Weierstrass sigma function: Differentiation (formula 09.15.20.0008)

WeierstrassSigma

$Mathematica Notation$

Elliptic Functions

WeierstrassSigma[z,{g₂,g₃}]

Differentiation

Low-order differentiation

With respect to g₂

http://functions.wolfram.com/09.15.20.0008.01

Input Form

D[WeierstrassSigma[z, {Subscript[g, 2], Subscript[g, 3]}], {Subscript[g, 2], 2}] == (1/(256 (Subscript[g, 2]^3 - 27 Subscript[g, 3]^2)^2)) (WeierstrassSigma[z, {Subscript[g, 2], Subscript[g, 3]}] (-24 z^2 Subscript[g, 2]^3 Subscript[g, 3] (-4 + z WeierstrassZeta[z, {Subscript[g, 2], Subscript[g, 3]}]) - 16 Subscript[g, 2]^4 (-5 + z^2 WeierstrassP[z, {Subscript[g, 2], Subscript[g, 3]}] + 5 z WeierstrassZeta[z, {Subscript[g, 2], Subscript[g, 3]}] - z^2 WeierstrassZeta[z, {Subscript[g, 2], Subscript[g, 3]}]^2) - 216 Subscript[g, 2] Subscript[g, 3]^2 (-20 + z^2 WeierstrassP[z, {Subscript[g, 2], Subscript[g, 3]}] + 14 z WeierstrassZeta[z, {Subscript[g, 2], Subscript[g, 3]}] - z^2 WeierstrassZeta[z, {Subscript[g, 2], Subscript[g, 3]}]^2) + 1296 Subscript[g, 3]^2 (z^2 Subscript[g, 3] - 3 WeierstrassP[z, {Subscript[g, 2], Subscript[g, 3]}]^2 - 4 WeierstrassPPrime[z, {Subscript[g, 2], Subscript[g, 3]}] WeierstrassZeta[z, {Subscript[g, 2], Subscript[g, 3]}] - 6 WeierstrassP[z, {Subscript[g, 2], Subscript[g, 3]}] WeierstrassZeta[z, {Subscript[g, 2], Subscript[g, 3]}]^2 + WeierstrassZeta[z, {Subscript[g, 2], Subscript[g, 3]}]^4) + 9 Subscript[g, 2]^2 Subscript[g, 3] (z^4 Subscript[g, 3] + 32 (z WeierstrassPPrime[z, {Subscript[g, 2], Subscript[g, 3]}] + WeierstrassZeta[z, {Subscript[g, 2], Subscript[g, 3]}]^2 (6 - z WeierstrassZeta[z, {Subscript[g, 2], Subscript[g, 3]}]) + 3 WeierstrassP[z, {Subscript[g, 2], Subscript[g, 3]}] (-2 + z WeierstrassZeta[z, {Subscript[g, 2], Subscript[g, 3]}])))))

Standard Form

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MathML Form

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<mo> ⁢ </mo> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <msup> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> ; </mo> <msub> <mi> g </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[RowBox[List[TagBox["z", Rule[Editable, True]], ";", TagBox[SubscriptBox["g", "2"], Rule[Editable, True]]]], ",", TagBox[SubscriptBox["g", "3"], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[WeierstrassZeta[Slot[1], List[Slot[2], Slot[3]]]]]] </annotation> </semantics> <mn> 2 </mn> </msup> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mi> ℘ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> ; </mo> <msub> <mi> g </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> g 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Date Added to functions.wolfram.com (modification date)

2001-10-29

WeierstrassSigma[n,z,{g₂,g₃}]