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 ArcSech

 http://functions.wolfram.com/01.30.06.0042.01

 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]] + (z - Subscript[z, 0])/ (Subscript[z, 0]^2 Sqrt[(-1 + Subscript[z, 0]^2)/Subscript[z, 0]^2]) - ((-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 + \[Ellipsis])) /; (z -> Subscript[z, 0])

 Standard Form

 Cell[BoxData[RowBox[List[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["(", RowBox[List[RowBox[List["ArcSec", "[", SubscriptBox["z", "0"], "]"]], "+", FractionBox[RowBox[List["z", "-", SubscriptBox["z", "0"]]], RowBox[List[SubsuperscriptBox["z", "0", "2"], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", SubsuperscriptBox["z", "0", "2"]]], SubsuperscriptBox["z", "0", "2"]]]]]], "-", RowBox[List[FractionBox[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"]]], " ", "+", "\[Ellipsis]"]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", 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 ) + z - z 0 z 0 2 z 0 2 - 1 z 0 2 - ( 2 z 0 2 - 1 ) ( z - z 0 ) 2 2 1 - 1 z 0 2 z 0 3 ( z 0 2 - 1 ) + ) ) /; ( z "\[Rule]" z 0 ) Condition 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 z -1 Subscript z 0 Subscript z 0 2 Subscript z 0 2 -1 Subscript z 0 2 -1 1 2 -1 -1 2 Subscript z 0 2 -1 z -1 Subscript z 0 2 2 1 -1 1 Subscript z 0 2 -1 1 2 Subscript z 0 3 Subscript z 0 2 -1 -1 Rule z Subscript z 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSech", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[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["(", 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"]]], ")"]]]]], "+", "\[Ellipsis]"]], ")"]]]]]], ")"]]]], SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SubscriptBox["zz", "0"]]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", SubscriptBox["zz", "0"]]], ")"]]]]]]]]

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

 2007-05-02