Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











JacobiAmplitude






Mathematica Notation

Traditional Notation









Elliptic Functions > JacobiAmplitude[z,m] > Specific values > Specialized values > Derivatives with respect to m > For m==0





http://functions.wolfram.com/09.24.03.0021.01









  


  










Input Form





Derivative[0, 6][JacobiAmplitude][z, 0] == (3/1048576) (-4133520 z - 16 z (447465 - 65600 z^2 + 512 z^4) Cos[2 z] + 80 z (-16983 + 3968 z^2) Cos[4 z] + 7680 z (-19 + 3 z^2) Cos[6 z] - 8880 z Cos[8 z] - 240 z Cos[10 z] + 80 (58953 - 48960 z^2 + 1792 z^4) Sin[2 z] + 5 (143865 - 193920 z^2 + 8192 z^4) Sin[4 z] - 320 (-235 + 306 z^2) Sin[6 z] - 120 (-45 + 32 z^2) Sin[8 z] + 240 Sin[10 z] + 5 Sin[12 z])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox["JacobiAmplitude", TagBox[RowBox[List["(", RowBox[List["0", ",", "6"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["z", ",", "0"]], "]"]], "\[Equal]", RowBox[List[FractionBox["3", "1048576"], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4133520"]], " ", "z"]], "-", RowBox[List["16", " ", "z", " ", RowBox[List["(", RowBox[List["447465", "-", RowBox[List["65600", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["512", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "z"]], "]"]]]], "+", RowBox[List["80", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "16983"]], "+", RowBox[List["3968", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["4", " ", "z"]], "]"]]]], "+", RowBox[List["7680", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "19"]], "+", RowBox[List["3", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["6", " ", "z"]], "]"]]]], "-", RowBox[List["8880", " ", "z", " ", RowBox[List["Cos", "[", RowBox[List["8", " ", "z"]], "]"]]]], "-", RowBox[List["240", " ", "z", " ", RowBox[List["Cos", "[", RowBox[List["10", " ", "z"]], "]"]]]], "+", RowBox[List["80", " ", RowBox[List["(", RowBox[List["58953", "-", RowBox[List["48960", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["1792", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "z"]], "]"]]]], "+", RowBox[List["5", " ", RowBox[List["(", RowBox[List["143865", "-", RowBox[List["193920", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["8192", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["4", " ", "z"]], "]"]]]], "-", RowBox[List["320", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "235"]], "+", RowBox[List["306", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["6", " ", "z"]], "]"]]]], "-", RowBox[List["120", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "45"]], "+", RowBox[List["32", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["8", " ", "z"]], "]"]]]], "+", RowBox[List["240", " ", RowBox[List["Sin", "[", RowBox[List["10", " ", "z"]], "]"]]]], "+", RowBox[List["5", " ", RowBox[List["Sin", "[", RowBox[List["12", " ", "z"]], "]"]]]]]], ")"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mi> am </mi> <semantics> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;0&quot;, &quot;,&quot;, &quot;6&quot;]], &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 1048576 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 16 </mn> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 512 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 65600 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 447465 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mn> 80 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3968 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 16983 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mn> 7680 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 19 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <mn> 8880 </mn> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <mn> 240 </mn> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 10 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <mn> 4133520 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mn> 80 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1792 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 48960 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 58953 </mn> </mrow> <mo> ) </mo> </mrow> <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> <mo> + </mo> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 8192 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 193920 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 143865 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 320 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 306 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 235 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 120 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 32 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 45 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 240 </mn> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 10 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 0 </cn> <cn type='integer'> 6 </cn> </list> <ci> am </ci> </apply> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <times /> <cn type='rational'> 3 <sep /> 1048576 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -16 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 512 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 65600 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 447465 </cn> </apply> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 80 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 3968 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -16983 </cn> </apply> <apply> <cos /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> z </ci> </apply> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 7680 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -19 </cn> </apply> <apply> <cos /> <apply> <times /> <cn type='integer'> 6 </cn> <ci> z </ci> </apply> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8880 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 8 </cn> <ci> z </ci> </apply> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 240 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 10 </cn> <ci> z </ci> </apply> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4133520 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 80 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 1792 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 48960 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 58953 </cn> </apply> <apply> <sin /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 8192 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 193920 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 143865 </cn> </apply> <apply> <sin /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 320 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 306 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -235 </cn> </apply> <apply> <sin /> <apply> <times /> <cn type='integer'> 6 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 120 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 32 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -45 </cn> </apply> <apply> <sin /> <apply> <times /> <cn type='integer'> 8 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 240 </cn> <apply> <sin /> <apply> <times /> <cn type='integer'> 10 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <sin /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["JacobiAmplitude", TagBox[RowBox[List["(", RowBox[List["0", ",", "6"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["z_", ",", "0"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["3", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4133520"]], " ", "z"]], "-", RowBox[List["16", " ", "z", " ", RowBox[List["(", RowBox[List["447465", "-", RowBox[List["65600", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["512", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "z"]], "]"]]]], "+", RowBox[List["80", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "16983"]], "+", RowBox[List["3968", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["4", " ", "z"]], "]"]]]], "+", RowBox[List["7680", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "19"]], "+", RowBox[List["3", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["6", " ", "z"]], "]"]]]], "-", RowBox[List["8880", " ", "z", " ", RowBox[List["Cos", "[", RowBox[List["8", " ", "z"]], "]"]]]], "-", RowBox[List["240", " ", "z", " ", RowBox[List["Cos", "[", RowBox[List["10", " ", "z"]], "]"]]]], "+", RowBox[List["80", " ", RowBox[List["(", RowBox[List["58953", "-", RowBox[List["48960", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["1792", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "z"]], "]"]]]], "+", RowBox[List["5", " ", RowBox[List["(", RowBox[List["143865", "-", RowBox[List["193920", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["8192", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["4", " ", "z"]], "]"]]]], "-", RowBox[List["320", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "235"]], "+", RowBox[List["306", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["6", " ", "z"]], "]"]]]], "-", RowBox[List["120", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "45"]], "+", RowBox[List["32", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["8", " ", "z"]], "]"]]]], "+", RowBox[List["240", " ", RowBox[List["Sin", "[", RowBox[List["10", " ", "z"]], "]"]]]], "+", RowBox[List["5", " ", RowBox[List["Sin", "[", RowBox[List["12", " ", "z"]], "]"]]]]]], ")"]]]], "1048576"]]]]]










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





2007-05-02





© 1998-2014 Wolfram Research, Inc.