( z , 1 , τ ) {\displaystyle \wp (z,\tau ):=\wp (z,1,\tau )} . With that definition, we have ℘ ( z , ω 1 , ω 2 ) = ω 1 − 2 ℘ ( z / ω 1 , ω 2 / ω 1 ) {\displaystyle \wp (z,\omega _{1},\omega _{2})=\omega _{1}^{-2}\wp (z/\omega _{1},\omega _{2}/\omega _{1})} .
Let r := min { | λ | : 0 ≠ λ ∈ Λ } {\displaystyle r:=\min\{{|\lambda }|:0\neq \lambda \in \Lambda \}} . Then for 0 < | z | < r {\displaystyle 0<|z|<r} the ℘ {\displaystyle \wp } -function has the following Laurent expansion ℘ ( z ) = 1 z 2 + ∑ n = 1 ∞ ( 2 n + 1 ) G 2 n + 2 z 2 n {\displaystyle \wp (z)={\frac {1}{z^{2}}}+\sum _{n=1}^{\infty }(2n+1)G_{2n+2}z^{2n}} where G n = ∑ 0 ≠ λ ∈ Λ λ − n {\displaystyle G_{n}=\sum _{0\neq \lambda \in \Lambda }\lambda ^{-n}} for n ≥ 3 {\displaystyle n\geq 3} are so called Eisenstein series.[6]
Differential equation[edit]Set g 2 = 60 G 4 {\displaystyle g_{2}=60G_{4}} and g 3 = 140 G 6 {\displaystyle g_{3}=140G_{6}} . Then the ℘ {\displaystyle \wp } -function satisfies the differential equation[6] ℘ ′ 2 ( z ) = 4 ℘ 3 ( z ) − g 2 ℘ ( z ) − g 3 . {\displaystyle \wp '^{2}(z)=4\wp ^{3}(z)-g_{2}\wp (z)-g_{3}.} This relation can be verified by forming a linear combination of powers of ℘ {\displaystyle \wp } and ℘ ′ {\displaystyle \wp '} to eliminate the pole at z = 0 {\displaystyle z=0} . This yields an entire elliptic function that has to be constant by Liouville's theorem.[6]
The real part of the invariant g3 as a function of the square of the nome q on the unit disk. The imaginary part of the invariant g3 as a function of the square of the nome q on the unit disk.The coefficients of the above differential equation g 2 {\displaystyle g_{2}} and g 3 {\displaystyle g_{3}} are known as the invariants. Because they depend on the lattice Λ {\displaystyle \Lambda } they can be viewed as functions in ω 1 {\displaystyle \omega _{1}} and ω 2 {\displaystyle \omega _{2}} .
The series expansion suggests that g 2 {\displaystyle g_{2}} and g 3 {\displaystyle g_{3}} are homogeneous functions of degree − 4 {\displaystyle -4} and − 6 {\displaystyle -6} . That is[7] g 2 ( λ ω 1 , λ ω 2 ) = λ − 4 g 2 ( ω 1 , ω 2 ) {\displaystyle g_{2}(\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-4}g_{2}(\omega _{1},\omega _{2})} g 3 ( λ ω 1 , λ ω 2 ) = λ − 6 g 3 ( ω 1 , ω 2 ) {\displaystyle g_{3}(\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-6}g_{3}(\omega _{1},\omega _{2})} for λ ≠ 0 {\displaystyle \lambda \neq 0} .
If ω 1 {\displaystyle \omega _{1}} and ω 2 {\displaystyle \omega _{2}} are chosen in such a way that Im ( ω 2 ω 1 ) > 0 {\displaystyle \operatorname {Im} \left({\tfrac {\omega _{2}}{\omega _{1}}}\right)>0} , g 2 {\displaystyle g_{2}} and g 3 {\displaystyle g_{3}} can be interpreted as functions on the upper half-plane H := { z ∈ C : Im ( z ) > 0 } {\displaystyle \mathbb {H} :=\{z\in \mathbb {C} :\operatorname {Im} (z)>0\}} .
Let τ = ω 2 ω 1 {\displaystyle \tau ={\tfrac {\omega _{2}}{\omega _{1}}}} . One has:[8] g 2 ( 1 , τ ) = ω 1 4 g 2 ( ω 1 , ω 2 ) , {\displaystyle g_{2}(1,\tau )=\omega _{1}^{4}g_{2}(\omega _{1},\omega _{2}),} g 3 ( 1 , τ ) = ω 1 6 g 3 ( ω 1 , ω 2 ) . {\displaystyle g_{3}(1,\tau )=\omega _{1}^{6}g_{3}(\omega _{1},\omega _{2}).} That means g2 and g3 are only scaled by doing this. Set g 2 ( τ ) := g 2 ( 1 , τ ) {\displaystyle g_{2}(\tau ):=g_{2}(1,\tau )} and g 3 ( τ ) := g 3 ( 1 , τ ) . {\displaystyle g_{3}(\tau ):=g_{3}(1,\tau ).} As functions of τ ∈ H {\displaystyle \tau \in \mathbb {H} } , g 2 {\displaystyle g_{2}} and g 3 {\displaystyle g_{3}} are so called modular forms.
The Fourier series for g 2 {\displaystyle g_{2}} and g 3 {\displaystyle g_{3}} are given as follows:[9] g 2 ( τ ) = 4 3 π 4 [ 1 + 240 ∑ k = 1 ∞ σ 3 ( k ) q 2 k ] {\displaystyle g_{2}(\tau )={\frac {4}{3}}\pi ^{4}\left[1+240\sum _{k=1}^{\infty }\sigma _{3}(k)q^{2k}\right]} g 3 ( τ ) = 8 27 π 6 [ 1 − 504 ∑ k = 1 ∞ σ 5 ( k ) q 2 k ] {\displaystyle g_{3}(\tau )={\frac {8}{27}}\pi ^{6}\left[1-504\sum _{k=1}^{\infty }\sigma _{5}(k)q^{2k}\right]} where σ m ( k ) := ∑ d ∣ k d m {\displaystyle \sigma _{m}(k):=\sum _{d\mid {k}}d^{m}} is the divisor function and q = e π i τ {\displaystyle q=e^{\pi i\tau }} is the nome.
Modular discriminant[edit] The real part of the discriminant as a function of the square of the nome q on the unit disk.The modular discriminant Δ {\displaystyle \Delta } is defined as the discriminant of the characteristic polynomial of the differential equation ℘ ′ 2 ( z ) = 4 ℘ 3 ( z ) − g 2 ℘ ( z ) − g 3 {\displaystyle \wp '^{2}(z)=4\wp ^{3}(z)-g_{2}\wp (z)-g_{3}} as follows: Δ = g 2 3 − 27 g 3 2 . {\displaystyle \Delta =g_{2}^{3}-27g_{3}^{2}.} The discriminant is a modular form of weight 12 {\displaystyle 12} . That is, under the action of the modular group, it transforms as Δ ( a τ + b c τ + d ) = ( c τ + d ) 12 Δ ( τ ) {\displaystyle \Delta \left({\frac {a\tau +b}{c\tau +d}}\right)=\left(c\tau +d\right)^{12}\Delta (\tau )} where a , b , d , c ∈ Z {\displaystyle a,b,d,c\in \mathbb {Z} } with a d − b c = 1 {\displaystyle ad-bc=1} .[10]
Note that Δ = ( 2 π ) 12 η 24 {\displaystyle \Delta =(2\pi )^{12}\eta ^{24}} where η {\displaystyle \eta } is the Dedekind eta function.[11]
For the Fourier coefficients of Δ {\displaystyle \Delta } , see Ramanujan tau function.
The constants e1, e2 and e3[edit]e 1 {\displaystyle e_{1}} , e 2 {\displaystyle e_{2}} and e 3 {\displaystyle e_{3}} are usually used to denote the values of the ℘ {\displaystyle \wp } -function at the half-periods. e 1 ≡ ℘ ( ω 1 2 ) {\displaystyle e_{1}\equiv \wp \left({\frac {\omega _{1}}{2}}\right)} e 2 ≡ ℘ ( ω 2 2 ) {\displaystyle e_{2}\equiv \wp \left({\frac {\omega _{2}}{2}}\right)} e 3 ≡ ℘ ( ω 1 + ω 2 2 ) {\displaystyle e_{3}\equiv \wp \left({\frac {\omega _{1}+\omega _{2}}{2}}\right)} They are pairwise distinct and only depend on the lattice Λ {\displaystyle \Lambda } and not on its generators.[12]
e 1 {\displaystyle e_{1}} , e 2 {\displaystyle e_{2}} and e 3 {\displaystyle e_{3}} are the roots of the cubic polynomial 4 ℘ ( z ) 3 − g 2 ℘ ( z ) − g 3 {\displaystyle 4\wp (z)^{3}-g_{2}\wp (z)-g_{3}} and are related by the equation: e 1 + e 2 + e 3 = 0. {\displaystyle e_{1}+e_{2}+e_{3}=0.} Because those roots are distinct the discriminant Δ {\displaystyle \Delta } does not vanish on the upper half plane.[13] Now we can rewrite the differential equation: ℘ ′ 2 ( z ) = 4 ( ℘ ( z ) − e 1 ) ( ℘ ( z ) − e 2 ) ( ℘ ( z ) − e 3 ) . {\displaystyle \wp '^{2}(z)=4(\wp (z)-e_{1})(\wp (z)-e_{2})(\wp (z)-e_{3}).} That means the half-periods are zeros of ℘ ′ {\displaystyle \wp '} .
The invariants g 2 {\displaystyle g_{2}} and g 3 {\displaystyle g_{3}} can be expressed in terms of these constants in the following way:[14] g 2 = − 4 ( e 1 e 2 + e 1 e 3 + e 2 e 3 ) {\displaystyle g_{2}=-4(e_{1}e_{2}+e_{1}e_{3}+e_{2}e_{3})} g 3 = 4 e 1 e 2 e 3 {\displaystyle g_{3}=4e_{1}e_{2}e_{3}} e 1 {\displaystyle e_{1}} , e 2 {\displaystyle e_{2}} and e 3 {\displaystyle e_{3}} are related to the modular lambda function: λ ( τ ) = e 3 − e 2 e 1 − e 2 , τ = ω 2 ω 1 . {\displaystyle \lambda (\tau )={\frac {e_{3}-e_{2}}{e_{1}-e_{2}}},\quad \tau ={\frac {\omega _{2}}{\omega _{1}}}.}
Relation to Jacobi's elliptic functions[edit]For numerical work, it is often convenient to calculate the Weierstrass elliptic function in terms of Jacobi's elliptic functions.
The basic relations are:[15] ℘ ( z ) = e 3 + e 1 − e 3 sn 2 w = e 2 + ( e 1 − e 3 ) dn 2 w sn 2 w = e 1 + ( e 1 − e 3 ) cn 2 w sn 2 w {\displaystyle \wp (z)=e_{3}+{\frac {e_{1}-e_{3}}{\operatorname {sn} ^{2}w}}=e_{2}+(e_{1}-e_{3}){\frac {\operatorname {dn} ^{2}w}{\operatorname {sn} ^{2}w}}=e_{1}+(e_{1}-e_{3}){\frac {\operatorname {cn} ^{2}w}{\operatorname {sn} ^{2}w}}} where e 1 , e 2 {\displaystyle e_{1},e_{2}} and e 3 {\displaystyle e_{3}} are the three roots described above and where the modulus k of the Jacobi functions equals k = e 2 − e 3 e 1 − e 3 {\displaystyle k={\sqrt {\frac {e_{2}-e_{3}}{e_{1}-e_{3}}}}} and their argument w equals w = z e 1 − e 3 . {\displaystyle w=z{\sqrt {e_{1}-e_{3}}}.}
Relation to Jacobi's theta functions[edit]The function ℘ ( z , τ ) = ℘ ( z , 1 , ω 2 / ω 1 ) {\displaystyle \wp (z,\tau )=\wp (z,1,\omega _{2}/\omega _{1})} can be represented by Jacobi's theta functions: ℘ ( z , τ ) = ( π θ 2 ( 0 , q ) θ 3 ( 0 , q ) θ 4 ( π z , q ) θ 1 ( π z , q ) ) 2 − π 2 3 ( θ 2 4 ( 0 , q ) + θ 3 4 ( 0 , q ) ) {\displaystyle \wp (z,\tau )=\left(\pi \theta _{2}(0,q)\theta _{3}(0,q){\frac {\theta _{4}(\pi z,q)}{\theta _{1}(\pi z,q)}}\right)^{2}-{\frac {\pi ^{2}}{3}}\left(\theta _{2}^{4}(0,q)+\theta _{3}^{4}(0,q)\right)} where q = e π i τ {\displaystyle q=e^{\pi i\tau }} is the nome and τ {\displaystyle \tau } is the period ratio ( τ ∈ H ) {\displaystyle (\tau \in \mathbb {H} )} .[16] This also provides a very rapid algorithm for computing ℘ ( z , τ ) {\displaystyle \wp (z,\tau )} .
Relation to elliptic curves[edit]Consider the embedding of the cubic curve in the complex projective plane
where O {\displaystyle O} is a point lying on the line at infinity P 1 ( C ) {\displaystyle \mathbb {P} _{1}(\mathbb {C} )} . For this cubic there exists no rational parameterization, if Δ ≠ 0 {\displaystyle \Delta \neq 0} .[1] In this case it is also called an elliptic curve. Nevertheless there is a parameterization in homogeneous coordinates that uses the ℘ {\displaystyle \wp } -function and its derivative ℘ ′ {\displaystyle \wp '} :[17]
Now the map φ {\displaystyle \varphi } is bijective and parameterizes the elliptic curve C ¯ g 2 , g 3 C {\displaystyle {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }} .
C / Λ {\displaystyle \mathbb {C} /\Lambda } is an abelian group and a topological space, equipped with the quotient topology.
It can be shown that every Weierstrass cubic is given in such a way. That is to say that for every pair g 2 , g 3 ∈ C {\displaystyle g_{2},g_{3}\in \mathbb {C} } with Δ = g 2 3 − 27 g 3 2 ≠ 0 {\displaystyle \Delta =g_{2}^{3}-27g_{3}^{2}\neq 0} there exists a lattice Z ω 1 + Z ω 2 {\displaystyle \mathbb {Z} \omega _{1}+\mathbb {Z} \omega _{2}} , such that
g 2 = g 2 ( ω 1 , ω 2 ) {\displaystyle g_{2}=g_{2}(\omega _{1},\omega _{2})} and g 3 = g 3 ( ω 1 , ω 2 ) {\displaystyle g_{3}=g_{3}(\omega _{1},\omega _{2})} .[18]
The statement that elliptic curves over Q {\displaystyle \mathbb {Q} } can be parameterized over Q {\displaystyle \mathbb {Q} } , is known as the modularity theorem. This is an important theorem in number theory. It was part of Andrew Wiles' proof (1995) of Fermat's Last Theorem.
The addition theorem states[19] that if z , w , {\displaystyle z,w,} and z + w {\displaystyle z+w} do not belong to Λ {\displaystyle \Lambda } , then det [ 1 ℘ ( z ) ℘ ′ ( z ) 1 ℘ ( w ) ℘ ′ ( w ) 1 ℘ ( z + w ) − ℘ ′ ( z + w ) ] = 0. {\displaystyle \det {\begin{bmatrix}1&\wp (z)&\wp '(z)\\1&\wp (w)&\wp '(w)\\1&\wp (z+w)&-\wp '(z+w)\end{bmatrix}}=0.} This states that the points P = ( ℘ ( z ) , ℘ ′ ( z ) ) , {\displaystyle P=(\wp (z),\wp '(z)),} Q = ( ℘ ( w ) , ℘ ′ ( w ) ) , {\displaystyle Q=(\wp (w),\wp '(w)),} and R = ( ℘ ( z + w ) , − ℘ ′ ( z + w ) ) {\displaystyle R=(\wp (z+w),-\wp '(z+w))} are collinear, the geometric form of the group law of an elliptic curve.
This can be proven[20] by considering constants A , B {\displaystyle A,B} such that ℘ ′ ( z ) = A ℘ ( z ) + B , ℘ ′ ( w ) = A ℘ ( w ) + B . {\displaystyle \wp '(z)=A\wp (z)+B,\quad \wp '(w)=A\wp (w)+B.} Then the elliptic function ℘ ′ ( ζ ) − A ℘ ( ζ ) − B {\displaystyle \wp '(\zeta )-A\wp (\zeta )-B} has a pole of order three at zero, and therefore three zeros whose sum belongs to Λ {\displaystyle \Lambda } . Two of the zeros are z {\displaystyle z} and w {\displaystyle w} , and thus the third is congruent to − z − w {\displaystyle -z-w} .
The addition theorem can be put into the alternative form, for z , w , z − w , z + w ∉ Λ {\displaystyle z,w,z-w,z+w\not \in \Lambda } :[21] ℘ ( z + w ) = 1 4 [ ℘ ′ ( z ) − ℘ ′ ( w ) ℘ ( z ) − ℘ ( w ) ] 2 − ℘ ( z ) − ℘ ( w ) . {\displaystyle \wp (z+w)={\frac {1}{4}}\left[{\frac {\wp '(z)-\wp '(w)}{\wp (z)-\wp (w)}}\right]^{2}-\wp (z)-\wp (w).}
As well as the duplication formula:[21] ℘ ( 2 z ) = 1 4 [ ℘ ″ ( z ) ℘ ′ ( z ) ] 2 − 2 ℘ ( z ) . {\displaystyle \wp (2z)={\frac {1}{4}}\left[{\frac {\wp ''(z)}{\wp '(z)}}\right]^{2}-2\wp (z).}
This can be proven from the addition theorem shown above. The points P = ( ℘ ( u ) , ℘ ′ ( u ) ) , Q = ( ℘ ( v ) , ℘ ′ ( v ) ) , {\displaystyle P=(\wp (u),\wp '(u)),Q=(\wp (v),\wp '(v)),} and R = ( ℘ ( u + v ) , − ℘ ′ ( u + v ) ) {\displaystyle R=(\wp (u+v),-\wp '(u+v))} are collinear and lie on the curve y 2 = 4 x 3 − g 2 x − g 3 {\displaystyle y^{2}=4x^{3}-g_{2}x-g_{3}} . The slope of that line is m = y P − y Q x P − x Q = ℘ ′ ( u ) − ℘ ′ ( v ) ℘ ( u ) − ℘ ( v ) . {\displaystyle m={\frac {y_{P}-y_{Q}}{x_{P}-x_{Q}}}={\frac {\wp '(u)-\wp '(v)}{\wp (u)-\wp (v)}}.} So x = x P = ℘ ( u ) {\displaystyle x=x_{P}=\wp (u)} , x = x Q = ℘ ( v ) {\displaystyle x=x_{Q}=\wp (v)} , and x = x R = ℘ ( u + v ) {\displaystyle x=x_{R}=\wp (u+v)} all satisfy a cubic ( m x + q ) 2 = 4 x 3 − g 2 x − g 3 , {\displaystyle (mx+q)^{2}=4x^{3}-g_{2}x-g_{3},} where q {\displaystyle q} is a constant. This becomes 4 x 3 − m 2 x 2 − ( 2 m q + g 2 ) x − g 3 − q 2 = 0. {\displaystyle 4x^{3}-m^{2}x^{2}-(2mq+g_{2})x-g_{3}-q^{2}=0.} Thus x P + x Q + x R = m 2 4 {\displaystyle x_{P}+x_{Q}+x_{R}={\frac {m^{2}}{4}}} which provides the wanted formula ℘ ( u + v ) + ℘ ( u ) + ℘ ( v ) = 1 4 [ ℘ ′ ( u ) − ℘ ′ ( v ) ℘ ( u ) − ℘ ( v ) ] 2 . {\displaystyle \wp (u+v)+\wp (u)+\wp (v)={\frac {1}{4}}\left[{\frac {\wp '(u)-\wp '(v)}{\wp (u)-\wp (v)}}\right]^{2}.}
A direct proof is as follows.[22] Any elliptic function f {\displaystyle f} can be expressed as: f ( u ) = c ∏ i = 1 n σ ( u − a i ) σ ( u − b i ) c ∈ C {\displaystyle f(u)=c\prod _{i=1}^{n}{\frac {\sigma (u-a_{i})}{\sigma (u-b_{i})}}\quad c\in \mathbb {C} } where σ {\displaystyle \sigma } is the Weierstrass sigma function and a i , b i {\displaystyle a_{i},b_{i}} are the respective zeros and poles in the period parallelogram. Considering the function ℘ ( u ) − ℘ ( v ) {\displaystyle \wp (u)-\wp (v)} as a function of u {\displaystyle u} , we have ℘ ( u ) − ℘ ( v ) = c σ ( u + v ) σ ( u − v ) σ ( u ) 2 . {\displaystyle \wp (u)-\wp (v)=c{\frac {\sigma (u+v)\sigma (u-v)}{\sigma (u)^{2}}}.} Multiplying both sides by u 2 {\displaystyle u^{2}} and letting u → 0 {\displaystyle u\to 0} , we have 1 = − c σ ( v ) 2 {\displaystyle 1=-c\sigma (v)^{2}} , so c = − 1 σ ( v ) 2 ⟹ ℘ ( u ) − ℘ ( v ) = − σ ( u + v ) σ ( u − v ) σ ( u ) 2 σ ( v ) 2 . {\displaystyle c=-{\frac {1}{\sigma (v)^{2}}}\implies \wp (u)-\wp (v)=-{\frac {\sigma (u+v)\sigma (u-v)}{\sigma (u)^{2}\sigma (v)^{2}}}.}
By definition the Weierstrass zeta function: d d z ln σ ( z ) = ζ ( z ) {\displaystyle {\frac {d}{dz}}\ln \sigma (z)=\zeta (z)} therefore we logarithmically differentiate both sides with respect to u {\displaystyle u} obtaining: ℘ ′ ( u ) ℘ ( u ) − ℘ ( v ) = ζ ( u + v ) − 2 ζ ( u ) − ζ ( u − v ) {\displaystyle {\frac {\wp '(u)}{\wp (u)-\wp (v)}}=\zeta (u+v)-2\zeta (u)-\zeta (u-v)} Once again by definition ζ ′ ( z ) = − ℘ ( z ) {\displaystyle \zeta '(z)=-\wp (z)} thus by differentiating once more on both sides and rearranging the terms we obtain − ℘ ( u + v ) = − ℘ ( u ) + 1 2 ℘ ″ ( v ) [ ℘ ( u ) − ℘ ( v ) ] − ℘ ′ ( u ) [ ℘ ′ ( u ) − ℘ ′ ( v ) ] [ ℘ ( u ) − ℘ ( v ) ] 2 {\displaystyle -\wp (u+v)=-\wp (u)+{\frac {1}{2}}{\frac {\wp ''(v)[\wp (u)-\wp (v)]-\wp '(u)[\wp '(u)-\wp '(v)]}{[\wp (u)-\wp (v)]^{2}}}} Knowing that ℘ ″ {\displaystyle \wp ''} has the following differential equation 2 ℘ ″ = 12 ℘ 2 − g 2 {\displaystyle 2\wp ''=12\wp ^{2}-g_{2}} and rearranging the terms one gets the wanted formula ℘ ( u + v ) = 1 4 [ ℘ ′ ( u ) − ℘ ′ ( v ) ℘ ( u ) − ℘ ( v ) ] 2 − ℘ ( u ) − ℘ ( v ) . {\displaystyle \wp (u+v)={\frac {1}{4}}\left[{\frac {\wp '(u)-\wp '(v)}{\wp (u)-\wp (v)}}\right]^{2}-\wp (u)-\wp (v).}
The Weierstrass's elliptic function is usually written with a rather special, lower case script letter ℘, which was Weierstrass's own notation introduced in his lectures of 1862–1863.[footnote 1] It should not be confused with the normal mathematical script letters P: 𝒫 and 𝓅.
In computing, the letter ℘ is available as \wp in TeX. In Unicode the code point is U+2118 ℘ SCRIPT CAPITAL P (℘, ℘), with the more correct alias weierstrass elliptic function.[footnote 2] In HTML, it can be escaped as ℘.
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