html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Coth

 http://functions.wolfram.com/01.22.21.0220.01

 Input Form

 Integrate[z^n Sin[a z]^m Cosh[b z]^u Coth[c z], z] == (-2^(-m - u)) Binomial[m, m/2] Binomial[u, u/2] n! (1 - Mod[m, 2]) (1 - Mod[u, 2]) (z^(1 + n)/(1 + n)! + 2 E^(2 c z) Sum[(1/(-j + n)!) ((-1)^j 2^(-1 - j) c^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[b, 1], \[Ellipsis], Subscript[b, 2 + j]}, {1 + Subscript[b, 1], \[Ellipsis], 1 + Subscript[b, 1 + j]}, E^(2 c z)]), {j, 0, n}]) - 2^(-m - u) Binomial[u, u/2] n! (1 - Mod[u, 2]) Sum[(-1)^k Binomial[m, k] (E^((I m Pi)/2) (Sum[(1/(-j + n)!) ((-1)^j ((-I) a (-2 k + m))^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[e, 1], \[Ellipsis], Subscript[e, 1 + j], 1}, {1 + Subscript[e, 1], \[Ellipsis], 1 + Subscript[e, 1 + j]}, E^(2 c z)]), {j, 0, n}]/ E^(I a (-2 k + m) z) + E^((2 c - I a (-2 k + m)) z) Sum[(1/(-j + n)!) ((-1)^j (2 c - I a (-2 k + m))^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[f, 1], \[Ellipsis], Subscript[f, 1 + j], 1}, {1 + Subscript[f, 1], \[Ellipsis], 1 + Subscript[f, 1 + j]}, E^(2 c z)]), {j, 0, n}]) + (E^(I a (-2 k + m) z) Sum[(1/(-j + n)!) ((-1)^j (I a (-2 k + m))^ (-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[h, 1], \[Ellipsis], Subscript[h, 1 + j], 1}, {1 + Subscript[h, 1], \[Ellipsis], 1 + Subscript[h, 1 + j]}, E^(2 c z)]), {j, 0, n}] + E^((2 c + I a (-2 k + m)) z) Sum[(1/(-j + n)!) ((-1)^j (2 c + I a (-2 k + m))^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[i, 1], \[Ellipsis], Subscript[i, 1 + j], 1}, {1 + Subscript[i, 1], \[Ellipsis], 1 + Subscript[i, 1 + j]}, E^(2 c z)]), {j, 0, n}])/ E^((1/2) I m Pi)), {k, 0, Floor[(1/2) (-1 + m)]}] - 2^(-m - u) Binomial[m, m/2] n! (1 - Mod[m, 2]) Sum[Binomial[u, i] (Sum[(1/(-j + n)!) ((-1)^j ((-b) (-2 i + u))^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[k, 1], \[Ellipsis], Subscript[k, 1 + j], 1}, {1 + Subscript[k, 1], \[Ellipsis], 1 + Subscript[k, 1 + j]}, E^(2 c z)]), {j, 0, n}]/ E^(b (-2 i + u) z) + E^((2 c - b (-2 i + u)) z) Sum[(1/(-j + n)!) ((-1)^j (2 c - b (-2 i + u))^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[l, 1], \[Ellipsis], Subscript[l, 1 + j], 1}, {1 + Subscript[l, 1], \[Ellipsis], 1 + Subscript[l, 1 + j]}, E^(2 c z)]), {j, 0, n}] + E^(b (-2 i + u) z) Sum[(1/(-j + n)!) ((-1)^j (b (-2 i + u))^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[n, 1], \[Ellipsis], Subscript[n, 1 + j], 1}, {1 + Subscript[n, 1], \[Ellipsis], 1 + Subscript[n, 1 + j]}, E^(2 c z)]), {j, 0, n}] + E^((2 c + b (-2 i + u)) z) Sum[(1/(-j + n)!) ((-1)^j (2 c + b (-2 i + u))^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[o, 1], \[Ellipsis], Subscript[o, 1 + j], 1}, {1 + Subscript[o, 1], \[Ellipsis], 1 + Subscript[o, 1 + j]}, E^(2 c z)]), {j, 0, n}]), {i, 0, Floor[(1/2) (-1 + u)]}] - 2^(-m - u) n! Sum[(-1)^k Binomial[m, k] Binomial[u, i] (E^((1/2) I Pi m) (E^(((-I) a (-2 k + m) - b (-2 i + u)) z) Sum[(1/(-j + n)!) ((-1)^j ((-I) a (-2 k + m) - b (-2 i + u))^ (-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[q, 1], \[Ellipsis], Subscript[q, 1 + j], 1}, {1 + Subscript[q, 1], \[Ellipsis], 1 + Subscript[q, 1 + j]}, E^(2 c z)]), {j, 0, n}] + E^((2 c - I a (-2 k + m) - b (-2 i + u)) z) Sum[(1/(-j + n)!) ((-1)^j (2 c - I a (-2 k + m) - b (-2 i + u))^ (-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[r, 1], \[Ellipsis], Subscript[r, 1 + j], 1}, {1 + Subscript[r, 1], \[Ellipsis], 1 + Subscript[r, 1 + j]}, E^(2 c z)]), {j, 0, n}]) + (E^((I a (-2 k + m) - b (-2 i + u)) z) Sum[(1/(-j + n)!) ((-1)^j (I a (-2 k + m) - b (-2 i + u))^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[t, 1], \[Ellipsis], Subscript[t, 1 + j], 1}, {1 + Subscript[t, 1], \[Ellipsis], 1 + Subscript[t, 1 + j]}, E^(2 c z)]), {j, 0, n}] + E^((2 c + I a (-2 k + m) - b (-2 i + u)) z) Sum[(1/(-j + n)!) ((-1)^j (2 c + I a (-2 k + m) - b (-2 i + u))^ (-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[u, 1], \[Ellipsis], Subscript[u, 1 + j], 1}, {1 + Subscript[u, 1], \[Ellipsis], 1 + Subscript[u, 1 + j]}, E^(2 c z)]), {j, 0, n}])/E^((1/2) I Pi m) + E^((1/2) I Pi m) (E^(((-I) a (-2 k + m) + b (-2 i + u)) z) Sum[(1/(-j + n)!) ((-1)^j ((-I) a (-2 k + m) + b (-2 i + u))^ (-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[w, 1], \[Ellipsis], Subscript[w, 1 + j], 1}, {1 + Subscript[w, 1], \[Ellipsis], 1 + Subscript[w, 1 + j]}, E^(2 c z)]), {j, 0, n}] + E^((2 c - I a (-2 k + m) + b (-2 i + u)) z) Sum[(1/(-j + n)!) ((-1)^j (2 c - I a (-2 k + m) + b (-2 i + u))^ (-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[x, 1], \[Ellipsis], Subscript[x, 1 + j], 1}, {1 + Subscript[x, 1], \[Ellipsis], 1 + Subscript[x, 1 + j]}, E^(2 c z)]), {j, 0, n}]) + (E^((I a (-2 k + m) + b (-2 i + u)) z) Sum[(1/(-j + n)!) ((-1)^j (I a (-2 k + m) + b (-2 i + u))^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[z, 1], \[Ellipsis], Subscript[z, 1 + j], 1}, {1 + Subscript[z, 1], \[Ellipsis], 1 + Subscript[z, 1 + j]}, E^(2 c z)]), {j, 0, n}] + E^((2 c + I a (-2 k + m) + b (-2 i + u)) z) Sum[(1/(-j + n)!) ((-1)^j (2 c + I a (-2 k + m) + b (-2 i + u))^ (-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[\[Alpha], 1], \[Ellipsis], Subscript[\[Alpha], 1 + j], 1}, {1 + Subscript[\[Alpha], 1], \[Ellipsis], 1 + Subscript[ \[Alpha], 1 + j]}, E^(2 c z)]), {j, 0, n}])/ E^((1/2) I Pi m)), {i, 0, Floor[(1/2) (-1 + u)]}, {k, 0, Floor[(1/2) (-1 + m)]}] /; Subscript[b, 1] == Subscript[b, 2] == \[Ellipsis] == Subscript[b, n + 2] == 1 && Subscript[e, 1] == Subscript[e, 2] == \[Ellipsis] == Subscript[e, n + 1] == ((-I) a (-2 k + m))/(2 c) && Subscript[f, 1] == Subscript[f, 2] == \[Ellipsis] == Subscript[f, n + 1] == ((-I) a (-2 k + m) + 2 c)/(2 c) && Subscript[h, 1] == Subscript[h, 2] == \[Ellipsis] == Subscript[h, n + 1] == (I a (-2 k + m))/(2 c) && Subscript[i, 1] == Subscript[i, 2] == \[Ellipsis] == Subscript[i, n + 1] == (I a (-2 k + m) + 2 c)/(2 c) && Subscript[k, 1] == Subscript[k, 2] == \[Ellipsis] == Subscript[k, n + 1] == ((-b) (-2 i + u))/(2 c) && Subscript[l, 1] == Subscript[l, 2] == \[Ellipsis] == Subscript[l, n + 1] == ((-b) (-2 i + u) + 2 c)/(2 c) && Subscript[n, 1] == Subscript[n, 2] == \[Ellipsis] == Subscript[n, n + 1] == (b (-2 i + u))/(2 c) && Subscript[o, 1] == Subscript[o, 2] == \[Ellipsis] == Subscript[o, n + 1] == (b (-2 i + u) + 2 c)/(2 c) && Subscript[q, 1] == Subscript[q, 2] == \[Ellipsis] == Subscript[q, n + 1] == ((-I) a (-2 k + m) - b (-2 i + u))/ (2 c) && Subscript[r, 1] == Subscript[r, 2] == \[Ellipsis] == Subscript[r, n + 1] == ((-I) a (-2 k + m) - b (-2 i + u) + 2 c)/(2 c) && Subscript[t, 1] == Subscript[t, 2] == \[Ellipsis] == Subscript[t, n + 1] == (I a (-2 k + m) - b (-2 i + u))/(2 c) && Subscript[u, 1] == Subscript[u, 2] == \[Ellipsis] == Subscript[u, n + 1] == (I a (-2 k + m) - b (-2 i + u) + 2 c)/(2 c) && Subscript[w, 1] == Subscript[w, 2] == \[Ellipsis] == Subscript[w, n + 1] == ((-I) a (-2 k + m) + b (-2 i + u))/(2 c) && Subscript[x, 1] == Subscript[x, 2] == \[Ellipsis] == Subscript[x, n + 1] == ((-I) a (-2 k + m) + b (-2 i + u) + 2 c)/(2 c) && Subscript[z, 1] == Subscript[z, 2] == \[Ellipsis] == Subscript[z, n + 1] == (I a (-2 k + m) + b (-2 i + u))/(2 c) && Subscript[\[Alpha], 1] == Subscript[\[Alpha], 2] == \[Ellipsis] == Subscript[\[Alpha], n + 1] == (I a (-2 k + m) + b (-2 i + u) + 2 c)/(2 c) && Element[n, Integers] && n >= 0 && Element[m, Integers] && m > 0 && Element[u, Integers] && u > 0

 Standard Form

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 MathML Form

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n m + u + Condition z z n a z m b z u c z -1 2 -1 m -1 u Binomial m m 2 -1 Binomial u u 2 -1 n 1 -1 \$CellContext`m 2 1 -1 \$CellContext`u 2 z n 1 n 1 -1 2 2 c z j 0 n -1 j 2 -1 j -1 c -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ 1 1 2 2 2 c z -1 2 -1 m -1 u Binomial u u 2 -1 n 1 -1 \$CellContext`u 2 k 0 m -1 2 -1 -1 k Binomial m k m 2 -1 -1 a m -1 2 k z j 0 n -1 j -1 a m -1 2 k -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ -1 a m -1 2 k 2 c -1 -1 a m -1 2 k 2 c -1 1 1 -1 a m -1 2 k 2 c -1 1 -1 a m -1 2 k 2 c -1 2 c z 2 c -1 a m -1 2 k z j 0 n -1 j 2 c -1 a m -1 2 k -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ 2 c -1 a m -1 2 k 2 c -1 2 c -1 a m -1 2 k 2 c -1 1 2 c -1 a m -1 2 k 2 c -1 1 2 c -1 a m -1 2 k 2 c -1 1 2 c z -1 1 2 m a m -1 2 k z j 0 n -1 j a m -1 2 k -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ a m -1 2 k 2 c -1 a m -1 2 k 2 c -1 1 a m -1 2 k 2 c -1 1 a m -1 2 k 2 c -1 1 2 c z 2 c a m -1 2 k z j 0 n -1 j 2 c a m -1 2 k -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ 2 c a m -1 2 k 2 c -1 2 c a m -1 2 k 2 c -1 1 2 c a m -1 2 k 2 c -1 1 2 c a m -1 2 k 2 c -1 1 2 c z -1 2 -1 m -1 u Binomial m m 2 -1 n 1 -1 \$CellContext`m 2 i 0 u -1 2 -1 Binomial u i -1 b u -1 2 i z j 0 n -1 j -1 b u -1 2 i -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ -1 b u -1 2 i 2 c -1 -1 b u -1 2 i 2 c -1 1 1 -1 b u -1 2 i 2 c -1 1 -1 b u -1 2 i 2 c -1 2 c z b u -1 2 i z j 0 n -1 j b u -1 2 i -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ b u -1 2 i 2 c -1 b u -1 2 i 2 c -1 1 b u -1 2 i 2 c -1 1 b u -1 2 i 2 c -1 1 2 c z 2 c -1 b u -1 2 i z j 0 n -1 j 2 c -1 b u -1 2 i -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ 2 c -1 b u -1 2 i 2 c -1 2 c -1 b u -1 2 i 2 c -1 1 2 c -1 b u -1 2 i 2 c -1 1 2 c -1 b u -1 2 i 2 c -1 1 2 c z 2 c b u -1 2 i z j 0 n -1 j 2 c b u -1 2 i -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ 2 c b u -1 2 i 2 c -1 2 c b u -1 2 i 2 c -1 1 2 c b u -1 2 i 2 c -1 1 2 c b u -1 2 i 2 c -1 1 2 c z -1 2 -1 m -1 u n k 0 m -1 2 -1 i 0 u -1 2 -1 -1 k Binomial m k Binomial u i m 2 -1 -1 a m -1 2 k -1 b u -1 2 i z j 0 n -1 j -1 a m -1 2 k -1 b u -1 2 i -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ -1 a m -1 2 k -1 b u -1 2 i 2 c -1 -1 a m -1 2 k -1 b u -1 2 i 2 c -1 1 -1 a m -1 2 k -1 b u -1 2 i 2 c -1 1 -1 a m -1 2 k -1 b u -1 2 i 2 c -1 1 2 c z 2 c -1 a m -1 2 k -1 b u -1 2 i z j 0 n -1 j 2 c -1 a m -1 2 k -1 b u -1 2 i -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ 2 c -1 a m -1 2 k -1 b u -1 2 i 2 c -1 2 c -1 a m -1 2 k -1 b u -1 2 i 2 c -1 1 2 c -1 a m -1 2 k -1 b u -1 2 i 2 c -1 1 2 c -1 a m -1 2 k -1 b u -1 2 i 2 c -1 1 2 c z -1 1 2 m a m -1 2 k -1 b u -1 2 i z j 0 n -1 j a m -1 2 k -1 b u -1 2 i -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ a m -1 2 k -1 b u -1 2 i 2 c -1 a m -1 2 k -1 b u -1 2 i 2 c -1 1 a m -1 2 k -1 b u -1 2 i 2 c -1 1 a m -1 2 k -1 b u -1 2 i 2 c -1 1 2 c z 2 c a m -1 2 k -1 b u -1 2 i z j 0 n -1 j 2 c a m -1 2 k -1 b u -1 2 i -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ 2 c a m -1 2 k -1 b u -1 2 i 2 c -1 2 c a m -1 2 k -1 b u -1 2 i 2 c -1 1 2 c a m -1 2 k -1 b u -1 2 i 2 c -1 1 2 c a m -1 2 k -1 b u -1 2 i 2 c -1 1 2 c z m 2 -1 b u -1 2 i -1 a m -1 2 k z j 0 n -1 j b u -1 2 i -1 a m -1 2 k -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ b u -1 2 i -1 a m -1 2 k 2 c -1 b u -1 2 i -1 a m -1 2 k 2 c -1 1 b u -1 2 i -1 a m -1 2 k 2 c -1 1 b u -1 2 i -1 a m -1 2 k 2 c -1 1 2 c z 2 c -1 a m -1 2 k b u -1 2 i z j 0 n -1 j 2 c -1 a m -1 2 k b u -1 2 i -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ 2 c -1 a m -1 2 k b u -1 2 i 2 c -1 2 c -1 a m -1 2 k b u -1 2 i 2 c -1 1 2 c -1 a m -1 2 k b u -1 2 i 2 c -1 1 2 c -1 a m -1 2 k b u -1 2 i 2 c -1 1 2 c z -1 1 2 m a m -1 2 k b u -1 2 i z j 0 n -1 j a m -1 2 k b u -1 2 i -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ a m -1 2 k b u -1 2 i 2 c -1 a m -1 2 k b u -1 2 i 2 c -1 1 a m -1 2 k b u -1 2 i 2 c -1 1 a m -1 2 k b u -1 2 i 2 c -1 1 2 c z 2 c a m -1 2 k b u -1 2 i z j 0 n -1 j 2 c a m -1 2 k b u -1 2 i -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ 2 c a m -1 2 k b u -1 2 i 2 c -1 2 c a m -1 2 k b u -1 2 i 2 c -1 1 2 c a m -1 2 k b u -1 2 i 2 c -1 1 2 c a m -1 2 k b u -1 2 i 2 c -1 1 2 c z n m SuperPlus u SuperPlus [/itex]

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

 2002-12-18