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 InverseJacobiND

 http://functions.wolfram.com/09.44.20.0010.01

 Input Form

 D[InverseJacobiND[z, m], {m, \[Alpha]}] == I Sum[(1/(k! Gamma[k - \[Alpha] + 1])) Pochhammer[1/2, k]^2 (PolyGamma[k + 1] - PolyGamma[1/2 + k]) m^k, {k, 0, Infinity}] - (I/(m^\[Alpha] 2)) Sum[(Pochhammer[1/2, k]^2 FDLogConstant[m, k, \[Alpha]] m^k)/k!^2, {k, 0, Infinity}] - ((I z Sqrt[Pi])/(m^\[Alpha] (2 Sqrt[1 - z^2]))) HypergeometricPFQRegularized[{{1/2}, {1/2}, {1/2, 1}}, {{3/2}, {}, {1 - \[Alpha]}}, z^2/(z^2 - 1), (m z^2)/(z^2 - 1)] /; -1 < z < 1 && -1 < m < 1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["m", ",", "\[Alpha]"]], "}"]]], RowBox[List["InverseJacobiND", "[", RowBox[List["z", ",", "m"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox["1", RowBox[List[RowBox[List["k", "!"]], RowBox[List["Gamma", "[", RowBox[List["k", "-", "\[Alpha]", "+", "1"]], "]"]]]]], " ", SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "-", " ", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k"]], "]"]]]], ")"]], SuperscriptBox["m", "k"]]]]]]], "-", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["m", RowBox[List["-", "\[Alpha]"]]]]], "2"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[" ", RowBox[List[SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], "2"], RowBox[List["FDLogConstant", "[", RowBox[List["m", ",", "k", ",", "\[Alpha]"]], "]"]], SuperscriptBox["m", "k"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["k", "!"]], ")"]], "2"]]]]]], "-", " ", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "z", " ", SuperscriptBox["m", RowBox[List["-", "\[Alpha]"]]], " ", SqrtBox["\[Pi]"]]], RowBox[List["2", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", "1"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox["3", "2"], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["1", "-", "\[Alpha]"]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]], ",", FractionBox[RowBox[List["m", " ", SuperscriptBox["z", "2"]]], RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "<", "z", "<", "1"]], "\[And]", RowBox[List[RowBox[List["-", "1"]], "<", "m", "<", "1"]]]]]]]]

 MathML Form

 α nd - 1 ( z m ) m α - z m - α π 2 1 - z 2 F ~ 1 0 1 1 1 2 ( 1 2 ; 1 2 ; 1 2 , 1 ; 3 2 ; ; 1 - α ; z 2 z 2 - 1 , m z 2 z 2 - 1 ) + k = 0 ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] 2 ( ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( k + 1 2 ) ) m k k ! Γ ( k - α + 1 ) - m - α 2 k = 0 ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] 2 ℱ𝒞 log ( α ) ( m , k ) m k k ! 2 /; - 1 < z < 1 - 1 < m < 1 FormBox RowBox RowBox FractionBox RowBox SuperscriptBox α RowBox SuperscriptBox nd RowBox - 1 ( RowBox z m ) RowBox SuperscriptBox m α RowBox RowBox RowBox - FractionBox RowBox z SuperscriptBox m RowBox - α SqrtBox π RowBox 2 SqrtBox RowBox 1 - SuperscriptBox z 2 RowBox SubsuperscriptBox OverscriptBox F ~ RowBox 1 0 1 RowBox 1 1 2 [ RowBox GridBox ErrorBox RowBox RowBox FractionBox 1 2 ; FractionBox 1 2 ; FractionBox 1 2 , RowBox 1 ; RowBox RowBox FractionBox 3 2 ; ; RowBox 1 - α ; FractionBox SuperscriptBox z 2 RowBox SuperscriptBox z 2 - 1 , FractionBox RowBox m SuperscriptBox z 2 RowBox SuperscriptBox z 2 - 1 ] + RowBox RowBox UnderoverscriptBox RowBox k = 0 FractionBox RowBox SuperscriptBox TagBox SubscriptBox RowBox ( FractionBox 1 2 ) k Pochhammer 2 RowBox ( RowBox RowBox TagBox ψ PolyGamma ( RowBox k + 1 ) - RowBox TagBox ψ PolyGamma ( RowBox k + FractionBox 1 2 ) ) SuperscriptBox m k RowBox RowBox k ! RowBox Γ ( RowBox k - α + 1 ) - RowBox FractionBox RowBox SuperscriptBox m RowBox - α 2 RowBox UnderoverscriptBox RowBox k = 0 FractionBox RowBox SuperscriptBox TagBox SubscriptBox RowBox ( FractionBox 1 2 ) k Pochhammer 2 RowBox SubsuperscriptBox ℱ𝒞 log RowBox ( α ) ( RowBox m , k ) SuperscriptBox m k SuperscriptBox RowBox k ! 2 /; RowBox RowBox RowBox - 1 < z < 1 RowBox RowBox - 1 < m < 1 TraditionalForm [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["m_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["InverseJacobiND", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k"]], "]"]]]], ")"]], " ", SuperscriptBox["m", "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "-", "\[Alpha]", "+", "1"]], "]"]]]]]]]]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["m", RowBox[List["-", "\[Alpha]"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], "2"], " ", RowBox[List["FDLogConstant", "[", RowBox[List["m", ",", "k", ",", "\[Alpha]"]], "]"]], " ", SuperscriptBox["m", "k"]]], SuperscriptBox[RowBox[List["(", RowBox[List["k", "!"]], ")"]], "2"]]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "z", " ", SuperscriptBox["m", RowBox[List["-", "\[Alpha]"]]], " ", SqrtBox["\[Pi]"]]], ")"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", "1"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox["3", "2"], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["1", "-", "\[Alpha]"]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]], ",", FractionBox[RowBox[List["m", " ", SuperscriptBox["z", "2"]]], RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]]]], "]"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "<", "z", "<", "1"]], "&&", RowBox[List[RowBox[List["-", "1"]], "<", "m", "<", "1"]]]]]]]]]]

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

 2001-10-29