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 ArcCos

 http://functions.wolfram.com/01.13.06.0070.01

 Input Form

 ArcCos[z]^2 == -(Log[-4 z^2]^2/4) - (Pi z Log[-4 z^2])/(2 Sqrt[-z^2]) + Pi^2/4 + (1/(4 z^2)) ((Pi z)/Sqrt[-z^2] + Log[-z^2]) HypergeometricPFQ[{3/2, 1, 1}, {2, 2}, 1/z^2] + (1/(2 z^2)) HypergeometricPFQ[{3/2, 1, 1, 1}, {2, 2, 2}, 1/z^2] - (1/(4 z^2)) Sum[(Pochhammer[3/2, k] ((PolyGamma[-(1/2) - k] - PolyGamma[k + 1])/((k + 1)^2 k!)))/z^(2 k), {k, 0, Infinity}] /; Abs[z] > 1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["ArcCos", "[", "z", "]"]], "2"], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["z", "2"]]], "]"]], "2"], "4"]]], "-", FractionBox[RowBox[List["\[Pi]", " ", "z", " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["z", "2"]]], "]"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]]], "+", FractionBox[SuperscriptBox["\[Pi]", "2"], "4"], "+", RowBox[List[FractionBox[RowBox[List["1", " "]], RowBox[List["4", " ", SuperscriptBox["z", "2"]]]], RowBox[List["(", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "z"]], SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]], "+", RowBox[List["Log", "[", RowBox[List["-", SuperscriptBox["z", "2"]]], "]"]]]], ")"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", "1", ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List["2", ",", "2"]], "}"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["2", SuperscriptBox["z", "2"]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", "1", ",", "1", ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List["2", ",", "2", ",", "2"]], "}"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]], "-", RowBox[List[FractionBox["1", RowBox[List["4", SuperscriptBox["z", "2"]]]], RowBox[List["Sum", "[", RowBox[List[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], FractionBox[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "k"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "2"], " ", RowBox[List["k", "!"]]]]], SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "k"]]]]], ",", RowBox[List["{", RowBox[List["k", ",", "0", ",", "\[Infinity]"]], "}"]]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], ">", "1"]]]]]]

 MathML Form

 cos - 1 ( z ) 2 - 1 4 log 2 ( - 4 z 2 ) - π z log ( - 4 z 2 ) 2 - z 2 + π 2 4 + 1 2 z 2 4 F 3 ( 3 2 , 1 , 1 , 1 ; 2 , 2 , 2 ; 1 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["4", TraditionalForm]], SubscriptBox["F", FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox["2", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["2", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["2", HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox["1", SuperscriptBox["z", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + 1 4 z 2 ( π z - z 2 + log ( - z 2 ) ) 3 F 2 ( 3 2 , 1 , 1 ; 2 , 2 ; 1 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[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox["2", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["2", HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox["1", SuperscriptBox["z", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] - 1 4 z 2 k = 0 ( 3 2 ) k ( ψ TagBox["\[Psi]", PolyGamma] ( - k - 1 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) ) z - 2 k ( k + 1 ) 2 k ! /; "\[LeftBracketingBar]" z "\[RightBracketingBar]" > 1 Condition z 2 -1 1 4 -4 z 2 2 -1 z -4 z 2 2 -1 z 2 1 2 -1 2 4 -1 1 2 z 2 -1 HypergeometricPFQ 3 2 1 1 1 2 2 2 1 z 2 -1 1 4 z 2 -1 z -1 z 2 1 2 -1 -1 z 2 HypergeometricPFQ 3 2 1 1 2 2 1 z 2 -1 -1 1 4 z 2 -1 k 0 Subscript 3 2 k PolyGamma -1 k -1 1 2 -1 PolyGamma k 1 z -2 k k 1 2 k -1 z 1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["ArcCos", "[", "z_", "]"]], "2"], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["z", "2"]]], "]"]], "2"]]], "-", FractionBox[RowBox[List["\[Pi]", " ", "z", " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["z", "2"]]], "]"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]]], "+", FractionBox[SuperscriptBox["\[Pi]", "2"], "4"], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "z"]], SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]], "+", RowBox[List["Log", "[", RowBox[List["-", SuperscriptBox["z", "2"]]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", "1", ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List["2", ",", "2"]], "}"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]], RowBox[List["4", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", "1", ",", "1", ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List["2", ",", "2", ",", "2"]], "}"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]], RowBox[List["2", " ", SuperscriptBox["z", "2"]]]], "-", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "k"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "2"], " ", RowBox[List["k", "!"]]]]]]], RowBox[List["4", " ", SuperscriptBox["z", "2"]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], ">", "1"]]]]]]]]

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

 2007-05-02