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 ArcCos

 http://functions.wolfram.com/01.13.06.0021.01

 Input Form

 ArcCos[z] == (1/(1 - Subscript[z, 0]))^ ((1/2) Floor[Arg[-z + Subscript[z, 0]]/(2 Pi)]) (1 - Subscript[z, 0])^((1/2) Floor[Arg[-z + Subscript[z, 0]]/(2 Pi)]) (-2 Pi I I^Floor[Arg[z - Subscript[z, 0]]/(2 Pi)] Floor[Arg[z - Subscript[z, 0]]/(2 Pi)] Floor[(Pi + Arg[1 + Subscript[z, 0]])/(2 Pi)] + (1/(1 + Subscript[z, 0]))^((1/2) Floor[Arg[z - Subscript[z, 0]]/(2 Pi)]) (1 + Subscript[z, 0])^((1/2) Floor[Arg[z - Subscript[z, 0]]/(2 Pi)]) (Pi - (Sqrt[Pi]/2) Sum[(1/((k - j)! j!)) Pochhammer[-(1/2), -j + k] (1 - Subscript[z, 0])^(1/2 + j - k) (1 + Subscript[z, 0])^(1/2 - j) Hypergeometric2F1Regularized[1, 1, 3/2 - j, (1/2) (1 + Subscript[z, 0])] (z - Subscript[z, 0])^k, {k, 0, Infinity}, {j, 0, k}]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["ArcCos", "[", "z", "]"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["1", "-", SubscriptBox["z", "0"]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["z", "0"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "\[Pi]", " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ImaginaryI]", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]], RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["Arg", "[", RowBox[List["1", "+", SubscriptBox["z", "0"]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["1", "+", SubscriptBox["z", "0"]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SubscriptBox["z", "0"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List[FractionBox[SqrtBox["\[Pi]"], "2"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox["1", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " ", RowBox[List["j", "!"]]]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List[RowBox[List["-", "j"]], "+", "k"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["z", "0"]]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "j", "-", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SubscriptBox["z", "0"]]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "j"]]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["1", ",", "1", ",", RowBox[List[FractionBox["3", "2"], "-", "j"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["z", "0"]]], ")"]]]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "k"]]]]]]]]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 cos - 1 ( z ) ( 1 1 - z 0 ) 1 2 arg ( z 0 - z ) 2 π ( 1 - z 0 ) 1 2 arg ( z 0 - z ) 2 π ( - 2 π arg ( z - z 0 ) 2 π arg ( z - z 0 ) 2 π arg ( z 0 + 1 ) + π 2 π + ( 1 z 0 + 1 ) 1 2 arg ( z - z 0 ) 2 π ( z 0 + 1 ) 1 2 arg ( z - z 0 ) 2 π ( π - π 2 k = 0 j = 0 k ( - 1 2 ) k - j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], RowBox[List["k", "-", "j"]]], Pochhammer] ( 1 - z 0 ) j - k + 1 2 ( z 0 + 1 ) 1 2 - j ( k - j ) ! j ! 2 F ~ 1 ( 1 , 1 ; 3 2 - j ; 1 2 ( z 0 + 1 ) ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox["1", Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["3", "2"], "-", "j"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SubscriptBox["z", "0"], "+", "1"]], ")"]]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] ( z - z 0 ) k ) ) z 1 1 -1 Subscript z 0 -1 1 2 Subscript z 0 -1 z 2 -1 1 -1 Subscript z 0 1 2 Subscript z 0 -1 z 2 -1 -2 z -1 Subscript z 0 2 -1 z -1 Subscript z 0 2 -1 Subscript z 0 1 2 -1 1 Subscript z 0 1 -1 1 2 z -1 Subscript z 0 2 -1 Subscript z 0 1 1 2 z -1 Subscript z 0 2 -1 -1 1 2 2 -1 j 0 k k 0 Pochhammer -1 1 2 k -1 j 1 -1 Subscript z 0 j -1 k 1 2 Subscript z 0 1 1 2 -1 j k -1 j j -1 Hypergeometric2F1Regularized 1 1 3 2 -1 j 1 2 Subscript z 0 1 z -1 Subscript z 0 k [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCos", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["1", "-", SubscriptBox["zz", "0"]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["zz", "0"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ImaginaryI]", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["Arg", "[", RowBox[List["1", "+", SubscriptBox["zz", "0"]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["1", "+", SubscriptBox["zz", "0"]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SubscriptBox["zz", "0"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List[FractionBox["1", "2"], " ", SqrtBox["\[Pi]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List[RowBox[List["-", "j"]], "+", "k"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["zz", "0"]]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "j", "-", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SubscriptBox["zz", "0"]]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "j"]]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["1", ",", "1", ",", RowBox[List[FractionBox["3", "2"], "-", "j"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["zz", "0"]]], ")"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " ", RowBox[List["j", "!"]]]]]]]]]]]]], ")"]]]]]], ")"]]]]]]]]

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

 2007-05-02