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Csch






Mathematica Notation

Traditional Notation









Elementary Functions > Csch[z] > Integration > Indefinite integration > Involving functions of the direct function, hyperbolic, exponential and trigonometric functions > Involving powers of the direct function, hyperbolic, exponential and trigonometric functions > Involving powers of sin, powers of sinh and exp > Involving ep z sinm(a z) sinhu(b z) cschnu(c z)





http://functions.wolfram.com/01.23.21.0545.01









  


  










Input Form





Integrate[E^(p z) Sin[a z]^m Sinh[c z]^\[Mu] Csch[c z]^\[Nu], z] == (2^(-m - \[Mu])/(p + c (-\[Mu] + \[Nu]))) E^(p z) (-E^((-c) z) + E^(c z))^\[Mu] (1 - E^(2 c z))^(-\[Mu] + \[Nu]) Binomial[m, m/2] Hypergeometric2F1[(p - c \[Mu] + c \[Nu])/(2 c), -\[Mu] + \[Nu], (p + c (2 - \[Mu] + \[Nu]))/(2 c), E^(2 c z)] (1 - Mod[m, 2]) Csch[c z]^\[Nu] + 2^(-m - \[Mu]) (-E^((-c) z) + E^(c z))^\[Mu] (1 - E^(2 c z))^(-\[Mu] + \[Nu]) Csch[c z]^\[Nu] Sum[(-1)^k Binomial[m, k] (E^((I m Pi)/2 + (I a (2 k - m) + p) z) (Hypergeometric2F1[(I a (2 k - m) + p - c \[Mu] + c \[Nu])/(2 c), -\[Mu] + \[Nu], (I a (2 k - m) + p + c (2 - \[Mu] + \[Nu]))/(2 c), E^(2 c z)]/(I a (2 k - m) + p + c (-\[Mu] + \[Nu]))) + E^((-(1/2)) I m Pi + (I a (-2 k + m) + p) z) (Hypergeometric2F1[(I a (-2 k + m) + p - c \[Mu] + c \[Nu])/(2 c), -\[Mu] + \[Nu], (I a (-2 k + m) + p + c (2 - \[Mu] + \[Nu]))/(2 c), E^(2 c z)]/(I a (-2 k + m) + p + c (-\[Mu] + \[Nu])))), {k, 0, Floor[(1/2) (-1 + m)]}] /; Element[m, Integers] && m > 0










Standard Form





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MathML Form







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</mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mi> &#957; </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> <mo> ; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List[RowBox[List[&quot;a&quot;, &quot; &quot;, &quot;\[ImaginaryI]&quot;, &quot; &quot;, RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;k&quot;]], &quot;-&quot;, &quot;m&quot;]], &quot;)&quot;]]]], &quot;+&quot;, &quot;p&quot;, &quot;-&quot;, RowBox[List[&quot;c&quot;, &quot; 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</ci> </apply> </apply> <apply> <times /> <ci> c </ci> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> a </ci> <imaginaryi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <ci> p </ci> <apply> <times /> <ci> c </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> a </ci> <imaginaryi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <ci> p </ci> <apply> <times /> <ci> c </ci> <apply> <plus /> <ci> &#957; 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</ci> </apply> </apply> <apply> <times /> <ci> c </ci> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> a </ci> <imaginaryi /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> <ci> p </ci> <apply> <times /> <ci> c </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> a </ci> <imaginaryi /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> <ci> p </ci> <apply> <times /> <ci> c </ci> <apply> <plus /> <ci> &#957; 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Rule Form





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





2002-12-18