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Coth






Mathematica Notation

Traditional Notation









Elementary Functions > Coth[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving hyperbolic, exponential and a power functions > Involving sinh, exp and power > Involving zn ep z sinh(a+b z) coth( c z)





http://functions.wolfram.com/01.22.21.0191.01









  


  










Input Form





Integrate[(z^n Sinh[b z] Coth[c z])/E^(b z), z] == (-(1/2)) (z^(1 + n)/(1 + n) + 2 E^(2 c z) n! Sum[(1/(-j + n)!) (-1)^j 2^(-1 - j) c^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, j + 2]}, {1 + Subscript[a, 1], \[Ellipsis], 1 + Subscript[a, j + 1]}, E^(2 c z)], {j, 0, n}] - n! (Sum[(1/(-j + n)!) (-1)^j (-2 b)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[b, 1], \[Ellipsis], Subscript[b, 1 + j], 1}, {1 + Subscript[b, 1], \[Ellipsis], 1 + Subscript[b, 1 + j]}, E^(2 c z)], {j, 0, n}]/E^(2 b z) + E^(2 (-b + c) z) Sum[(1/(-j + n)!) (-1)^j (-2 b + 2 c)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[c, 1], \[Ellipsis], Subscript[c, 1 + j], 1}, {1 + Subscript[c, 1], \[Ellipsis], 1 + Subscript[c, 1 + j]}, E^(2 c z)], {j, 0, n}])) /; Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, n + 2] == 1 && Subscript[b, 1] == Subscript[b, 2] == \[Ellipsis] == Subscript[b, n + 1] == -(b/c) && Subscript[c, 1] == Subscript[c, 2] == \[Ellipsis] == Subscript[c, n + 1] == (-b + c)/c && Element[n, Integers] && n >= 0










Standard Form





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MathML Form







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</ci> <cn type='integer'> 1 </cn> </list> <list> <cn type='integer'> 2 </cn> <ci> &#8230; </ci> <cn type='integer'> 2 </cn> </list> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <factorial /> <ci> n </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> b </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> &#8230; 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Rule Form





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





2002-12-18