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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 and trigonometric functions > Involving powers of sin and powers of sinh > Involving sinm(a z) sinhu(b z) coth(c z)





http://functions.wolfram.com/01.22.21.0176.01









  


  










Input Form





Integrate[Sin[a z]^m Sinh[c z]^\[Mu] Coth[c z], z] == (-(1/(c (-2 + \[Mu]) \[Mu]))) ((2^(-m - \[Mu]) (-E^((-c) z) + E^(c z))^ \[Mu] Binomial[m, m/2] (E^(2 c z) \[Mu] Hypergeometric2F1[1 - \[Mu]/2, 1 - \[Mu], 2 - \[Mu]/2, E^(2 c z)] + (-2 + \[Mu]) Hypergeometric2F1[-(\[Mu]/2), 1 - \[Mu], 1 - \[Mu]/2, E^(2 c z)]) (-1 + Mod[m, 2]))/(1 - E^(2 c z))^\[Mu]) - (2^(-m - \[Mu]) (-E^((-c) z) + E^(c z))^\[Mu] Sum[(-1)^k Binomial[m, k] ((E^((I m Pi)/2 - I a (-2 k + m) z) Hypergeometric2F1[((-I) a (-2 k + m) - c \[Mu])/(2 c), 1 - \[Mu], ((-I) a (-2 k + m) - c (\[Mu] - 2))/(2 c), E^(2 c z)])/ ((-I) a (-2 k + m) - c \[Mu]) + (E^((I m Pi)/2 + (2 c - I a (-2 k + m)) z) Hypergeometric2F1[ ((-I) a (-2 k + m) - c (\[Mu] - 2))/(2 c), 1 - \[Mu], ((-I) a (-2 k + m) - c (\[Mu] - 4))/(2 c), E^(2 c z)])/ (I a (2 k - m) - c (\[Mu] - 2)) + (E^((-(1/2)) I m Pi + I a (-2 k + m) z) Hypergeometric2F1[ (I a (-2 k + m) - c \[Mu])/(2 c), 1 - \[Mu], (I a (-2 k + m) - c (\[Mu] - 2))/(2 c), E^(2 c z)])/ (I a (-2 k + m) - c \[Mu]) + (E^((-(1/2)) I m Pi + (2 c + I a (-2 k + m)) z) Hypergeometric2F1[ (I a (-2 k + m) - c (\[Mu] - 2))/(2 c), 1 - \[Mu], (I a (-2 k + m) - c (\[Mu] - 4))/(2 c), E^(2 c z)])/ (I a (-2 k + m) - c (\[Mu] - 2))), {k, 0, Floor[(1/2) (-1 + m)]}])/ (1 - E^(2 c z))^\[Mu] /; Element[m, Integers] && m > 0










Standard Form





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MathML Form







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</ci> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> <apply> <ci> Binomial </ci> <ci> m </ci> <apply> <times /> <ci> m </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> -2 </cn> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#956; 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Rule Form





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





2002-12-18





© 1998- Wolfram Research, Inc.