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 ArcSec

 http://functions.wolfram.com/01.18.06.0020.01

 Input Form

 ArcSec[z] \[Proportional] (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 I I^Floor[Arg[(-z + Subscript[z, 0])/(z Subscript[z, 0])]/(2 Pi)] 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)]) (ArcSec[Subscript[z, 0]] + (1/(Subscript[z, 0]^2 Sqrt[(-1 + Subscript[z, 0]^2)/Subscript[z, 0]^2])) (z - Subscript[z, 0]) - ((-1 + 2 Subscript[z, 0]^2)/ (2 Sqrt[1 - 1/Subscript[z, 0]^2] Subscript[z, 0]^3 (-1 + Subscript[z, 0]^2))) (z - Subscript[z, 0])^2 + O[(z - Subscript[z, 0])^3]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["ArcSec", "[", "z", "]"]], "\[Proportional]", 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"]], " ", "\[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]"]]], "]"]]], " ", "\[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[RowBox[List["ArcSec", "[", SubscriptBox["z", "0"], "]"]], "+", RowBox[List[FractionBox["1", RowBox[List[SubsuperscriptBox["z", "0", "2"], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", SubsuperscriptBox["z", "0", "2"]]], SubsuperscriptBox["z", "0", "2"]]]]]], RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", SubsuperscriptBox["z", "0", "2"]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", FractionBox["1", SubsuperscriptBox["z", "0", "2"]]]]], " ", SubsuperscriptBox["z", "0", "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubsuperscriptBox["z", "0", "2"]]], ")"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "2"]]], "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "3"], "]"]]]], ")"]]]]]], ")"]]]]]]]]

 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 arg ( z 0 - z z z 0 ) 2 π 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 ( sec - 1 ( z 0 ) + 1 z 0 2 z 0 2 - 1 z 0 2 ( z - z 0 ) - ( 2 z 0 2 - 1 ) 2 1 - 1 z 0 2 z 0 3 ( z 0 2 - 1 ) ( z - z 0 ) 2 + O ( ( z - z 0 ) 3 ) ) ) Proportional 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 Subscript z 0 1 Subscript z 0 2 Subscript z 0 2 -1 Subscript z 0 2 -1 1 2 -1 z -1 Subscript z 0 -1 2 Subscript z 0 2 -1 2 1 -1 1 Subscript z 0 2 -1 1 2 Subscript z 0 3 Subscript z 0 2 -1 -1 z -1 Subscript z 0 2 O z -1 Subscript z 0 3 [/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"]], " ", "\[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]"]]], "]"]]], " ", "\[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[RowBox[List["ArcSec", "[", SubscriptBox["zz", "0"], "]"]], "+", FractionBox[RowBox[List["z", "-", SubscriptBox["zz", "0"]]], RowBox[List[SubsuperscriptBox["zz", "0", "2"], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", SubsuperscriptBox["zz", "0", "2"]]], SubsuperscriptBox["zz", "0", "2"]]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", SubsuperscriptBox["zz", "0", "2"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "2"]]], RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", FractionBox["1", SubsuperscriptBox["zz", "0", "2"]]]]], " ", SubsuperscriptBox["zz", "0", "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubsuperscriptBox["zz", "0", "2"]]], ")"]]]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], "3"]]], ")"]]]]]], ")"]]]]]]]]

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

 2007-05-02