Associated Weierstrass sigma function: Differentiation (formula 09.16.20.0003)

WeierstrassSigma

$Mathematica Notation$

Elliptic Functions

WeierstrassSigma[n,z,{g₂,g₃}]

Differentiation

Symbolic differentiation

With respect to z

http://functions.wolfram.com/09.16.20.0003.01

Input Form

D[WeierstrassSigma[m, z, {Subscript[g, 2], Subscript[g, 3]}], {z, n}] == (-Subscript[\[Eta], m]) WeierstrassSigma[m, z, {Subscript[g, 2], Subscript[g, 3]}] + (((2 Subscript[\[Omega], 1])^(1 - n) Pi^(n - 1/2) Exp[(-Subscript[\[Eta], m]) z])/WeierstrassSigma[ Subscript[\[Omega], m], {Subscript[g, 2], Subscript[g, 3]}]) Product[1/(1 - q^(2 k)), {k, 1, Infinity}]^3 Sum[HypergeometricPFQRegularized[{1/2, 1}, {(1 - j)/2, (2 - j)/2}, ((z + Subscript[\[Omega], m])^2/(2 Subscript[\[Omega], 1])) WeierstrassZeta[Subscript[\[Omega], 1], {Subscript[g, 2], Subscript[g, 3]}]] ((4 Subscript[\[Omega], 1])/ (Pi (z + Subscript[\[Omega], m])))^j Binomial[n, j] Sum[(-1)^k q^(k (k + 1)) (1 + 2 k)^(n - j) Sin[(Pi ((z + Subscript[\[Omega], m]) (1 + 2 k) + (n - j) Subscript[\[Omega], 1]))/(2 Subscript[\[Omega], 1])], {k, 0, Infinity}], {j, 0, n}] /; Element[n, Integers] && n > 0

Standard Form

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MathML Form

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<ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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Date Added to functions.wolfram.com (modification date)

2001-10-29

WeierstrassSigma[z,{g₂,g₃}]