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 InverseJacobiCN

 http://functions.wolfram.com/09.38.06.0012.01

 Input Form

 InverseJacobiCN[z, m] == InverseJacobiCN[Subscript[z, 0], m] - (Pi/(JacobiDN[InverseJacobiCN[Subscript[z, 0], m], m] JacobiSN[InverseJacobiCN[Subscript[z, 0], m], m])) Sum[((2 Subscript[z, 0])^(k - 1)/k) Sum[(m^(k - j - 1) ((1 - m + m Subscript[z, 0]^2)^(1 + j - k)/ (j! (k - j - 1)! Gamma[1/2 - j] Gamma[3/2 + j - k])) Hypergeometric2F1[(1 - j)/2, -(j/2), 1/2 - j, 1 - 1/Subscript[z, 0]^2] Hypergeometric2F1[(2 + j - k)/2, (1 + j - k)/2, 3/2 + j - k, 1 + (1 - m)/(m Subscript[z, 0]^2)] (z - Subscript[z, 0])^k)/(Subscript[z, 0]^2 - 1)^j, {j, 0, k - 1}], {k, 1, Infinity}]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["InverseJacobiCN", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["InverseJacobiCN", "[", RowBox[List[SubscriptBox["z", "0"], ",", "m"]], "]"]], "-", " ", RowBox[List[FractionBox["\[Pi]", RowBox[List[RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["InverseJacobiCN", "[", RowBox[List[SubscriptBox["z", "0"], ",", "m"]], "]"]], ",", "m"]], "]"]], " ", RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["InverseJacobiCN", "[", RowBox[List[SubscriptBox["z", "0"], ",", "m"]], "]"]], ",", "m"]], "]"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["2", SubscriptBox["z", "0"]]], ")"]], RowBox[List["k", "-", "1"]]], "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["k", "-", "1"]]], " ", RowBox[List[SuperscriptBox["m", RowBox[List["k", "-", "j", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SubsuperscriptBox["z", "0", "2"], "-", "1"]], ")"]], RowBox[List["-", "j"]]], " ", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "m", "+", RowBox[List["m", " ", SubsuperscriptBox["z", "0", "2"]]]]], ")"]], RowBox[List["1", "+", "j", "-", "k"]]], "/", RowBox[List["(", RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j", "-", "1"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "j"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "j", "-", "k"]], "]"]]]], ")"]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "j"]], "2"], ",", RowBox[List["-", FractionBox["j", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "-", "j"]], ",", RowBox[List["1", "-", FractionBox["1", SubsuperscriptBox["z", "0", "2"]]]]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["2", "+", "j", "-", "k"]], "2"], ",", FractionBox[RowBox[List["1", "+", "j", "-", "k"]], "2"], ",", RowBox[List[FractionBox["3", "2"], "+", "j", "-", "k"]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["1", "-", "m"]], RowBox[List["m", " ", SubsuperscriptBox["z", "0", "2"]]]]]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "k"]]]]]]]]]]]]]]]]]

 MathML Form

 cn - 1 ( z m ) cn - 1 ( z 0 m ) - π dn ( cn - 1 ( z 0 m ) m ) sn ( cn - 1 ( z 0 m ) m ) k = 1 ( 2 z 0 ) k - 1 k j = 0 k - 1 m k - j - 1 ( z 0 2 - 1 ) - j ( m z 0 2 - m + 1 ) j - k + 1 j ! ( k - j - 1 ) ! Γ ( 1 2 - j ) Γ ( j - k + 3 2 ) 2 F 1 ( 1 - j 2 , - j 2 ; 1 2 - j ; 1 - 1 z 0 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["1", "-", "j"]], "2"], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["j", "2"]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], "-", "j"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List["1", "-", FractionBox["1", SubsuperscriptBox["z", "0", "2"]]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] 2 F 1 ( 1 2 ( j - k + 2 ) , 1 2 ( j - k + 1 ) ; j - k + 3 2 ; 1 - m m z 0 2 + 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["j", "-", "k", "+", "2"]], ")"]]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["j", "-", "k", "+", "1"]], ")"]]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["j", "-", "k", "+", FractionBox["3", "2"]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List[FractionBox[RowBox[List["1", "-", "m"]], RowBox[List["m", " ", SubsuperscriptBox["z", "0", "2"]]]], "+", "1"]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ( z - z 0 ) k InverseJacobiCN z m InverseJacobiCN Subscript z 0 m -1 JacobiDN InverseJacobiCN Subscript z 0 m m JacobiSN InverseJacobiCN Subscript z 0 m m -1 k 1 2 Subscript z 0 k -1 k -1 j 0 k -1 m k -1 j -1 Subscript z 0 2 -1 -1 j m Subscript z 0 2 -1 m 1 j -1 k 1 j k -1 j -1 Gamma 1 2 -1 j Gamma j -1 k 3 2 -1 Hypergeometric2F1 1 -1 j 2 -1 -1 j 2 -1 1 2 -1 j 1 -1 1 Subscript z 0 2 -1 Hypergeometric2F1 1 2 j -1 k 2 1 2 j -1 k 1 j -1 k 3 2 1 -1 m m Subscript z 0 2 -1 1 z -1 Subscript z 0 k [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["InverseJacobiCN", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["InverseJacobiCN", "[", RowBox[List[SubscriptBox["zz", "0"], ",", "m"]], "]"]], "-", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", SubscriptBox["zz", "0"]]], ")"]], RowBox[List["k", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["k", "-", "1"]]], FractionBox[RowBox[List[SuperscriptBox["m", RowBox[List["k", "-", "j", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SubsuperscriptBox["zz", "0", "2"], "-", "1"]], ")"]], RowBox[List["-", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "m", "+", RowBox[List["m", " ", SubsuperscriptBox["zz", "0", "2"]]]]], ")"]], RowBox[List["1", "+", "j", "-", "k"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "j"]], "2"], ",", RowBox[List["-", FractionBox["j", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "-", "j"]], ",", RowBox[List["1", "-", FractionBox["1", SubsuperscriptBox["zz", "0", "2"]]]]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "j", "-", "k"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "j", "-", "k"]], ")"]]]], ",", RowBox[List[FractionBox["3", "2"], "+", "j", "-", "k"]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["1", "-", "m"]], RowBox[List["m", " ", SubsuperscriptBox["zz", "0", "2"]]]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j", "-", "1"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "j"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "j", "-", "k"]], "]"]]]]]]]]], "k"]]]]], RowBox[List[RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["InverseJacobiCN", "[", RowBox[List[SubscriptBox["zz", "0"], ",", "m"]], "]"]], ",", "m"]], "]"]], " ", RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["InverseJacobiCN", "[", RowBox[List[SubscriptBox["zz", "0"], ",", "m"]], "]"]], ",", "m"]], "]"]]]]]]]]]]]

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

 2007-05-02