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 ArcSech

 http://functions.wolfram.com/01.30.06.0045.01

 Input Form

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

 Standard Form

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

 MathML Form

 sech - 1 ( z ) 1 z 0 - 1 ( 1 - z 0 z 0 ) 1 z 0 ( z 0 1 - z 0 ) 1 2 arg ( z 0 - z z z 0 ) 2 π ( 1 - z 0 z 0 ) 1 2 arg ( z 0 - z 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 sec - 1 ( z 0 ) ) ( 1 + O ( z - z 0 ) ) Proportional z 1 Subscript z 0 -1 1 2 -1 1 -1 Subscript z 0 1 2 Subscript z 0 1 2 1 Subscript z 0 -1 1 2 Subscript z 0 1 -1 Subscript z 0 -1 1 2 Subscript z 0 -1 z z Subscript z 0 -1 2 -1 1 -1 Subscript z 0 Subscript z 0 -1 1 2 Subscript z 0 -1 z 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 O z -1 Subscript z 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSech", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", SubscriptBox["zz", "0"]]]], " ", SqrtBox[SubscriptBox["zz", "0"]]]], ")"]], " ", SqrtBox[FractionBox["1", SubscriptBox["zz", "0"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SubscriptBox["zz", "0"], RowBox[List["1", "-", SubscriptBox["zz", "0"]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List[SubscriptBox["zz", "0"], "-", "z"]], RowBox[List["z", " ", SubscriptBox["zz", "0"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", SubscriptBox["zz", "0"]]], SubscriptBox["zz", "0"]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List[SubscriptBox["zz", "0"], "-", "z"]], 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[SubscriptBox["zz", "0"], "-", "z"]], RowBox[List["z", " ", SubscriptBox["zz", "0"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List[SubscriptBox["zz", "0"], "-", "z"]], RowBox[List["z", " ", SubscriptBox["zz", "0"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["Arg", "[", FractionBox[RowBox[List["1", "+", SubscriptBox["zz", "0"]]], SubscriptBox["zz", "0"]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[SubscriptBox["zz", "0"], RowBox[List["1", "+", SubscriptBox["zz", "0"]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List[SubscriptBox["zz", "0"], "-", "z"]], RowBox[List["z", " ", SubscriptBox["zz", "0"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "+", SubscriptBox["zz", "0"]]], SubscriptBox["zz", "0"]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List[SubscriptBox["zz", "0"], "-", "z"]], RowBox[List["z", " ", SubscriptBox["zz", "0"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["ArcSec", "[", SubscriptBox["zz", "0"], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]]]], ")"]]]], SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SubscriptBox["zz", "0"]]]]]]]]]

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

 2007-05-02