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 ArcCsch

 http://functions.wolfram.com/01.29.06.0055.01

 Input Form

 ArcCsch[z]^2 == Subscript[F, Infinity][z] /; Subscript[F, n][z] == (-((Pi I)/2) + 2 I Sqrt[(-I) (z - I)] Sum[((I^k Pochhammer[1/2, k])/k!) Hypergeometric2F1[1/2, 3/2 + k, 3/2, -1] (z - I)^k, {k, 0, n}])^2 == (ArcCsch[z] + ((2 I^n Pochhammer[1/2, n + 1])/(n + 1)!) (z - I)^(n + 1) Sqrt[(-I) (z - I)] HypergeometricPFQ[{{5/2 + n}, {1/2}, {1, 3/2 + n}}, {{}, {3/2}, {2 + n, 5/2 + n}}, -1, I (z - I)])^2 && Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["ArcCsch", "[", "z", "]"]], "2"], "\[Equal]", RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", "z", "]"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", "z", "]"]], "\[Equal]", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["z", "-", "\[ImaginaryI]"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ImaginaryI]", "k"], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]]]], RowBox[List["k", "!"]]], RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[FractionBox["3", "2"], "+", "k"]], ",", FractionBox["3", "2"], ",", RowBox[List["-", "1"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "\[ImaginaryI]"]], ")"]], "k"]]]]]]]]], ")"]], "2"], "\[Equal]", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["ArcCsch", "[", "z", "]"]], "+", " ", RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ImaginaryI]", "n"], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["n", "+", "1"]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "\[ImaginaryI]"]], ")"]], RowBox[List["n", "+", "1"]]], SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["z", "-", "\[ImaginaryI]"]], ")"]]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["5", "2"], "+", "n"]], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["3", "2"], "+", "n"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List[FractionBox["5", "2"], "+", "n"]]]], "}"]]]], "}"]], ",", RowBox[List["-", "1"]], ",", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["z", "-", "\[ImaginaryI]"]], ")"]]]]]], "]"]]]]]], ")"]], "2"]]], StyleBox[")", Rule[FontWeight, "Plain"]]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]], ")"]]]]]]

 MathML Form

 csch - 1 ( z ) 2 F ( z ) /; ( ( F n ( z ) ( - π 2 + 2 - ( z - ) k = 0 n k ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] k ! 2 F 1 ( 1 2 , k + 3 2 ; 3 2 ; - 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["k", "+", FractionBox["3", "2"]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox["3", "2"], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List["-", "1"]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ( z - ) k ) 2 ( csch - 1 ( z ) + 2 n ( 1 2 ) n + 1 TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List["n", "+", "1"]]], Pochhammer] ( n + 1 ) ! ( z - ) n + 1 - ( z - ) F 0 1 2 1 1 2 ( n + 5 2 ; 1 2 ; 1 , n + 3 2 ; ; 3 2 ; n + 2 , n + 5 2 ; - 1 , ( z - ) ) ) 2 ) n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] ) FormBox RowBox RowBox SuperscriptBox RowBox SuperscriptBox csch RowBox - 1 ( z ) 2 RowBox SubscriptBox F ( z ) /; RowBox ( RowBox RowBox ( RowBox RowBox SubscriptBox F n ( z ) SuperscriptBox RowBox ( RowBox RowBox - FractionBox RowBox π 2 + RowBox 2 SqrtBox RowBox RowBox - RowBox ( RowBox z - ) RowBox UnderoverscriptBox RowBox k = 0 n RowBox FractionBox RowBox SuperscriptBox k TagBox SubscriptBox RowBox ( FractionBox 1 2 ) k Pochhammer RowBox k ! TagBox TagBox RowBox RowBox SubscriptBox FormBox 2 TraditionalForm SubscriptBox F FormBox 1 TraditionalForm RowBox ( RowBox TagBox TagBox RowBox TagBox FractionBox 1 2 Hypergeometric2F1 Rule Editable , TagBox RowBox k + FractionBox 3 2 Hypergeometric2F1 Rule Editable InterpretTemplate Function SlotSequence 1 Hypergeometric2F1 Rule Editable ; TagBox TagBox TagBox FractionBox 3 2 Hypergeometric2F1 Rule Editable InterpretTemplate Function SlotSequence 1 Hypergeometric2F1 Rule Editable ; TagBox RowBox - 1 Hypergeometric2F1 Rule Editable ) InterpretTemplate Function HypergeometricPFQ Slot 1 Slot 2 Slot 3 Rule Editable Hypergeometric2F1 SuperscriptBox RowBox ( RowBox z - ) k ) 2 SuperscriptBox RowBox ( RowBox RowBox SuperscriptBox csch RowBox - 1 ( z ) + RowBox FractionBox RowBox 2 SuperscriptBox n TagBox SubscriptBox RowBox ( FractionBox 1 2 ) RowBox n + 1 Pochhammer RowBox RowBox ( RowBox n + 1 ) ! SuperscriptBox RowBox ( RowBox z - ) RowBox n + 1 SqrtBox RowBox RowBox - RowBox ( RowBox z - ) RowBox SubsuperscriptBox F RowBox 0 1 2 RowBox 1 1 2 [ RowBox GridBox ErrorBox RowBox RowBox RowBox n + FractionBox 5 2 ; FractionBox 1 2 ; 1 , RowBox RowBox n + FractionBox 3 2 ; RowBox RowBox ; FractionBox 3 2 ; RowBox n + 2 , RowBox RowBox n + FractionBox 5 2 ; - 1 , RowBox RowBox ( RowBox z - ) ] ) 2 ) RowBox n TagBox Function ) TraditionalForm [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["ArcCsch", "[", "z_", "]"]], "2"], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", "z", "]"]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", "z", "]"]], "\[Equal]", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["z", "-", "\[ImaginaryI]"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ImaginaryI]", "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[FractionBox["3", "2"], "+", "k"]], ",", FractionBox["3", "2"], ",", RowBox[List["-", "1"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "\[ImaginaryI]"]], ")"]], "k"]]], RowBox[List["k", "!"]]]]]]]]], ")"]], "2"], "\[Equal]", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["ArcCsch", "[", "z", "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SuperscriptBox["\[ImaginaryI]", "n"], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["n", "+", "1"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "\[ImaginaryI]"]], ")"]], RowBox[List["n", "+", "1"]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["z", "-", "\[ImaginaryI]"]], ")"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["5", "2"], "+", "n"]], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["3", "2"], "+", "n"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List[FractionBox["5", "2"], "+", "n"]]]], "}"]]]], "}"]], ",", RowBox[List["-", "1"]], ",", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["z", "-", "\[ImaginaryI]"]], ")"]]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]]]]], ")"]], "2"]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02