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InverseJacobiSC






Mathematica Notation

Traditional Notation









Elliptic Functions > InverseJacobiSC[z,m] > Differentiation > Low-order differentiation > With respect to m





http://functions.wolfram.com/09.46.20.0017.01









  


  










Input Form





D[InverseJacobiSC[z, m], {m, 3}] == ((-(1 + z^2)) (-1 + (-1 + m) z^2) ((8 - 23 m + 23 m^2) (-1 + (-1 + m) z^2) EllipticE[JacobiAmplitude[InverseJacobiSC[z, m], m], m] + (-1 + m) (m z Sqrt[(1 + z^2 - m z^2)/(1 + z^2)] + (7 - 11 m + 7 z^2 - 18 m z^2 + 11 m^2 z^2) EllipticF[JacobiAmplitude[InverseJacobiSC[z, m], m], m])) - 15 (-1 + m)^3 (1 + z^2) (-1 + (-1 + m) z^2)^2 InverseJacobiSC[z, m] + m z (23 m^4 z^4 + 5 (1 + z^2)^2 - m^3 z^2 (35 + 59 z^2) + 3 m^2 (5 + 20 z^2 + 18 z^4) - m (12 + 35 z^2 + 23 z^4)) JacobiND[InverseJacobiSC[z, m], m])/(8 (-1 + m)^3 m^3 (1 + z^2) (-1 + (-1 + m) z^2)^2)










Standard Form





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MathML Form







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<mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> m </ci> <degree> <cn type='integer'> 3 </cn> </degree> </bvar> <apply> <ci> InverseJacobiSC </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -15 </cn> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> InverseJacobiSC </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 23 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 23 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> 8 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <ci> EllipticE </ci> <apply> <ci> JacobiAmplitude </ci> <apply> <ci> InverseJacobiSC </ci> <ci> z </ci> <ci> m </ci> </apply> <ci> m </ci> </apply> <ci> m </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 11 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 18 </cn> <ci> m </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 11 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> 7 </cn> </apply> <apply> <ci> EllipticF </ci> <apply> <ci> JacobiAmplitude </ci> <apply> <ci> InverseJacobiSC </ci> <ci> z </ci> <ci> m </ci> </apply> <ci> m </ci> </apply> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <ci> m </ci> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 23 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 59 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 35 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 18 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 20 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 23 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 35 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 12 </cn> </apply> </apply> </apply> </apply> <apply> <ci> JacobiND </ci> <apply> <ci> InverseJacobiSC </ci> <ci> z </ci> <ci> m </ci> </apply> <ci> m </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2007-05-02





© 1998- Wolfram Research, Inc.