Hyperbolic cosecant: Integration (formula 01.23.21.0241)

Csch

$Mathematica Notation$

Elementary Functions

Csch[z]

Integration

Indefinite integration

Involving one direct function and elementary functions

Involving hyperbolic, exponential and a power functions

Involving coth, exp and power

Involving zⁿ e^{p z}coth(c z) csch( c z)

http://functions.wolfram.com/01.23.21.0241.01

Input Form

Integrate[z^n E^(p z) Coth[c z] Csch[c z], z] == 2 E^(2 c z) n! (E^((-c + p) z) Sum[(1/(-j + n)!) (-1)^j z^(-j + n) (p + c)^(-1 - j) HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, j + 1], 2}, {1 + Subscript[a, 1], \[Ellipsis], 1 + Subscript[a, j + 1]}, E^(2 c z)], {j, 0, n}] + E^((c + p) z) Sum[(1/(-j + n)!) (-1)^j z^(-j + n) (p + 3 c)^(-1 - j) HypergeometricPFQ[{Subscript[b, 1], \[Ellipsis], Subscript[b, j + 1], 2}, {1 + Subscript[b, 1], \[Ellipsis], 1 + Subscript[b, j + 1]}, E^(2 c z)], {j, 0, n}]) /; Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, n + 1] == (p + c)/(2 c) && Subscript[b, 1] == Subscript[b, 2] == \[Ellipsis] == Subscript[b, n + 1] == (p + 3 c)/(2 c) && Element[n, Integers] && n >= 0 && p != -c && p != -3 c

Standard Form

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MathML Form

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<apply> <plus /> <apply> <times /> <apply> <plus /> <ci> p </ci> <apply> <times /> <cn type='integer'> 3 </cn> <ci> c </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </list> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2002-12-18