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 ArcSec

 http://functions.wolfram.com/01.18.06.0021.01

 Input Form

 ArcSec[z] == (Subscript[z, 0]/(Subscript[z, 0] - 1))^ ((1/2) Floor[Arg[(z - Subscript[z, 0])/(z Subscript[z, 0])]/(2 Pi)]) ((Subscript[z, 0] - 1)/Subscript[z, 0])^ ((1/2) Floor[Arg[(z - Subscript[z, 0])/(z Subscript[z, 0])]/(2 Pi)]) (-2 Pi I I^Floor[Arg[(-z + Subscript[z, 0])/(z Subscript[z, 0])]/(2 Pi)] Floor[Arg[(-z + Subscript[z, 0])/(z Subscript[z, 0])]/(2 Pi)] Floor[(Pi + Arg[(Subscript[z, 0] + 1)/Subscript[z, 0]])/(2 Pi)] + (Subscript[z, 0]/(Subscript[z, 0] + 1))^ ((1/2) Floor[Arg[(-z + Subscript[z, 0])/(z Subscript[z, 0])]/(2 Pi)]) ((Subscript[z, 0] + 1)/Subscript[z, 0])^ ((1/2) Floor[Arg[(-z + Subscript[z, 0])/(z Subscript[z, 0])]/(2 Pi)]) (Pi/2 + Sum[(-1)^(k - 1) Subscript[z, 0]^(-k - 1) HypergeometricPFQ[{1/2, (1 + k)/2, 1 + k/2}, {1, 3/2}, 1/Subscript[z, 0]^2] (z - Subscript[z, 0])^k, {k, 0, Infinity}]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["ArcSec", "[", "z", "]"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[SubscriptBox["z", "0"], RowBox[List[SubscriptBox["z", "0"], "-", "1"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List["z", "-", SubscriptBox["z", "0"]]], RowBox[List["z", " ", SubscriptBox["z", "0"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List[SubscriptBox["z", "0"], "-", "1"]], SubscriptBox["z", "0"]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List["z", "-", SubscriptBox["z", "0"]]], 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", "[", FractionBox[RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["z", "0"]]], RowBox[List["z", " ", SubscriptBox["z", "0"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["z", "0"]]], RowBox[List["z", " ", SubscriptBox["z", "0"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["Arg", "[", FractionBox[RowBox[List[SubscriptBox["z", "0"], "+", "1"]], SubscriptBox["z", "0"]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[SubscriptBox["z", "0"], RowBox[List[SubscriptBox["z", "0"], "+", "1"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["z", "0"]]], RowBox[List["z", " ", SubscriptBox["z", "0"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List[SubscriptBox["z", "0"], "+", "1"]], SubscriptBox["z", "0"]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["z", "0"]]], RowBox[List["z", " ", SubscriptBox["z", "0"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[FractionBox["\[Pi]", "2"], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], " ", SubsuperscriptBox["z", "0", RowBox[List[RowBox[List["-", "k"]], "-", "1"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox[RowBox[List["1", "+", "k"]], "2"], ",", RowBox[List["1", "+", FractionBox["k", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", ",", FractionBox["3", "2"]]], "}"]], ",", FractionBox["1", SubsuperscriptBox["z", "0", "2"]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "k"]]]]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 sec - 1 ( z ) ( z 0 z 0 - 1 ) 1 2 arg ( z - z 0 z z 0 ) 2 π ( z 0 - 1 z 0 ) 1 2 arg ( z - z 0 z z 0 ) 2 π ( - 2 π arg ( z 0 - z z z 0 ) 2 π arg ( z 0 - z z z 0 ) 2 π arg ( z 0 + 1 z 0 ) + π 2 π + ( z 0 z 0 + 1 ) 1 2 arg ( z 0 - z z z 0 ) 2 π ( z 0 + 1 z 0 ) 1 2 arg ( z 0 - z z z 0 ) 2 π ( k = 0 ( - 1 ) k - 1 z 0 - k - 1 3 F 2 ( 1 2 , k + 1 2 , k 2 + 1 ; 1 , 3 2 ; 1 z 0 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["1", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", "1"]], "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["k", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox["1", SubsuperscriptBox["z", "0", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ( z - z 0 ) k + π 2 ) ) z Subscript z 0 Subscript z 0 -1 -1 1 2 z -1 Subscript z 0 z Subscript z 0 -1 2 -1 Subscript z 0 -1 Subscript z 0 -1 1 2 z -1 Subscript z 0 z Subscript z 0 -1 2 -1 -2 Subscript z 0 -1 z z Subscript z 0 -1 2 -1 Subscript z 0 -1 z z Subscript z 0 -1 2 -1 Subscript z 0 1 Subscript z 0 -1 2 -1 Subscript z 0 Subscript z 0 1 -1 1 2 Subscript z 0 -1 z z Subscript z 0 -1 2 -1 Subscript z 0 1 Subscript z 0 -1 1 2 Subscript z 0 -1 z z Subscript z 0 -1 2 -1 k 0 -1 k -1 Subscript z 0 -1 k -1 HypergeometricPFQ 1 2 k 1 2 -1 k 2 -1 1 1 3 2 1 Subscript z 0 2 -1 z -1 Subscript z 0 k 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSec", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[SubscriptBox["zz", "0"], RowBox[List[SubscriptBox["zz", "0"], "-", "1"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List["z", "-", SubscriptBox["zz", "0"]]], RowBox[List["z", " ", SubscriptBox["zz", "0"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List[SubscriptBox["zz", "0"], "-", "1"]], SubscriptBox["zz", "0"]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List["z", "-", SubscriptBox["zz", "0"]]], 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", "[", FractionBox[RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["zz", "0"]]], RowBox[List["z", " ", SubscriptBox["zz", "0"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["zz", "0"]]], RowBox[List["z", " ", SubscriptBox["zz", "0"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["Arg", "[", FractionBox[RowBox[List[SubscriptBox["zz", "0"], "+", "1"]], SubscriptBox["zz", "0"]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[SubscriptBox["zz", "0"], RowBox[List[SubscriptBox["zz", "0"], "+", "1"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["zz", "0"]]], RowBox[List["z", " ", SubscriptBox["zz", "0"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List[SubscriptBox["zz", "0"], "+", "1"]], SubscriptBox["zz", "0"]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["zz", "0"]]], RowBox[List["z", " ", SubscriptBox["zz", "0"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[FractionBox["\[Pi]", "2"], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], " ", SubsuperscriptBox["zz", "0", RowBox[List[RowBox[List["-", "k"]], "-", "1"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox[RowBox[List["1", "+", "k"]], "2"], ",", RowBox[List["1", "+", FractionBox["k", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", ",", FractionBox["3", "2"]]], "}"]], ",", FractionBox["1", SubsuperscriptBox["zz", "0", "2"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]]]]]], ")"]]]]]], ")"]]]]]]]]

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

 2007-05-02