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 ArcCsch

 http://functions.wolfram.com/01.29.06.0047.01

 Input Form

 ArcCsch[z]^2 == Subscript[F, Infinity][z] /; Subscript[F, n][z] == (1/4) Log[4/z^2]^2 + (z^2/4) Log[4/z^2] Sum[(((-1)^k Pochhammer[3/2, k])/((k + 1)^2 k!)) z^(2 k), {k, 0, n}] + (z^4/16) Sum[((-1)^k Pochhammer[3/2, k] z^(2 k))/((k + 1)^2 k!), {k, 0, n}]^2 == (-(z^(4 (2 + n))/(16 (2 + n)^4 (1 + n)!^2))) ((4 I (-1)^n (2 + n)^2 ArcCsch[z] (1 + n)!)/z^(2 (2 + n)) + 3 Pochhammer[5/2, n] HypergeometricPFQ[{1, 2 + n, 5/2 + n}, {3 + n, 3 + n}, -z^2])^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]", RowBox[List[RowBox[List[FractionBox["1", "4"], SuperscriptBox[RowBox[List["Log", "[", FractionBox["4", SuperscriptBox["z", "2"]], "]"]], "2"]]], "+", RowBox[List[FractionBox[SuperscriptBox["z", "2"], "4"], RowBox[List["Log", "[", FractionBox["4", SuperscriptBox["z", "2"]], "]"]], RowBox[List["Sum", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "2"], " ", RowBox[List["k", "!"]]]]], SuperscriptBox["z", RowBox[List["2", "k"]]]]], ",", RowBox[List["{", RowBox[List["k", ",", "0", ",", "n"]], "}"]]]], "]"]]]], "+", RowBox[List[FractionBox[SuperscriptBox["z", "4"], "16"], SuperscriptBox[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "2"], " ", RowBox[List["k", "!"]]]]]]], ")"]], "2"]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["z", RowBox[List["4", " ", RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]]]]], RowBox[List["16", " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], "4"], " ", SuperscriptBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]], "!"]], "2"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["4", " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], SuperscriptBox[RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], "2"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]]]]], " ", RowBox[List["ArcCsch", "[", "z", "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]], "!"]]]], "+", RowBox[List["3", " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["5", "2"], ",", "n"]], "]"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["2", "+", "n"]], ",", RowBox[List[FractionBox["5", "2"], "+", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["3", "+", "n"]], ",", RowBox[List["3", "+", "n"]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]]]]]], ")"]], "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 ) 1 4 log 2 ( 4 z 2 ) + z 2 4 log 2 ( 4 z 2 ) k = 0 n ( - 1 ) k ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] z 2 k ( k + 1 ) 2 k ! + z 4 16 ( k = 0 n ( - 1 ) k ( 3 2 ) k TagBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"]]], Pochhammer] z 2 k ( k + 1 ) 2 k ! ) 2 - z 4 ( n + 2 ) 16 ( n + 2 ) 4 ( n + 1 ) ! 2 ( 4 ( - 1 ) n ( n + 2 ) 2 csch - 1 ( z ) ( n + 1 ) ! z - 2 ( n + 2 ) + 3 ( 5 2 ) n 3 F 2 ( 1 , n + 2 , n + 5 2 ; n + 3 , n + 3 ; - z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", FractionBox["5", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["n", "+", "3"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", "3"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", SuperscriptBox["z", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) 2 ) n ) Condition z 2 Subscript F z Subscript F n z 1 4 4 z 2 -1 2 z 2 4 -1 4 z 2 -1 2 k 0 n -1 k Pochhammer 3 2 k z 2 k k 1 2 k -1 z 4 16 -1 k 0 n Pochhammer -1 k Subscript 3 2 k z 2 k k 1 2 k -1 2 -1 z 4 n 2 16 n 2 4 n 1 2 -1 4 -1 n n 2 2 z n 1 z -2 n 2 3 Subscript 5 2 n HypergeometricPFQ 1 n 2 n 5 2 n 3 n 3 -1 z 2 2 n [/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]", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["Log", "[", FractionBox["4", SuperscriptBox["z", "2"]], "]"]], "2"]]], "+", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox["z", "2"], " ", RowBox[List["Log", "[", FractionBox["4", SuperscriptBox["z", "2"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "2"], " ", RowBox[List["k", "!"]]]]]]]]], "+", RowBox[List[FractionBox["1", "16"], " ", SuperscriptBox["z", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "2"], " ", RowBox[List["k", "!"]]]]]]], ")"]], "2"]]]]], "\[Equal]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["4", " ", RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["4", " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], "2"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]]]]], " ", RowBox[List["ArcCsch", "[", "z", "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]], "!"]]]], "+", RowBox[List["3", " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["5", "2"], ",", "n"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["2", "+", "n"]], ",", RowBox[List[FractionBox["5", "2"], "+", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["3", "+", "n"]], ",", RowBox[List["3", "+", "n"]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]]]]]], ")"]], "2"]]], RowBox[List["16", " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]], "!"]], ")"]], "2"]]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02