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; }

 ArcSech

 http://functions.wolfram.com/01.30.27.0489.01

 Input Form

 ArcSech[1/(2 z Sqrt[1 - z^2])] == (Sqrt[2 z Sqrt[1 - z^2] - 1]/Sqrt[1 - 2 z Sqrt[1 - z^2]]) (((2 Sqrt[1 - 2 z^2] Sqrt[-z^2 + z^4])/(Sqrt[-z^2] Sqrt[-1 + z^2] Sqrt[-1 + 2 z^2])) (Pi/2 - ArcCos[z]) + ((Pi Sqrt[1 - 2 z^2] Sqrt[-z^2 + z^4])/(2 Sqrt[-z^2] Sqrt[-1 + z^2] Sqrt[-1 + 2 z^2])) ((-Sqrt[-(1/z)]) Sqrt[-z] Sqrt[1/(1 - Sqrt[2] z)] Sqrt[1 - Sqrt[2] z] + Sqrt[1/z] Sqrt[z] Sqrt[1/(1 + Sqrt[2] z)] Sqrt[1 + Sqrt[2] z] - Sqrt[z^2]/z - Sqrt[-z^2 + z^4]/ (z Sqrt[-1 + z^2])) + Pi/2)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["ArcSech", "[", FractionBox["1", RowBox[List["2", " ", "z", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]], "]"]], "\[Equal]", RowBox[List[FractionBox[SqrtBox[RowBox[List[RowBox[List["2", " ", "z", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]], "-", "1"]]], SqrtBox[RowBox[List["1", "-", RowBox[List["2", " ", "z", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]]]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List["2", SqrtBox[RowBox[List["1", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], "+", SuperscriptBox["z", "4"]]]]]], RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]]]]]], RowBox[List["(", RowBox[List[FractionBox["\[Pi]", "2"], "-", RowBox[List["ArcCos", "[", "z", "]"]]]], ")"]]]], "+", " ", RowBox[List[FractionBox[RowBox[List["\[Pi]", SqrtBox[RowBox[List["1", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], "+", SuperscriptBox["z", "4"]]]]]], RowBox[List["2", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SqrtBox[RowBox[List["-", FractionBox["1", "z"]]]]]], " ", SqrtBox[RowBox[List["-", "z"]]], " ", SqrtBox[FractionBox["1", RowBox[List["1", "-", RowBox[List[SqrtBox["2"], " ", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List[SqrtBox["2"], " ", "z"]]]]]]], "+", RowBox[List[SqrtBox[FractionBox["1", "z"]], " ", SqrtBox["z"], " ", SqrtBox[FractionBox["1", RowBox[List["1", "+", RowBox[List[SqrtBox["2"], " ", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "+", RowBox[List[SqrtBox["2"], " ", "z"]]]]]]], "-", FractionBox[SqrtBox[SuperscriptBox["z", "2"]], "z"], "-", FractionBox[SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], "+", SuperscriptBox["z", "4"]]]], RowBox[List["z", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]]]]]]], ")"]]]], "+", FractionBox["\[Pi]", "2"]]], ")"]]]]]]]]

 MathML Form

 sech - 1 ( 1 2 z 1 - z 2 ) 2 z 1 - z 2 - 1 1 - 2 z 1 - z 2 ( π 1 - 2 z 2 z 4 - z 2 2 - z 2 z 2 - 1 2 z 2 - 1 ( - z 2 z + 1 z z 1 2 z + 1 2 z + 1 - - 1 z - z 1 1 - 2 z 1 - 2 z - z 4 - z 2 z z 2 - 1 ) + 2 1 - 2 z 2 z 4 - z 2 - z 2 z 2 - 1 2 z 2 - 1 ( π 2 - cos - 1 ( z ) ) + π 2 ) 1 2 z 1 -1 z 2 1 2 -1 2 z 1 -1 z 2 1 2 -1 1 2 1 -1 2 z 1 -1 z 2 1 2 1 2 -1 1 -1 2 z 2 1 2 z 4 -1 z 2 1 2 2 -1 z 2 1 2 z 2 -1 1 2 2 z 2 -1 1 2 -1 -1 z 2 1 2 z -1 1 z -1 1 2 z 1 2 1 2 1 2 z 1 -1 1 2 2 1 2 z 1 1 2 -1 -1 1 z -1 1 2 -1 z 1 2 1 1 -1 2 1 2 z -1 1 2 1 -1 2 1 2 z 1 2 -1 z 4 -1 z 2 1 2 z z 2 -1 1 2 -1 2 1 -1 2 z 2 1 2 z 4 -1 z 2 1 2 -1 z 2 1 2 z 2 -1 1 2 2 z 2 -1 1 2 -1 2 -1 -1 z 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSech", "[", FractionBox["1", RowBox[List["2", " ", "z_", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z_", "2"]]]]]]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", "z", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]], "-", "1"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], "+", SuperscriptBox["z", "4"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox["\[Pi]", "2"], "-", RowBox[List["ArcCos", "[", "z", "]"]]]], ")"]]]], RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Pi]", " ", SqrtBox[RowBox[List["1", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], "+", SuperscriptBox["z", "4"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SqrtBox[RowBox[List["-", FractionBox["1", "z"]]]]]], " ", SqrtBox[RowBox[List["-", "z"]]], " ", SqrtBox[FractionBox["1", RowBox[List["1", "-", RowBox[List[SqrtBox["2"], " ", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List[SqrtBox["2"], " ", "z"]]]]]]], "+", RowBox[List[SqrtBox[FractionBox["1", "z"]], " ", SqrtBox["z"], " ", SqrtBox[FractionBox["1", RowBox[List["1", "+", RowBox[List[SqrtBox["2"], " ", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "+", RowBox[List[SqrtBox["2"], " ", "z"]]]]]]], "-", FractionBox[SqrtBox[SuperscriptBox["z", "2"]], "z"], "-", FractionBox[SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], "+", SuperscriptBox["z", "4"]]]], RowBox[List["z", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]]]]]], "+", FractionBox["\[Pi]", "2"]]], ")"]]]], SqrtBox[RowBox[List["1", "-", RowBox[List["2", " ", "z", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]]]]]]]]]

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

 2003-08-21