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http://functions.wolfram.com/01.17.27.0985.01
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ArcCsc[Sqrt[(2 z)/(z - Sqrt[z^2 - 1])]] == (Sqrt[I z]/2) Sqrt[-(I/z)]
(Sqrt[z^2] Sqrt[1/z^2] (Pi/2 - ArcSec[z]) - (Pi/2) (Sqrt[z^2]/z - 1))
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Cell[BoxData[RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox[FractionBox[RowBox[List["2", "z"]], RowBox[List["z", "-", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]]]]]], "]"]], "\[Equal]", RowBox[List[FractionBox[SqrtBox[RowBox[List["\[ImaginaryI]", " ", "z"]]], "2"], SqrtBox[RowBox[List["-", FractionBox["\[ImaginaryI]", "z"]]]], RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], SqrtBox[FractionBox["1", SuperscriptBox["z", "2"]]], RowBox[List["(", RowBox[List[FractionBox["\[Pi]", "2"], "-", RowBox[List["ArcSec", "[", "z", "]"]]]], ")"]]]], "-", RowBox[List[FractionBox["\[Pi]", "2"], " ", RowBox[List["(", RowBox[List[FractionBox[SqrtBox[SuperscriptBox["z", "2"]], "z"], "-", "1"]], ")"]]]]]], ")"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mi> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mrow> <mi> z </mi> <mo> - </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> </msqrt> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <msqrt> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msqrt> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mi> ⅈ </mi> <mi> z </mi> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mtext> </mtext> <mrow> <mrow> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mo> ⁢ </mo> <msqrt> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <msup> <mi> sec </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mi> z </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arccsc /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arcsec /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCsc", "[", SqrtBox[FractionBox[RowBox[List["2", " ", "z_"]], RowBox[List["z_", "-", SqrtBox[RowBox[List[SuperscriptBox["z_", "2"], "-", "1"]]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "z"]]], " ", SqrtBox[RowBox[List["-", FractionBox["\[ImaginaryI]", "z"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], " ", SqrtBox[FractionBox["1", SuperscriptBox["z", "2"]]], " ", RowBox[List["(", RowBox[List[FractionBox["\[Pi]", "2"], "-", RowBox[List["ArcSec", "[", "z", "]"]]]], ")"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox[SqrtBox[SuperscriptBox["z", "2"]], "z"], "-", "1"]], ")"]]]]]], ")"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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