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Csch






Mathematica Notation

Traditional 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





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MathML Form







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</mo> <mtext> &#8201; </mtext> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#960; </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, 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> &#945; </ci> </list> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <imaginaryi /> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> <apply> <power /> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </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> &#945; </ci> </apply> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> <apply> <ci> PolyGamma </ci> <ci> &#945; </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> &#945; </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> &#945; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </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> &#945; </ci> </apply> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> <apply> <ci> PolyGamma </ci> <ci> &#945; </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> &#945; </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> &#945; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <limit /> <bvar> <ci> &#957; </ci> </bvar> <condition> <apply> <tendsto /> <ci> &#957; </ci> <ci> &#945; </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> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </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> &#957; 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Rule Form





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





2007-05-02





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