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 ArcCosh

 http://functions.wolfram.com/01.26.06.0096.01

 Input Form

 ArcCosh[z]^2 == Subscript[F, Infinity][z] /; Subscript[F, n][z] == -Pi^2 + Pi 2 Sqrt[2] Sqrt[z + 1] Sum[(Pochhammer[1/2, k] (z + 1)^k)/(2^k (2 k + 1) k!), {k, 0, n}] - 2 (z + 1) Sum[(Pochhammer[1/2, k] (z + 1)^k)/(2^k (2 k + 1) k!), {k, 0, n}]^2 == ArcCosh[z]^2 - (((1 + z)^(3 + 2 n) Gamma[3/2 + n]^4)/ (Pi^(3/2) (2 n + 3)!)) HypergeometricPFQ[{1, 3/2 + n, 3/2 + n}, {2 + n, 5/2 + n}, (1 + z)/2]^2 - ((2^(1/2 - n) (1 + z)^(3/2 + n) ArcCos[z] Gamma[3/2 + n])/ ((3 + 2 n) Sqrt[Pi] (n + 1)!)) HypergeometricPFQ[{1, 3/2 + n, 3/2 + n}, {2 + n, 5/2 + n}, (1 + z)/2] && Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["ArcCosh", "[", "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["-", SuperscriptBox["\[Pi]", "2"]]], "+", RowBox[List["\[Pi]", " ", "2", SqrtBox["2"], " ", SqrtBox[RowBox[List["z", "+", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "k"]]], RowBox[List[SuperscriptBox["2", "k"], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]], "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "k"]]], RowBox[List[SuperscriptBox["2", "k"], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", RowBox[List["k", "!"]]]]]]], ")"]], "2"]]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["ArcCosh", "[", "z", "]"]], "2"], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["3", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "n"]], "]"]], "4"]]], RowBox[List[SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "n"]], "+", "3"]], ")"]], "!"]]]]], SuperscriptBox[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List[FractionBox["3", "2"], "+", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List[FractionBox["5", "2"], "+", "n"]]]], "}"]], ",", FractionBox[RowBox[List["1", "+", "z"]], "2"]]], "]"]], "2"]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List[FractionBox["3", "2"], "+", "n"]]], " ", RowBox[List["ArcCos", "[", "z", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "n"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", SqrtBox["\[Pi]"], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List[FractionBox["3", "2"], "+", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List[FractionBox["5", "2"], "+", "n"]]]], "}"]], ",", FractionBox[RowBox[List["1", "+", "z"]], "2"]]], "]"]]]]]]]], ")"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]], ")"]]]]]]

 MathML Form

 cosh - 1 ( z ) 2 F ( z ) /; ( ( F n ( z ) - π 2 + π 2 2 z + 1 k = 0 n ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] ( z + 1 ) k 2 k ( 2 k + 1 ) k ! - 2 ( z + 1 ) ( k = 0 n ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] ( z + 1 ) k 2 k ( 2 k + 1 ) k ! ) 2 cosh - 1 ( z ) 2 - ( z + 1 ) 2 n + 3 Γ ( n + 3 2 ) 4 π 3 / 2 ( 2 n + 3 ) ! 3 F 2 ( 1 , n + 3 2 , n + 3 2 ; n + 2 , n + 5 2 ; z + 1 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", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[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[FractionBox[RowBox[List["z", "+", "1"]], "2"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] 2 - 2 1 2 - n ( z + 1 ) n + 3 2 cos - 1 ( z ) Γ ( n + 3 2 ) ( 2 n + 3 ) π ( n + 1 ) ! 3 F 2 ( 1 , n + 3 2 , n + 3 2 ; n + 2 , n + 5 2 ; z + 1 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", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[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[FractionBox[RowBox[List["z", "+", "1"]], "2"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) n ) Condition z 2 Subscript F z Subscript F n z -1 2 2 2 1 2 z 1 1 2 k 0 n Pochhammer 1 2 k z 1 k 2 k 2 k 1 k -1 -1 2 z 1 k 0 n Pochhammer 1 2 k z 1 k 2 k 2 k 1 k -1 2 z 2 -1 z 1 2 n 3 Gamma n 3 2 4 3 2 2 n 3 -1 HypergeometricPFQ 1 n 3 2 n 3 2 n 2 n 5 2 z 1 2 -1 2 -1 2 1 2 -1 n z 1 n 3 2 z Gamma n 3 2 2 n 3 1 2 n 1 -1 HypergeometricPFQ 1 n 3 2 n 3 2 n 2 n 5 2 z 1 2 -1 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["ArcCosh", "[", "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["-", SuperscriptBox["\[Pi]", "2"]]], "+", RowBox[List["\[Pi]", " ", "2", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List["z", "+", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "k"]]], RowBox[List[SuperscriptBox["2", "k"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]], "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "k"]]], RowBox[List[SuperscriptBox["2", "k"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", RowBox[List["k", "!"]]]]]]], ")"]], "2"]]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["ArcCosh", "[", "z", "]"]], "2"], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["3", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "n"]], "]"]], "4"]]], ")"]], " ", SuperscriptBox[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List[FractionBox["3", "2"], "+", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List[FractionBox["5", "2"], "+", "n"]]]], "}"]], ",", FractionBox[RowBox[List["1", "+", "z"]], "2"]]], "]"]], "2"]]], RowBox[List[SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "3"]], ")"]], "!"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List[FractionBox["3", "2"], "+", "n"]]], " ", RowBox[List["ArcCos", "[", "z", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "n"]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List[FractionBox["3", "2"], "+", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List[FractionBox["5", "2"], "+", "n"]]]], "}"]], ",", FractionBox[RowBox[List["1", "+", "z"]], "2"]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", SqrtBox["\[Pi]"], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]]]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02