Cosecant: Representations through equivalent functions (formula 01.10.27.0075)

Csc

$Mathematica Notation$

Elementary Functions

Csc[z]

Representations through equivalent functions

With inverse function

http://functions.wolfram.com/01.10.27.0075.01

Input Form

ArcCsc[Csc[z]] == Piecewise[ {{(-1)^Floor[(2 Re[z] + Pi)/(2 Pi)] (-z + Pi Floor[(2 Re[z] - Pi)/(2 Pi)]), Element[(2 Re[z] + Pi)/(2 Pi), Integers] && Im[z] <= 0}, {(-1)^Floor[(2 Re[z] + Pi)/(2 Pi)] (z - Pi Floor[(2 Re[z] + Pi)/(2 Pi)]), True}}]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["ArcCsc", "[", RowBox[List["Csc", "[", "z", "]"]], "]"]], "\[Equal]", RowBox[List["Piecewise", "[", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["2", RowBox[List["Re", "[", "z", "]"]]]], "+", "\[Pi]"]], RowBox[List["2", "\[Pi]"]]], "]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "z"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["2", RowBox[List["Re", "[", "z", "]"]]]], "-", "\[Pi]"]], RowBox[List["2", "\[Pi]"]]], "]"]]]]]], ")"]]]], ",", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["2", RowBox[List["Re", "[", "z", "]"]]]], "+", "\[Pi]"]], RowBox[List["2", "\[Pi]"]]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Im", "[", "z", "]"]], "\[LessEqual]", "0"]]]]]], "}"]], ",", "\[IndentingNewLine]", RowBox[List["{", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["2", RowBox[List["Re", "[", "z", "]"]]]], "+", "\[Pi]"]], RowBox[List["2", "\[Pi]"]]], "]"]]], " ", RowBox[List["(", RowBox[List["z", "-", RowBox[List["\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["2", RowBox[List["Re", "[", "z", "]"]]]], "+", "\[Pi]"]], RowBox[List["2", "\[Pi]"]]], "]"]]]]]], ")"]]]], ",", "True"]], "}"]]]], "}"]], "]"]]]]]]

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> <mrow> <mi> csc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mo>  </mo> <mtable> <mtr> <mtd> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mi> π </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mi> π </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mi> π </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ≤ </mo> <mn> 0 </mn> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mi> π </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mi> π </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mtd> <mtd> <mi> True </mi> </mtd> </mtr> </mtable> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arccsc /> <apply> <csc /> <ci> z </ci> </apply> </apply> <apply> <ci> Piecewise </ci> <list> <list> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <real /> <ci> z </ci> </apply> </apply> <pi /> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <pi /> <apply> <floor /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <real /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <real /> <ci> z </ci> </apply> </apply> <pi /> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <integers /> </apply> <apply> <leq /> <apply> <imaginary /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </list> <list> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <real /> <ci> z </ci> </apply> </apply> <pi /> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <floor /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <real /> <ci> z </ci> </apply> </apply> <pi /> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <true /> </list> </list> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCsc", "[", RowBox[List["Csc", "[", "z_", "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["\[Piecewise]", GridBox[List[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "z"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]], "-", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Im", "[", "z", "]"]], "\[LessEqual]", "0"]]]]], List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]], " ", RowBox[List["(", RowBox[List["z", "-", RowBox[List["\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]], ")"]]]], "True"]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]]]]]]

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

2007-05-02