Amplitude: Specific values (formula 09.24.03.0030)

JacobiAmplitude

$Mathematica Notation$

Derivatives with respect to m

For m==1

http://functions.wolfram.com/09.24.03.0030.01

Input Form

Derivative[0, 5][JacobiAmplitude][z, 1] == (-(1/1048576)) Sech[z]^5 (-2555580 z + 127200 z^3 - 14720 z^5 + 8 z (-457725 + 9360 z^2 + 1216 z^4) Cosh[2 z] - 16 z (78525 + 3510 z^2 + 8 z^4) Cosh[4 z] - 120 z (1237 + 32 z^2) Cosh[6 z] + 1740 z Cosh[8 z] + 40 (30099 - 3789 z^2 + 1780 z^4) Sinh[2 z] - 10 (-95697 + 2304 z^2 + 256 z^4) Sinh[4 z] + 5 (46743 + 6936 z^2 + 32 z^4) Sinh[6 z] - 45 (49 + 8 z^2) Sinh[8 z] + 3 Sinh[10 z])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox["JacobiAmplitude", TagBox[RowBox[List["(", RowBox[List["0", ",", "5"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["z", ",", "1"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", "1048576"]]], SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "5"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2555580"]], " ", "z"]], "+", RowBox[List["127200", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["14720", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["8", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "457725"]], "+", RowBox[List["9360", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["1216", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]], "-", RowBox[List["16", " ", "z", " ", RowBox[List["(", RowBox[List["78525", "+", RowBox[List["3510", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["8", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["4", " ", "z"]], "]"]]]], "-", RowBox[List["120", " ", "z", " ", RowBox[List["(", RowBox[List["1237", "+", RowBox[List["32", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["6", " ", "z"]], "]"]]]], "+", RowBox[List["1740", " ", "z", " ", RowBox[List["Cosh", "[", RowBox[List["8", " ", "z"]], "]"]]]], "+", RowBox[List["40", " ", RowBox[List["(", RowBox[List["30099", "-", RowBox[List["3789", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["1780", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["2", " ", "z"]], "]"]]]], "-", RowBox[List["10", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "95697"]], "+", RowBox[List["2304", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["256", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["4", " ", "z"]], "]"]]]], "+", RowBox[List["5", " ", RowBox[List["(", RowBox[List["46743", "+", RowBox[List["6936", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["32", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["6", " ", "z"]], "]"]]]], "-", RowBox[List["45", " ", RowBox[List["(", RowBox[List["49", "+", RowBox[List["8", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["8", " ", "z"]], "]"]]]], "+", RowBox[List["3", " ", RowBox[List["Sinh", "[", RowBox[List["10", " ", "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> 5 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", RowBox[List["0", ",", "5"]], ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 1048576 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> sech </mi> <mn> 5 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 14720 </mn> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 127200 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1216 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 9360 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 457725 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <mn> 16 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3510 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 78525 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <mn> 120 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 32 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 1237 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mn> 1740 </mn> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <mn> 2555580 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mn> 40 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1780 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 3789 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 30099 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 10 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 256 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2304 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 95697 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 32 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 6936 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 46743 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 45 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 49 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 10 </mn> <mo> ⁢ </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'> 5 </cn> </list> <ci> am </ci> </apply> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 1048576 </cn> </apply> <apply> <power /> <apply> <sech /> <ci> z </ci> </apply> <cn type='integer'> 5 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -14720 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 127200 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 1216 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 9360 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -457725 </cn> </apply> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3510 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 78525 </cn> </apply> <apply> <cosh /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> z </ci> </apply> </apply> <ci> z </ci> </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'> 1237 </cn> </apply> <apply> <cosh /> <apply> <times /> <cn type='integer'> 6 </cn> <ci> z </ci> </apply> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 1740 </cn> <apply> <cosh /> <apply> <times /> <cn type='integer'> 8 </cn> <ci> z </ci> </apply> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2555580 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 40 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 1780 </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'> 3789 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 30099 </cn> </apply> <apply> <sinh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 10 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 256 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2304 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -95697 </cn> </apply> <apply> <sinh /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 32 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 6936 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 46743 </cn> </apply> <apply> <sinh /> <apply> <times /> <cn type='integer'> 6 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 45 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 49 </cn> </apply> <apply> <sinh /> <apply> <times /> <cn type='integer'> 8 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <sinh /> <apply> <times /> <cn type='integer'> 10 </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", ",", "5"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["z_", ",", "1"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "5"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2555580"]], " ", "z"]], "+", RowBox[List["127200", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["14720", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["8", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "457725"]], "+", RowBox[List["9360", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["1216", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]], "-", RowBox[List["16", " ", "z", " ", RowBox[List["(", RowBox[List["78525", "+", RowBox[List["3510", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["8", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["4", " ", "z"]], "]"]]]], "-", RowBox[List["120", " ", "z", " ", RowBox[List["(", RowBox[List["1237", "+", RowBox[List["32", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["6", " ", "z"]], "]"]]]], "+", RowBox[List["1740", " ", "z", " ", RowBox[List["Cosh", "[", RowBox[List["8", " ", "z"]], "]"]]]], "+", RowBox[List["40", " ", RowBox[List["(", RowBox[List["30099", "-", RowBox[List["3789", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["1780", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["2", " ", "z"]], "]"]]]], "-", RowBox[List["10", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "95697"]], "+", RowBox[List["2304", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["256", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["4", " ", "z"]], "]"]]]], "+", RowBox[List["5", " ", RowBox[List["(", RowBox[List["46743", "+", RowBox[List["6936", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["32", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["6", " ", "z"]], "]"]]]], "-", RowBox[List["45", " ", RowBox[List["(", RowBox[List["49", "+", RowBox[List["8", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["8", " ", "z"]], "]"]]]], "+", RowBox[List["3", " ", RowBox[List["Sinh", "[", RowBox[List["10", " ", "z"]], "]"]]]]]], ")"]]]], "1048576"]]]]]]]

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

2007-05-02