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ArcCsc




Mathematica Notation

Traditional Notation









Elementary Functions > ArcCsc[z] > Representations through equivalent functions > With related functions > Involving cot-1 > Involving csc-1((a zc+1)1/2) > Involving csc-1((a zc+1)1/2) and cot-1(a1/2 zc/2)





http://functions.wolfram.com/01.17.27.0826.01









  


  








Input Form





ArcCsc[Sqrt[a z^c + 1]] == (Sqrt[a] z^(c/2) ArcCot[Sqrt[a] z^(c/2)])/ Sqrt[a z^c] - (1/2) I Pi (Sqrt[(-I) Sqrt[a] Sqrt[z^c] - 1]/ Sqrt[I Sqrt[a] Sqrt[z^c] + 1] + Sqrt[I Sqrt[a] Sqrt[z^c] - 1]/ Sqrt[1 - I Sqrt[a] Sqrt[z^c]])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox[RowBox[List[RowBox[List["a", " ", SuperscriptBox["z", "c"]]], "+", "1"]]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SqrtBox["a"], " ", SuperscriptBox["z", RowBox[List["c", "/", "2"]]], " ", RowBox[List["ArcCot", "[", RowBox[List[SqrtBox["a"], " ", SuperscriptBox["z", RowBox[List["c", "/", "2"]]]]], "]"]]]], SqrtBox[RowBox[List["a", " ", SuperscriptBox["z", "c"]]]]], "-", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox[SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SqrtBox["a"], " ", SqrtBox[SuperscriptBox["z", "c"]]]], "-", "1"]]], SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox["a"], " ", SqrtBox[SuperscriptBox["z", "c"]]]], "+", "1"]]]], "+", FractionBox[SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox["a"], " ", SqrtBox[SuperscriptBox["z", "c"]]]], "-", "1"]]], SqrtBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["a"], " ", SqrtBox[SuperscriptBox["z", "c"]]]]]]]]]], ")"]]]]]]]]]]








MathML Form







<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> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mrow> <msqrt> <mi> a </mi> </msqrt> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mi> c </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mtext> </mtext> </mrow> <msqrt> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> cot </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <msqrt> <mi> a </mi> </msqrt> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mi> c </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCsc", "[", SqrtBox[RowBox[List[RowBox[List["a_", " ", SuperscriptBox["z_", "c_"]]], "+", "1"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SqrtBox["a"], " ", SuperscriptBox["z", RowBox[List["c", "/", "2"]]], " ", RowBox[List["ArcCot", "[", RowBox[List[SqrtBox["a"], " ", SuperscriptBox["z", RowBox[List["c", "/", "2"]]]]], "]"]]]], SqrtBox[RowBox[List["a", " ", SuperscriptBox["z", "c"]]]]], "-", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox[SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SqrtBox["a"], " ", SqrtBox[SuperscriptBox["z", "c"]]]], "-", "1"]]], SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox["a"], " ", SqrtBox[SuperscriptBox["z", "c"]]]], "+", "1"]]]], "+", FractionBox[SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox["a"], " ", SqrtBox[SuperscriptBox["z", "c"]]]], "-", "1"]]], SqrtBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["a"], " ", SqrtBox[SuperscriptBox["z", "c"]]]]]]]]]], ")"]]]]]]]]]]


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





2003-08-21