Hyperbolic cosecant: Differentiation (formula 01.23.20.0008)

Csch

$Mathematica Notation$

Elementary Functions

Csch[z]

Differentiation

Fractional integro-differentiation

http://functions.wolfram.com/01.23.20.0008.01

Input Form

Derivative[\[Alpha]][Csch][c z] == (I^(\[Alpha] + 1) Pi^(-\[Alpha] - 1) ((((-I) c z)^\[Alpha] PolyGamma[\[Alpha], -((I c z)/(2 Pi))])/ (I c z)^\[Alpha] - PolyGamma[\[Alpha], (I c z)/(2 Pi)]))/2^\[Alpha] - I^(\[Alpha] + 1) Pi^(-\[Alpha] - 1) ((((-I) c z)^\[Alpha] PolyGamma[\[Alpha], -((I c z)/Pi)])/ (I c z)^\[Alpha] - PolyGamma[\[Alpha], (I c z)/Pi]) + I^(\[Alpha] + 1) Limit[(1/Gamma[-\[Nu]]) ((I c z)^(-1 - \[Nu]) (EulerGamma + 4 Log[2] + 2 Log[Pi] - Log[(-I) c z] + PolyGamma[-\[Nu]])), \[Nu] -> \[Alpha]]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", "\[Alpha]", "]"]], "[", "Csch", "]"]], "[", RowBox[List["c", " ", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["\[Alpha]", "+", "1"]]], " ", SuperscriptBox["2", RowBox[List["-", "\[Alpha]"]]], SuperscriptBox["\[Pi]", RowBox[List[RowBox[List["-", "\[Alpha]"]], "-", "1"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c", " ", "z"]], ")"]], "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], ")"]], RowBox[List["-", "\[Alpha]"]]], RowBox[List["PolyGamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], RowBox[List["2", "\[Pi]"]]]]]]], "]"]]]], "-", RowBox[List["PolyGamma", "[", RowBox[List["\[Alpha]", ",", FractionBox[RowBox[List[" ", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]]]], RowBox[List["2", "\[Pi]"]]]]], "]"]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["\[Alpha]", "+", "1"]]], " ", SuperscriptBox["\[Pi]", RowBox[List[RowBox[List["-", "\[Alpha]"]], "-", "1"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c", " ", "z"]], ")"]], "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], ")"]], RowBox[List["-", "\[Alpha]"]]], RowBox[List["PolyGamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], "\[Pi]"]]]]], "]"]]]], "-", RowBox[List["PolyGamma", "[", RowBox[List["\[Alpha]", ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], "\[Pi]"]]], "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["\[Alpha]", "+", "1"]]], " ", RowBox[List["Limit", "[", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List["EulerGamma", "+", RowBox[List["4", RowBox[List["Log", "[", "2", "]"]]]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", "\[Pi]", "]"]]]], "-", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c", " ", "z"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]], ")"]]]], ")"]]]], ",", RowBox[List["\[Nu]", "\[Rule]", "\[Alpha]"]]]], "]"]]]]]]]]]]

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> csch </mi> <semantics> <mrow> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", "\[Alpha]", ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mrow> <msup> <mi> ⅈ </mi> <mrow> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mi> α </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> π </mi> <mrow> <mrow> <mo> - </mo> <mi> α </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> α </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> α </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mi> ⅈ </mi> <mrow> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> π </mi> <mrow> <mrow> <mo> - </mo> <mi> α </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> α </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> α </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mi> π </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mi> π </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mi> ⅈ </mi> <mrow> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> ν </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> α </mi> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> π </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> D </ci> <apply> <csch /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <list> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> <ci> α </ci> </list> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <imaginaryi /> <apply> <plus /> <ci> α </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> <apply> <power /> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> c </ci> <ci> z </ci> </apply> <ci> α </ci> </apply> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> <apply> <ci> PolyGamma </ci> <ci> α </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <ci> α </ci> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <imaginaryi /> <apply> <plus /> <ci> α </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> c </ci> <ci> z </ci> </apply> <ci> α </ci> </apply> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> <apply> <ci> PolyGamma </ci> <ci> α </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> z </ci> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <ci> α </ci> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> z </ci> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <imaginaryi /> <apply> <plus /> <ci> α </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <limit /> <bvar> <ci> ν </ci> </bvar> <condition> <apply> <tendsto /> <ci> ν </ci> <ci> α </ci> </apply> </condition> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ln /> <pi /> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <eulergamma /> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2007-05-02