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JacobiZeta




Mathematica Notation

Traditional Notation









Elliptic Integrals > JacobiZeta[z,m] > Representations through more general functions > Through other functions > Involving some elliptic-type functions





http://functions.wolfram.com/08.07.26.0004.01









  


  








Input Form





JacobiZeta[z, m] == (1 - m) EllipticPi[m, z, m] - (EllipticE[m]/EllipticK[m]) EllipticPi[0, z, m] + (m Sin[2 z])/(2 Sqrt[1 - m Sin[z]^2])








Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> &#918; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mi> E </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#928; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> ; </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#928; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> ; </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> JacobiZeta </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> EllipticE </ci> <ci> m </ci> </apply> <apply> <ci> EllipticPi </ci> <cn type='integer'> 0 </cn> <ci> z </ci> <ci> m </ci> </apply> <apply> <power /> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <ci> EllipticPi </ci> <ci> m </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <ci> m </ci> <apply> <sin /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <sin /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29