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Sech






Mathematica Notation

Traditional Notation









Elementary Functions > Sech[z] > Integration > Indefinite integration > Involving functions of the direct function, hyperbolic and a power functions > Involving powers of the direct function, hyperbolic and a power functions > Involving cosh and power > Involving zncosh(a+b z) sechv( c z)





http://functions.wolfram.com/01.24.21.0486.01









  


  










Input Form





Integrate[z^n Cosh[q \[Nu] c z] Sech[c z]^\[Nu], z] == (n!/2) Sech[c z]^\[Nu] (1 + E^(2 c z))^\[Nu] (E^(q \[Nu] c z) Sum[(((-1)^j z^(-j + n))/((n - j)! (c \[Nu] (q + 1))^ (j + 1))) HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, j + 1], \[Nu]}, {1 + Subscript[a, 1], \[Ellipsis], 1 + Subscript[a, j + 1]}, -E^(2 c z)], {j, 0, n}] + ((-1)^((\[Nu] (q - 1))/2) Gamma[(\[Nu] (q + 1))/2] z^(n + 1))/ (E^(c \[Nu] z) (Gamma[1 + (\[Nu] (q - 1))/2] Gamma[\[Nu]] (n + 1)!)) - Sum[(((-1)^k Pochhammer[\[Nu], k] z^(-j + n))/ ((c (q \[Nu] - 2 k - \[Nu]))^(j + 1) (n - j)! k!)) E^(c z (2 k - q \[Nu])), {j, 0, n}, {k, 0, (\[Nu] (q - 1))/2 - 1}] + (-1)^((\[Nu] (q - 1))/2) Sum[((E^(c z (2 - \[Nu])) Pochhammer[\[Nu], 1 + (\[Nu] (q - 1))/2] z^(-j + n))/((n - j)! (-2 c)^(1 + j) (1 + (\[Nu] (q - 1))/2)!)) HypergeometricPFQ[{Subscript[b, 1], \[Ellipsis], Subscript[b, j + 2], 1 + (\[Nu] (q + 1))/2}, {1 + Subscript[b, 1], \[Ellipsis], 1 + Subscript[b, j + 1], 2 + (\[Nu] (q - 1))/2}, -E^(2 c z)], {j, 0, n}]) /; Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, n + 1] == (\[Nu] (q + 1))/2 && Subscript[b, 1] == Subscript[b, 2] == \[Ellipsis] == Subscript[b, n + 2] == 1 && Element[n, Integers] && n >= 0 && Element[(\[Nu] (q - 1))/2, Integers] && (\[Nu] (q - 1))/2 >= 0










Standard Form





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MathML Form







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</ci> <apply> <plus /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </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 /> <exponentiale /> <apply> <times /> <ci> c </ci> <ci> z </ci> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <apply> <ci> Pochhammer </ci> <ci> &#957; </ci> <apply> <plus /> <apply> <times /> <ci> &#957; </ci> <apply> <plus /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </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> <times /> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> c </ci> </apply> <apply> <plus /> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <factorial /> <apply> <plus /> <apply> <times /> <ci> &#957; 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Rule Form





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





2002-12-18