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; }

 Csch

 http://functions.wolfram.com/01.23.21.0278.01

 Input Form

 Integrate[z^n Sin[a z]^m Coth[c z]^u Csch[c z], z] == 2^(-m - u) (1 - E^(2 c z))^(1 + u) Binomial[m, m/2] Csch[c z]^(1 + u) n! (1 - Mod[m, 2]) (Binomial[u, u/2] (1 - Mod[u, 2]) Sum[(1/(-j + n)!) (-1)^j (c (1 + u))^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, 1 + j], 1 + u}, {1 + Subscript[a, 1], \[Ellipsis], 1 + Subscript[a, 1 + j]}, E^(2 c z)], {j, 0, n}] + Sum[Binomial[u, s] (Sum[(1/(-j + n)!) ((-1)^j (c (1 + 2 s))^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[b, 1], \[Ellipsis], Subscript[b, 1 + j], 1 + u}, {1 + Subscript[b, 1], \[Ellipsis], 1 + Subscript[b, 1 + j]}, E^(2 c z)]), {j, 0, n}]/ E^(c (-2 s + u) z) + E^(c (-2 s + u) z) Sum[(1/(-j + n)!) ((-1)^j (c (1 - 2 s + 2 u))^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[c, 1], \[Ellipsis], Subscript[c, 1 + j], 1 + u}, {1 + Subscript[c, 1], \[Ellipsis], 1 + Subscript[c, 1 + j]}, E^(2 c z)]), {j, 0, n}]), {s, 0, Floor[(1/2) (-1 + u)]}]) + 2^(-m - u) (1 - E^(2 c z))^(1 + u) Csch[c z]^(1 + u) n! Sum[(-1)^k Binomial[m, k] (E^((I m Pi)/2) ((Binomial[u, u/2] (1 - Mod[u, 2]) Sum[(1/(-j + n)!) ((-1)^j ((-I) a (-2 k + m) + c (1 + u))^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[d, 1], \[Ellipsis], Subscript[d, 1 + j], 1 + u}, {1 + Subscript[d, 1], \[Ellipsis], 1 + Subscript[d, 1 + j]}, E^(2 c z)]), {j, 0, n}])/E^(I a (-2 k + m) z) + Sum[Binomial[u, s] (E^(((-I) a (-2 k + m) - c (-2 s + u)) z) Sum[(1/(-j + n)!) ((-1)^j ((-I) a (-2 k + m) + c (1 + 2 s))^ (-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[e, 1], \[Ellipsis], Subscript[e, 1 + j], 1 + u}, {1 + Subscript[e, 1], \[Ellipsis], 1 + Subscript[e, 1 + j]}, E^(2 c z)]), {j, 0, n}] + E^(((-I) a (-2 k + m) + c (-2 s + u)) z) Sum[(1/(-j + n)!) ((-1)^j ((-I) a (-2 k + m) + c (1 - 2 s + 2 u))^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[f, 1], \[Ellipsis], Subscript[f, 1 + j], 1 + u}, {1 + Subscript[f, 1], \[Ellipsis], 1 + Subscript[f, 1 + j]}, E^(2 c z)]), {j, 0, n}]), {s, 0, Floor[(1/2) (-1 + u)]}]) + (E^(I a (-2 k + m) z) Binomial[u, u/2] (1 - Mod[u, 2]) Sum[(1/(-j + n)!) ((-1)^j (I a (-2 k + m) + c (1 + u))^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[g, 1], \[Ellipsis], Subscript[g, 1 + j], 1 + u}, {1 + Subscript[g, 1], \[Ellipsis], 1 + Subscript[g, 1 + j]}, E^(2 c z)]), {j, 0, n}] + Sum[Binomial[u, s] (E^((I a (-2 k + m) - c (-2 s + u)) z) Sum[(1/(-j + n)!) ((-1)^j (I a (-2 k + m) + c (1 + 2 s))^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[h, 1], \[Ellipsis], Subscript[h, 1 + j], 1 + u}, {1 + Subscript[h, 1], \[Ellipsis], 1 + Subscript[h, 1 + j]}, E^(2 c z)]), {j, 0, n}] + E^((I a (-2 k + m) + c (-2 s + u)) z) Sum[(1/(-j + n)!) ((-1)^j (I a (-2 k + m) + c (1 - 2 s + 2 u))^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[i, 1], \[Ellipsis], Subscript[i, 1 + j], 1 + u}, {1 + Subscript[i, 1], \[Ellipsis], 1 + Subscript[i, 1 + j]}, E^(2 c z)]), {j, 0, n}]), {s, 0, Floor[(1/2) (-1 + u)]}])/E^((1/2) I m Pi)), {k, 0, Floor[(1/2) (-1 + m)]}] /; Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, n + 1] == (u + 1)/2 && Subscript[b, 1] == Subscript[b, 2] == \[Ellipsis] == Subscript[b, n + 1] == (2 s + 1)/2 && Subscript[c, 1] == Subscript[c, 2] == \[Ellipsis] == Subscript[c, n + 1] == (2 u - 2 s + 1)/2 && Subscript[d, 1] == Subscript[d, 2] == \[Ellipsis] == Subscript[d, n + 1] == (c (u + 1) - I a (-2 k + m))/(2 c) && Subscript[e, 1] == Subscript[e, 2] == \[Ellipsis] == Subscript[e, n + 1] == (c (2 s + 1) - I a (-2 k + m))/(2 c) && Subscript[f, 1] == Subscript[f, 2] == \[Ellipsis] == Subscript[f, n + 1] == (c (2 u - 2 s + 1) - I a (-2 k + m))/(2 c) && Subscript[g, 1] == Subscript[g, 2] == \[Ellipsis] == Subscript[g, n + 1] == (c (u + 1) + I a (-2 k + m))/(2 c) && Subscript[h, 1] == Subscript[h, 2] == \[Ellipsis] == Subscript[h, n + 1] == (c (2 s + 1) + I a (-2 k + m))/(2 c) && Subscript[i, 1] == Subscript[i, 2] == \[Ellipsis] == Subscript[i, n + 1] == (c (2 u - 2 s + 1) + I a (-2 k + m))/(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

 z n sin m ( a z ) coth u ( c z ) csch ( c z ) z 2 - m - u ( 1 - 2 c z ) u + 1 ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["m", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] n ! ( 1 - m mod 2 \$CellContext`m 2 ) ( ( u u 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["u", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 1 - u mod 2 \$CellContext`u 2 ) j = 0 n ( - 1 ) j ( c ( u + 1 ) ) - j - 1 z n - j ( n - j ) ! j + 2 F j + 1 ( u + 1 2 , , u + 1 2 , u + 1 ; u + 1 2 + 1 , , u + 1 2 + 1 ; 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["u", "+", "1"]], "2"], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["u", "+", "1"]], "2"], HypergeometricPFQ], ",", TagBox[RowBox[List["u", "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["u", "+", "1"]], "2"], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["u", "+", "1"]], "2"], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] + s = 0 u - 1 2 ( u s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - c ( u - 2 s ) z j = 0 n ( - 1 ) j ( c ( 2 s + 1 ) ) - j - 1 z n - j ( n - j ) ! 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j + 2 F j + 1 ( 1 2 ( - 2 s + 2 u + 1 ) , , 1 2 ( - 2 s + 2 u + 1 ) , u + 1 ; 1 2 ( - 2 s + 2 u + 1 ) + 1 , , 1 2 ( - 2 s + 2 u + 1 ) + 1 ; 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", RowBox[List["2", " ", "u"]], "+", "1"]], ")"]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", RowBox[List["2", " ", "u"]], "+", "1"]], ")"]]]], HypergeometricPFQ], ",", TagBox[RowBox[List["u", "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", RowBox[List["2", " ", "u"]], "+", "1"]], ")"]]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", RowBox[List["2", " ", "u"]], "+", "1"]], ")"]]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) ) csch u + 1 ( c z ) + 2 - m - u ( 1 - 2 c z ) u + 1 n ! 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j + 2 F j + 1 ( c ( 2 s + 1 ) - a ( m - 2 k ) 2 c , , c ( 2 s + 1 ) - a ( m - 2 k ) 2 c , u + 1 ; c ( 2 s + 1 ) - a ( m - 2 k ) 2 c + 1 , , c ( 2 s + 1 ) - a ( m - 2 k ) 2 c + 1 ; 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "+", "1"]], ")"]]]], "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "+", "1"]], ")"]]]], "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox[RowBox[List["u", "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "+", "1"]], ")"]]]], "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "+", "1"]], ")"]]]], "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] + ( c ( u - 2 s ) - a ( m - 2 k ) ) z j = 0 n ( - 1 ) j ( c ( - 2 s + 2 u + 1 ) - a ( m - 2 k ) ) - j - 1 z n - j ( n - j ) ! 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 Rule Form

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

 2002-12-18