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| In [[mathematics]], '''Weierstrass's elliptic functions''' are [[elliptic function]]s that take a particularly simple form; they are named for [[Karl Weierstrass]]. This class of functions are also referred to as '''P-functions''' and generally written using the symbol ℘ (or <math>\wp</math>), and known as "[[Weierstrass P]]").
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| <div class="thumb tright">
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| <div style="width:131px;">
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| [[Image:Weierstrass p.svg|100px|Symbol for Weierstrass P function]]<div class="thumbcaption">
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| Symbol for Weierstrass P function
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| </div>
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| </div>
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| </div>
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| [[File:Modell der Weierstraßschen p-Funktion -Schilling, XIV, 7ab, 8 - 313, 314-.jpg|thumb|right|Model of Weierstrass P-function]]
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| ==Definitions==
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| [[File:Weierstrass elliptic function P.png|thumb|200px|Weierstrass P function defined over a subset of the complex plane using a standard visualization technique in which white corresponds to a pole, black to a zero, and maximal [[saturation (color theory)|saturation]] to <math>\left|f(z)\right|=\left|f(x+iy)\right|=1\;.</math> Note the regular lattice of poles, and two interleaving lattices of zeros.]]
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| The '''Weierstrass elliptic function''' can be defined in three closely related ways, each of which possesses certain advantages. One is as a function of a complex variable ''z'' and a [[lattice (group)|lattice]] Λ in the complex plane. Another is in terms of ''z'' and two complex numbers ω<sub>1</sub> and ω<sub>2</sub> defining a pair of generators, or periods, for the lattice. The third is in terms ''z'' and of a modulus τ in the [[upper half-plane]]. This is related to the previous definition by τ = ω<sub>2</sub>/ω<sub>1</sub>, which by the conventional choice on the pair of periods is in the upper half-plane. Using this approach, for fixed ''z'' the Weierstrass functions become [[modular function]]s of τ.
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| In terms of the two periods, '''Weierstrass's elliptic function''' is an elliptic function with periods ω<sub>1</sub> and ω<sub>2</sub> defined as
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| :<math>
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| \wp(z;\omega_1,\omega_2)=\frac{1}{z^2}+
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| \sum_{n^2+m^2 \ne 0}
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| \left\{
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| \frac{1}{(z+m\omega_1+n\omega_2)^2}-
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| \frac{1}{\left(m\omega_1+n\omega_2\right)^2}
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| \right\}.
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| </math>
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| Then <math>\Lambda=\{m\omega_1+n\omega_2:m,n\in\mathbb{Z}\}</math> are the points of the '''[[period lattice]]''', so that
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| :<math>\wp(z;\Lambda)=\wp(z;\omega_1,\omega_2)</math>
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| for any pair of generators of the lattice defines the Weierstrass function as a function of a complex variable and a lattice.
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| If <math>\tau</math> is a complex number in the upper half-plane, then
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| :<math>\wp(z;\tau) = \wp(z;1,\tau) = \frac{1}{z^2} + \sum_{n^2+m^2 \ne 0}\left\{
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| {1 \over (z+m+n\tau)^2} - {1 \over (m+n\tau)^2}\right\}.</math>
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| The above sum is homogeneous of degree minus two, from which we may define the Weierstrass ℘ function for any pair of periods, as | |
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| :<math>\wp(z;\omega_1,\omega_2) = \frac{\wp(\frac{z}{\omega_1}; \frac{\omega_2}{\omega_1})}{\omega_1^2}.</math>
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| We may compute ℘ very rapidly in terms of [[theta function]]s; because these converge so quickly, this is a more expeditious way of computing
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| ℘ than the series we used to define it. The formula here is
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| :<math>\wp(z; \tau) = \pi^2 \vartheta^2(0;\tau) \vartheta_{10}^2(0;\tau){\vartheta_{01}^2(z;\tau) \over \vartheta_{11}^2(z;\tau)}-{\pi^2 \over {3}}\left[\vartheta^4(0;\tau) + \vartheta_{10}^4(0;\tau)\right] </math>
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| There is a second-order [[pole (complex analysis)|pole]] at each point of the period lattice (including the origin). With these definitions, <math>\wp(z)</math> is an even function and its derivative with respect to ''z'', ℘′, an odd function.
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| Further development of the theory of elliptic functions shows that the condition on Weierstrass's function is determined up to addition of a constant and multiplication by a non-zero constant by the condition on the poles alone, amongst all [[meromorphic function]]s with the given period lattice.
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| ==Invariants==
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| [[Image:Gee three real.jpeg|thumb|The real part of the invariant ''g''<sub>3</sub> as a function of the nome ''q'' on the unit disk.]]
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| [[Image:Gee three imag.jpeg|thumb|The imaginary part of the invariant ''g''<sub>3</sub> as a function of the nome ''q'' on the unit disk.]]
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| In a deleted neighborhood of the origin, the [[Laurent series]] expansion of <math>\wp</math> is
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| :<math>
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| \wp(z;\omega_1,\omega_2)=z^{-2}+\frac{1}{20}g_2z^2+\frac{1}{28}g_3z^4+O(z^6)
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| </math>
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| where
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| :<math>g_2= 60\sum_{(m,n) \neq (0,0)} (m\omega_1+n\omega_2)^{-4} </math>
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| and
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| :<math> g_3=140\sum_{(m,n) \neq (0,0)} (m\omega_1+n\omega_2)^{-6}.</math>
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| The numbers ''g''<sub>2</sub> and ''g''<sub>3</sub> are known as the ''invariants''. The summations after the coefficients 60 and 140 are the first two [[Eisenstein series]], which are [[modular forms]] when considered as functions G<sub>4</sub>(τ) and G<sub>6</sub>(τ), respectively, of τ = ω<sub>2</sub>/ω<sub>1</sub> with Im(τ) > 0.
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| Note that ''g''<sub>2</sub> and ''g''<sub>3</sub> are [[homogeneous function]]s of degree −4 and −6; that is,
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| :<math>g_2(\lambda \omega_1, \lambda \omega_2) = \lambda^{-4} g_2(\omega_1, \omega_2)</math>
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| and
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| :<math>g_3(\lambda \omega_1, \lambda \omega_2) = \lambda^{-6} g_3(\omega_1, \omega_2).</math>
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| Thus, by convention, one frequently writes <math>g_2</math> and <math>g_3</math> in terms of the [[nome (mathematics)|period ratio]] <math>\tau=\omega_2/\omega_1</math> and take <math>\tau</math> to lie in the [[upper half-plane]]. Thus, <math>g_2(\tau)=g_2(1, \omega_2/\omega_1)</math> and <math>g_3(\tau)=g_3(1, \omega_2/\omega_1)</math>.
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| The [[Fourier series]] for <math>g_2</math> and <math>g_3</math> can be written in terms of the square of the [[nome (mathematics)|nome]] <math>q=\exp(i\pi\tau)</math> as
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| :<math>g_2(\tau)=\frac{4\pi^4}{3} \left[ 1+ 240\sum_{k=1}^\infty \sigma_3(k) q^{2k} \right] </math>
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| and
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| :<math>g_3(\tau)=\frac{8\pi^6}{27} \left[ 1- 504\sum_{k=1}^\infty \sigma_5(k) q^{2k} \right] </math>
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| where <math>\sigma_a(k)</math> is the [[divisor function]]. This formula may be rewritten in terms of [[Lambert series]].
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| The invariants may be expressed in terms of [[theta functions|Jacobi's theta functions]]. This method is very convenient for numerical calculation: the theta functions converge very quickly. In the notation of Abramowitz and Stegun, but denoting the primitive half-periods by <math>\omega_1,\omega_2</math>, the invariants satisfy
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| :<math>
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| g_2(\omega_1,\omega_2)=
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| \frac{\pi^4}{12\omega_1^4}
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| \left(
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| \theta_2(0,q)^8-\theta_3(0,q)^4\theta_2(0,q)^4+\theta_3(0,q)^8
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| \right)
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| </math>
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| and | |
| :<math>
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| g_3(\omega_1,\omega_2)=
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| \frac{\pi^6}{(2\omega_1)^6}\frac{8}{27}
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| \left[
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| \left(\theta_2(0,q)^{12}+\theta_3(0,q)^{12}\right)\right.
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| </math>
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| ::::<math>\left. {} -
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| 33\left(\theta_2(0,q)^4+\theta_3(0,q)^4\right)\cdot
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| \theta_2(0,q)^4\theta_3(0,q)^4
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| \right]
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| </math>
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| where <math>\tau=\omega_2/\omega_1</math> is the [[half-period ratio|period ratio]] and <math>q=e^{\pi i\tau}</math> is the nome.
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| === Special cases ===
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| If the invariants are ''g''<sub>2</sub> = 0, ''g''<sub>3</sub> = 1, then this is known as the [[equianharmonic]] case; ''g''<sub>2</sub> = 1, ''g''<sub>3</sub> = 0 is the [[lemniscatic elliptic function|lemniscatic]] case.
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| ==Differential equation==
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| With this notation, the ℘ function satisfies the following [[differential equation]]:
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| :<math> [\wp'(z)]^2 = 4[\wp(z)]^3-g_2\wp(z)-g_3, \, </math>
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| where dependence on <math>\omega_1</math> and <math>\omega_2</math> is suppressed.
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| This relation can be quickly verified by comparing the poles of both sides, for example, the pole at ''z'' = 0 of lhs is
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| :<math>
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| [\wp'(z)]^2|_{z=0}\sim \frac{4}{z^6}-\frac{24}{z^2}\sum \frac{1}{(m\omega_1+n\omega_2)^4}-80\sum \frac{1}{(m\omega_1+n\omega_2)^6} </math>
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| while the pole at ''z'' = 0 of
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| :<math>
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| [\wp(z)]^3|_{z=0}\sim \frac{1}{z^6}+\frac{9}{z^2}\sum \frac{1}{(m\omega_1+n\omega_2)^4}+15\sum \frac{1}{(m\omega_1+n\omega_2)^6}.</math>
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| Comparing these two yields the relation above.
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| ==Integral equation==
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| The Weierstrass elliptic function can be given as the inverse of an [[elliptic integral]]. Let
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| :<math>u = \int_y^\infty \frac {ds} {\sqrt{4s^3 - g_2s -g_3}}.</math>
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| Here, ''g''<sub>2</sub> and ''g''<sub>3</sub> are taken as constants. Then one has
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| :<math>y=\wp(u).</math>
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| The above follows directly by integrating the differential equation.
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| ==Modular discriminant==
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| [[Image:Discriminant real part.jpeg|thumb|The real part of the discriminant as a function of the nome ''q'' on the unit disk.]]
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| The ''modular discriminant'' Δ is defined as the quotient by 16 of the [[discriminant]] of the right-hand side of the above differential equation:
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| :<math> \Delta=g_2^3-27g_3^2. \, </math>
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| This is studied in its own right, as a [[cusp form]], in [[modular form]] theory (that is, as a ''function of the period lattice'').
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| Note that <math>\Delta=(2\pi)^{12}\eta^{24}</math> where <math>\eta</math> is the [[Dedekind eta function]].
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| The presence of [[24 (number)|24]] can be understood by connection with other occurrences, as in the eta function and the Leech lattice.
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| The discriminant is a modular form of weight 12. That is, under the action of the [[modular group]], it transforms as
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| :<math>\Delta \left( \frac {a\tau+b} {c\tau+d}\right) =
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| \left(c\tau+d\right)^{12} \Delta(\tau) </math>
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| with τ being the half-period ratio, and ''a'',''b'',''c'' and ''d'' being integers, with ''ad'' − ''bc'' = 1.
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| For the Fourier coefficients of <math>\Delta</math>, see [[Ramanujan tau function]].
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| ==The constants ''e''<sub>1</sub>, ''e''<sub>2</sub> and ''e''<sub>3</sub>==
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| Consider the [[cubic function|cubic polynomial equation]] 4''t''<sup>3</sup> − ''g''<sub>2</sub>''t'' − ''g''<sub>3</sub> = 0 with roots ''e''<sub>1</sub>, ''e''<sub>2</sub>, and ''e''<sub>3</sub>. Its discriminant is 16 times the modular discriminant Δ = ''g''<sub>2</sub><sup>3</sup> − 27''g''<sub>3</sub><sup>2</sup>. If it is not zero, no two of these roots are equal. Since the quadratic term of this cubic polynomial is zero, the roots are related by the equation
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| :<math>
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| e_1+e_2+e_3=0. \,
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| </math>
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| The linear and constant coefficients (''g''<sub>2</sub> and ''g''<sub>3</sub>, respectively) are related to the roots by the equations (see [[Elementary symmetric polynomial]]).<ref>Abramowitz and Stegun, p. 629</ref>
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|
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| :<math>
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| g_2 = -4 \left( e_1 e_2 + e_1 e_3 + e_2 e_3 \right) = 2 \left( e_1^2 + e_2^2 + e_3^2 \right) \,
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| </math>
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| :<math>
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| g_3 = 4 e_1 e_2 e_3. \,
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| </math>
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| In the case of real invariants, the sign of <math>\Delta</math> determines the nature of the roots. If <math>\Delta>0</math>, all three are real and it is conventional to name them so that <math>e_1>e_2>e_3</math>. If <math>\Delta<0</math>, it is conventional to write <math>e_1=-\alpha+\beta i</math> (where <math>\alpha\geq 0</math>, <math>\beta>0</math>), whence <math>e_3=\overline{e_1}</math> and <math>e_2</math> is real and non-negative.
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| The half-periods ω<sub>1</sub>/2 and ω<sub>2</sub>/2 of Weierstrass' elliptic function are related to the roots
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| :<math>
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| \wp(\omega_1/2)=e_1\qquad
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| \wp(\omega_2/2)=e_2\qquad
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| \wp(\omega_3/2)=e_3
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| </math>
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| where <math>\omega_3=-(\omega_1+\omega_2)</math>. Since the square of the derivative of Weierstrass's elliptic function equals the above cubic polynomial of the function's value, <math>\wp'(\omega_i/2)^2=\wp'(\omega_i/2)=0</math> for <math>i=1,2,3</math>. Conversely, if the function's value equals a root of the polynomial, the derivative is zero.
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| If ''g''<sub>2</sub> and ''g''<sub>3</sub> are real and Δ > 0, the ''e''<sub>''i''</sub> are all real, and <math>\wp()</math> is real on the perimeter of the rectangle with corners 0, ω<sub>3</sub>, ω<sub>1</sub> + ω<sub>3</sub>, and ω<sub>1</sub>. If the roots are ordered as above (''e''<sub>1</sub> > ''e''<sub>2</sub> > ''e''<sub>3</sub>), then the first half-period is completely real
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| :<math>
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| \omega_{1}/2 = \int_{e_{1}}^{\infty} \frac{dz}{\sqrt{4z^{3} - g_{2}z - g_{3}}}
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| </math>
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| whereas the third half-period is completely imaginary
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| :<math>
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| \omega_{3}/2 = i \int_{-e_{3}}^{\infty} \frac{dz}{\sqrt{4z^{3} - g_{2}z - g_{3}}}.
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| </math>
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| ==Addition theorems==
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| The Weierstrass elliptic functions have several properties that may be proved:
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| :<math>
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| \det\begin{bmatrix}
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| \wp(z) & \wp'(z) & 1\\
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| \wp(y) & \wp'(y) & 1\\
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| \wp(z+y) & -\wp'(z+y) & 1
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| \end{bmatrix}=0</math>
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| (a symmetrical version would be
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| :<math>
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| \det\begin{bmatrix}
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| \wp(u) & \wp'(u) & 1\\
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| \wp(v) & \wp'(v) & 1\\
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| \wp(w) & \wp'(w) & 1
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| \end{bmatrix}=0</math>
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| where ''u'' + ''v'' + ''w'' = 0).
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| Also | |
| :<math>
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| \wp(z+y)=\frac{1}{4}
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| \left\{
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| \frac{\wp'(z)-\wp'(y)}{\wp(z)-\wp(y)}
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| \right\}^2
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| -\wp(z)-\wp(y).</math>
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| and the ''duplication formula''
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| :<math>
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| \wp(2z)=
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| \frac{1}{4}\left\{
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| \frac{\wp''(z)}{\wp'(z)}\right\}^2-2\wp(z),</math>
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| unless 2''z'' is a period.
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| ==The case with 1 a basic half-period==
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| If <math>\omega_1=1</math>, much of the above theory becomes simpler; it is then conventional to
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| write <math>\tau</math> for <math>\omega_2</math>. For a fixed τ in the [[upper half-plane]], so that the imaginary part of τ is positive, we define the
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| '''Weierstrass ℘ function''' by
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| :<math>\wp(z;\tau) =\frac{1}{z^2} + \sum_{(m,n) \ne (0,0)}{1 \over (z+m+n\tau)^2} - {1 \over (m+n\tau)^2}.</math>
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| The sum extends over the [[lattice (group)|lattice]] {''n''+''m''τ : ''n'' and ''m'' in '''Z'''} with the origin omitted.
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| Here we regard τ as fixed and ℘ as a function of ''z''; fixing ''z'' and letting τ vary leads into the area of [[elliptic modular function]]s.
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| ==General theory==
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| ℘ is a [[meromorphic]] function in the complex plane with a double [[pole (complex analysis)|pole]] at each lattice points. It is doubly periodic with periods 1 and τ; this means that
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| ℘ satisfies
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| :<math>\wp(z+1) = \wp(z+\tau) = \wp(z).</math>
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| The above sum is homogeneous of degree minus two, and if ''c'' is any non-zero complex number,
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| :<math>\wp(cz;c\tau) = \wp(z;\tau)/c^2</math>
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| from which we may define the Weierstrass ℘ function for any pair of periods. We also may take the [[derivative]] (of course, with respect to ''z'') and obtain a function algebraically related to ℘ by
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| :<math>\wp'^2 = 4\wp^3 - g_2 \wp - g_3</math> | |
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| where <math>g_2</math> and <math>g_3</math> depend only on τ, being [[modular forms]]. The equation
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| :<math>Y^2 = 4 X^3 - g_2 X - g_3</math>
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| defines an [[elliptic curve]], and we see that <math>(\wp, \wp')</math> is a parametrization of that curve.
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| The totality of meromorphic doubly periodic functions with given periods defines an [[algebraic function field]], associated to that curve. It can be shown that this field is
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| :<math>\Bbb{C}(\wp, \wp'),</math>
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| so that all such functions are [[rational function]]s in the Weierstrass function and its derivative.
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| We can also wrap a single period parallelogram into a [[torus]], or donut-shaped [[Riemann surface]], and regard the elliptic functions associated to a given pair of periods to be functions defined on that Riemann surface.
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| The roots ''e''<sub>1</sub>, ''e''<sub>2</sub>, and ''e''<sub>3</sub> of the equation <math>4 X^3 - g_2 X - g_3</math> depend on τ and can be expressed in terms of [[theta function]]s; we have
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| :<math>e_1(\tau) = \tfrac{1}{3} \pi^2(\vartheta^4(0;\tau) + \vartheta_{01}^4(0;\tau)),</math>
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| :<math>e_2(\tau) = -\tfrac{1}{3} \pi^2(\vartheta^4(0;\tau) + \vartheta_{10}^4(0;\tau)),</math>
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| :<math>e_3(\tau) = \tfrac{1}{3} \pi^2(\vartheta_{10}^4(0;\tau) - \vartheta_{01}^4(0;\tau)).</math>
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| Since <math>g_2 = -4(e_1e_2+e_2e_3+e_3e_1)</math> and <math>g_3 = 4e_1e_2e_3</math> we have these in terms of theta functions also.
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| We may also express ℘ in terms of theta functions; because these converge very rapidly, this is a more expeditious way of computing ℘ than the series we used to define it.
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| :<math>\wp(z; \tau) = \pi^2 \vartheta^2(0;\tau) \vartheta_{10}^2(0;\tau){\vartheta_{01}^2(z;\tau) \over \vartheta_{11}^2(z;\tau)} + e_2(\tau).</math>
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| The function ℘ has two zeros ([[Modulo (jargon)|modulo]] periods) and the function ℘′ has three. The zeros of ℘′ are easy to find: since ℘′ is an odd function they must be at the half-period points. On the other hand it is very difficult to express the zeros of ℘ by [[closed formula]], except for special values of the modulus (e.g. when the period lattice is the [[Gaussian integer]]s). An expression was found, by [[Don Zagier|Zagier]] and [[Martin Eichler|Eichler]].<ref>{{cite journal |first=M. |last=Eichler |first2=D. |last2=Zagier |title=On the zeros of the Weierstrass ℘-Function |journal=[[Mathematische Annalen]] |volume=258 |issue=4 |year=1982 |pages=399–407 |doi=10.1007/BF01453974 }}</ref>
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| The Weierstrass theory also includes the [[Weierstrass zeta function]], which is an indefinite integral of ℘ and not doubly periodic, and a theta function called the [[Weierstrass sigma function]], of which his zeta-function is the [[log-derivative]]. The sigma-function has zeros at all the period points (only), and can be expressed in terms of [[Jacobi's elliptic functions|Jacobi's functions]]. This gives one way to convert between Weierstrass and Jacobi notations.
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| The Weierstrass sigma-function is an [[entire function]]; it played the role of 'typical' function in a theory of ''random entire functions'' of [[J. E. Littlewood]].
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| ==Relation to Jacobi elliptic functions==
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| For numerical work, it is often convenient to calculate the Weierstrass elliptic function in terms of the [[Jacobi's elliptic functions]]. The basic relations are<ref>{{cite book | author = Korn GA, Korn TM | year = 1961 | title = Mathematical Handbook for Scientists and Engineers | publisher = McGraw-Hill | location = New York | pages = 721 | lccn = 5914456}}</ref>
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| :<math>
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| \wp(z) = e_{3} + \frac{e_{1} - e_{3}}{\mathrm{sn}^{2}\,w}
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| = e_{2} + \left( e_{1} - e_{3} \right) \frac{\mathrm{dn}^{2}\,w}{\mathrm{sn}^{2}\,w}
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| = e_{1} + \left( e_{1} - e_{3} \right) \frac{\mathrm{cn}^{2}\,w}{\mathrm{sn}^{2}\,w}
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| </math>
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| where ''e''<sub>1–3</sub> are the three roots described above and where the modulus ''k'' of the Jacobi functions equals
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| :<math>
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| k \equiv \sqrt{\frac{e_{2} - e_{3}}{e_{1} - e_{3}}}
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| </math>
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| and their argument ''w'' equals
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| :<math>
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| w \equiv z \sqrt{e_{1} - e_{3}}.
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| </math>
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| ==Notes==
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| {{Reflist}}
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| == References ==
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| *{{AS ref|18|627}}
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| * [[Naum Akhiezer|N. I. Akhiezer]], ''Elements of the Theory of Elliptic Functions'', (1970) Moscow, translated into English as ''AMS Translations of Mathematical Monographs Volume 79'' (1990) AMS, Rhode Island ISBN 0-8218-4532-2
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| * [[Tom M. Apostol]], ''Modular Functions and Dirichlet Series in Number Theory, Second Edition'' (1990), Springer, New York ISBN 0-387-97127-0 (See chapter 1.)
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| * K. Chandrasekharan, ''Elliptic functions'' (1980), Springer-Verlag ISBN 0-387-15295-4
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| * [[Konrad Knopp]], ''Funktionentheorie II'' (1947), Dover; Republished in English translation as ''Theory of Functions'' (1996), Dover ISBN 0-486-69219-1
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| * [[Serge Lang]], ''Elliptic Functions'' (1973), Addison-Wesley, ISBN 0-201-04162-6
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| *{{dlmf|first=William P. |last=Reinhardt|first2=Peter L. |last2=Walker|id=23|title=Weierstrass Elliptic and Modular Functions}}
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| * [[E. T. Whittaker]] and [[G. N. Watson]], ''A course of modern analysis'', [[Cambridge University Press]], 1952, chapters 20 and 21
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| ==External links==
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| {{commonscat|Weierstrass's elliptic functions}}
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| * {{springer|title=Weierstrass elliptic functions|id=p/w097450}}
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| * [http://mathworld.wolfram.com/WeierstrassEllipticFunction.html Weierstrass's elliptic functions on Mathworld].
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| * [http://www.mai.liu.se/~halun/complex/elliptic/ Elliptic functions, Hans Lundmark's Complex analysis page].
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| [[Category:Modular forms]]
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| [[Category:Algebraic curves]]
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| [[Category:Elliptic functions]]
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