html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Sech

 http://functions.wolfram.com/01.24.27.0004.01

 Input Form

 ArcSech[Sech[z]] == (Sqrt[z^2] - ((Pi I)/2) E^(I Pi Floor[1/2 - Arg[z]/Pi]) (1 - (-1)^Floor[Im[z]/Pi] + 2 Floor[Im[z]/Pi])) (1 - KroneckerDelta[Re[z]]) + ((-1)^Floor[Im[z]/Pi] (z - Pi I Floor[Im[z]/Pi] - (Pi I)/2) + (Pi I)/2) KroneckerDelta[Re[z]] - Pi I UnitStep[Im[z]] KroneckerDelta[Re[z]] (1 + (-1)^(Floor[Im[z]/(2 Pi) + 1/2] + Floor[-(Im[z]/(2 Pi)) - 1/2])) + ((Pi I)/2) (E^(I Pi Floor[1/2 - Arg[z]/Pi]) + 1) (1 + (-1)^(Floor[Im[z]/(2 Pi) + 1/2] + Floor[-(Im[z]/(2 Pi)) - 1/2])) UnitStep[Re[z]]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["ArcSech", "[", RowBox[List["Sech", "[", "z", "]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], " ", "-", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " "]], "2"], SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", "z", "]"]], "\[Pi]"]]], "]"]]]]], RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["Im", "[", "z", "]"]], "\[Pi]"], "]"]]], "+", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Im", "[", "z", "]"]], "\[Pi]"], "]"]]]]]], ")"]]]]]], ")"]], RowBox[List["(", RowBox[List["1", "-", RowBox[List["KroneckerDelta", "[", RowBox[List["Re", "[", "z", "]"]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["Im", "[", "z", "]"]], "\[Pi]"], "]"]]], RowBox[List["(", RowBox[List["z", "-", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Im", "[", "z", "]"]], "\[Pi]"], "]"]]]], "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]]]], "+", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], RowBox[List["KroneckerDelta", "[", RowBox[List["Re", "[", "z", "]"]], "]"]]]], "-", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["UnitStep", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], RowBox[List["KroneckerDelta", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["Floor", "[", RowBox[List[FractionBox[RowBox[List["Im", "[", "z", "]"]], RowBox[List["2", " ", "\[Pi]"]]], "+", FractionBox["1", "2"]]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["Im", "[", "z", "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "-", FractionBox["1", "2"]]], "]"]]]]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", "z", "]"]], "\[Pi]"]]], "]"]]]]], "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["Floor", "[", RowBox[List[FractionBox[RowBox[List["Im", "[", "z", "]"]], RowBox[List["2", " ", "\[Pi]"]]], "+", FractionBox["1", "2"]]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["Im", "[", "z", "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "-", FractionBox["1", "2"]]], "]"]]]]]]], ")"]], " ", RowBox[List["UnitStep", "[", RowBox[List["Re", "[", "z", "]"]], "]"]]]]]]]]]]

 MathML Form

 sech - 1 ( sech ( z ) ) ( z 2 - π 2 π 1 2 - arg ( z ) π ( 2 Im ( z ) π - ( - 1 ) Im ( z ) π + 1 ) ) ( 1 - δ KroneckerDelta Re ( z ) ) - π θ UnitStep ( Im ( z ) ) ( 1 + ( - 1 ) Im ( z ) 2 π + 1 2 + - Im ( z ) 2 π - 1 2 ) δ KroneckerDelta Re ( z ) + ( ( - 1 ) Im ( z ) π ( z - π Im ( z ) π - π 2 ) + π 2 ) δ KroneckerDelta Re ( z ) + π 2 ( π 1 2 - arg ( z ) π + 1 ) ( 1 + ( - 1 ) Im ( z ) 2 π + 1 2 + - Im ( z ) 2 π - 1 2 ) θ UnitStep ( Re ( z ) ) z z 2 1 2 -1 2 -1 1 2 -1 z -1 2 z -1 -1 -1 z -1 1 1 -1 KroneckerDelta z -1 UnitStep z 1 -1 z 2 -1 1 2 -1 z 2 -1 -1 1 2 KroneckerDelta z -1 z -1 z -1 z -1 -1 2 -1 2 -1 KroneckerDelta z 2 -1 1 2 -1 z -1 1 1 -1 z 2 -1 1 2 -1 z 2 -1 -1 1 2 UnitStep z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSech", "[", RowBox[List["Sech", "[", "z_", "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", "z", "]"]], "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["Im", "[", "z", "]"]], "\[Pi]"], "]"]]], "+", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Im", "[", "z", "]"]], "\[Pi]"], "]"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["KroneckerDelta", "[", RowBox[List["Re", "[", "z", "]"]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["Im", "[", "z", "]"]], "\[Pi]"], "]"]]], " ", RowBox[List["(", RowBox[List["z", "-", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Im", "[", "z", "]"]], "\[Pi]"], "]"]]]], "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]]]], "+", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List["Re", "[", "z", "]"]], "]"]]]], "-", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["UnitStep", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["Floor", "[", RowBox[List[FractionBox[RowBox[List["Im", "[", "z", "]"]], RowBox[List["2", " ", "\[Pi]"]]], "+", FractionBox["1", "2"]]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["Im", "[", "z", "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "-", FractionBox["1", "2"]]], "]"]]]]]]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", "z", "]"]], "\[Pi]"]]], "]"]]]]], "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["Floor", "[", RowBox[List[FractionBox[RowBox[List["Im", "[", "z", "]"]], RowBox[List["2", " ", "\[Pi]"]]], "+", FractionBox["1", "2"]]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["Im", "[", "z", "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "-", FractionBox["1", "2"]]], "]"]]]]]]], ")"]], " ", RowBox[List["UnitStep", "[", RowBox[List["Re", "[", "z", "]"]], "]"]]]]]]]]]]

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

 2001-10-29