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http://functions.wolfram.com/09.24.06.0010.01
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JacobiAmplitude[z, m] \[Proportional] JacobiAmplitude[z, Subscript[m, 0]] +
(1/(2 (-1 + Subscript[m, 0]) Subscript[m, 0]))
(((-1 + Subscript[m, 0]) z + EllipticE[JacobiAmplitude[z,
Subscript[m, 0]], Subscript[m, 0]]) JacobiDN[z, Subscript[m, 0]] -
Subscript[m, 0] JacobiCN[z, Subscript[m, 0]]
JacobiSN[z, Subscript[m, 0]]) (m - Subscript[m, 0]) -
(1/(8 (-1 + Subscript[m, 0])^2 Subscript[m, 0]^2))
((-((-1 + Subscript[m, 0]) z + EllipticE[JacobiAmplitude[z,
Subscript[m, 0]], Subscript[m, 0]])) JacobiDN[z, Subscript[m, 0]]^2
Sqrt[1 - Subscript[m, 0] JacobiSN[z, Subscript[m, 0]]^2] +
Subscript[m, 0] JacobiCN[z, Subscript[m, 0]]
JacobiSN[z, Subscript[m, 0]] (-2 Subscript[m, 0] +
(-1 + Subscript[m, 0])^2 z^2 +
EllipticE[JacobiAmplitude[z, Subscript[m, 0]], Subscript[m, 0]]
(2 (-1 + Subscript[m, 0]) z + EllipticE[JacobiAmplitude[z,
Subscript[m, 0]], Subscript[m, 0]]) - Subscript[m, 0]
JacobiCN[z, Subscript[m, 0]]^2 + Subscript[m, 0]
JacobiSN[z, Subscript[m, 0]]^2) + JacobiDN[z, Subscript[m, 0]]
(EllipticE[JacobiAmplitude[z, Subscript[m, 0]], Subscript[m, 0]]
(-1 + 3 Subscript[m, 0] + Subscript[m, 0]
JacobiCN[z, Subscript[m, 0]]^2 - 2 Subscript[m, 0]
JacobiSN[z, Subscript[m, 0]]^2) + Subscript[m, 0]
JacobiCN[z, Subscript[m, 0]] JacobiSN[z, Subscript[m, 0]]
Sqrt[1 - Subscript[m, 0] JacobiSN[z, Subscript[m, 0]]^2] +
(-1 + Subscript[m, 0]) (2 (-1 + Subscript[m, 0]) z +
EllipticF[JacobiAmplitude[z, Subscript[m, 0]], Subscript[m, 0]] +
Subscript[m, 0] z (JacobiCN[z, Subscript[m, 0]]^2 -
2 JacobiSN[z, Subscript[m, 0]]^2)))) (m - Subscript[m, 0])^2 +
\[Ellipsis] /; (m -> Subscript[m, 0])
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List["z", ",", SubscriptBox["m", "0"]]], "]"]], "+", RowBox[List[FractionBox["1", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubscriptBox["m", "0"]]], ")"]], " ", SubscriptBox["m", "0"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubscriptBox["m", "0"]]], ")"]], " ", "z"]], "+", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List["z", ",", SubscriptBox["m", "0"]]], "]"]], ",", SubscriptBox["m", "0"]]], "]"]]]], ")"]], " ", RowBox[List["JacobiDN", "[", RowBox[List["z", ",", SubscriptBox["m", "0"]]], "]"]]]], "-", RowBox[List[SubscriptBox["m", "0"], " ", RowBox[List["JacobiCN", "[", RowBox[List["z", ",", SubscriptBox["m", "0"]]], "]"]], 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RowBox[List["z", ",", SubscriptBox["m", "0"]]], "]"]], ",", SubscriptBox["m", "0"]]], "]"]], "+", RowBox[List[SubscriptBox["m", "0"], " ", "z", " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["JacobiCN", "[", RowBox[List["z", ",", SubscriptBox["m", "0"]]], "]"]], "2"], "-", RowBox[List["2", " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["z", ",", SubscriptBox["m", "0"]]], "]"]], "2"]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", SubscriptBox["m", "0"]]], ")"]], "2"]]], "+", "\[Ellipsis]"]]]], "/;", RowBox[List["(", RowBox[List["m", "\[Rule]", SubscriptBox["m", "0"]]], ")"]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> am </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ∝ </mo> <mrow> <mrow> <mi> am </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> am </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> m </mi> <mn> 0 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> dn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msub> <mi> m </mi> <mn> 0 </mn> </msub> <mo> ⁢ </mo> <mrow> <mi> cn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> m </mi> <mn> 0 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msub> <mi> m </mi> <mn> 0 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msubsup> <mi> m </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> am </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> m </mi> <mn> 0 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msub> <mi> m </mi> <mn> 0 </mn> </msub> <mo> ⁢ </mo> <msup> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <msup> <mrow> <mi> dn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> am </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> m </mi> <mn> 0 </mn> </msub> <mo> ⁢ </mo> <msup> <mrow> <mi> cn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> <mo> ⁢ </mo> <msup> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> F </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> am </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> m </mi> <mn> 0 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mi> cn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> m </mi> <mn> 0 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> cn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msub> <mi> m </mi> <mn> 0 </mn> </msub> <mo> ⁢ </mo> <msup> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> dn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> cn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mi> cn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msub> <mi> m </mi> <mn> 0 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> am </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> am </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z 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Date Added to functions.wolfram.com (modification date)
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