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 Csch

 http://functions.wolfram.com/01.23.21.0244.01

 Input Form

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

 Standard Form

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 MathML Form

 z n p z coth u ( c z ) csch ( c z ) z - 2 1 - u ( 1 - 2 c z ) u ( u u 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity]], List[TagBox[FractionBox["u", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] csch u ( c z ) n ! ( 1 - u mod 2 \$CellContext`u 2 ) ( p + c ) z j = 0 n ( - 1 ) j z n - j ( p + c ( u + 1 ) ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( c ( u + 1 ) + p 2 c , , c ( u + 1 ) + p 2 c , 1 + u ; c ( u + 1 ) + p 2 c + 1 , , c ( u + 1 ) + p 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["u", "+", "1"]], ")"]]]], "+", "p"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List["u", "+", "1"]], ")"]]]], "+", "p"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox[RowBox[List["1", "+", "u"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List["u", "+", "1"]], ")"]]]], "+", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List["u", "+", "1"]], ")"]]]], "+", "p"]], 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] - 2 1 - u c z ( 1 - 2 c z ) u csch u ( c z ) n ! s = 0 u - 1 2 ( u s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( p - c ( - 2 s + u ) ) z j = 0 n 1 ( n - j ) ! ( - 1 ) j z n - j ( p + c ( 2 s + 1 ) ) - j - 1 j + 2 F j + 1 ( p + c ( 2 s + 1 ) 2 c , , p + c ( 2 s + 1 ) 2 c , u + 1 ; p + c ( 2 s + 1 ) 2 c + 1 , , p + c ( 2 s + 1 ) 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["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", "s"]], "+", "1"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", "s"]], "+", "1"]], ")"]]]]]], 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["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", "s"]], "+", "1"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", "s"]], "+", "1"]], ")"]]]]]], 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] + ( p + c ( - 2 s + u ) ) z j = 0 n 1 ( n - j ) ! ( - 1 ) j z n - j ( p + c ( 2 u - 2 s + 1 ) ) - j - 1 j + 2 F j + 1 ( p + c ( 2 u - 2 s + 1 ) 2 c , , p + c ( 2 u - 2 s + 1 ) 2 c , u + 1 ; p + c ( 2 u - 2 s + 1 ) 2 c + 1 , , p + c ( 2 u - 2 s + 1 ) 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["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", "u"]], "-", RowBox[List["2", "s"]], "+", "1"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", "u"]], "-", RowBox[List["2", "s"]], "+", "1"]], ")"]]]]]], 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["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", "u"]], "-", RowBox[List["2", "s"]], "+", "1"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", "u"]], "-", RowBox[List["2", "s"]], "+", "1"]], ")"]]]]]], 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] ) /; n u + Condition z z n p z c z u c z -1 2 1 -1 u 1 -1 2 c z u Binomial u u 2 -1 c z u n 1 -1 \$CellContext`u 2 p c z j 0 n -1 j z n -1 j p c u 1 -1 j -1 n -1 j -1 HypergeometricPFQ c u 1 p 2 c -1 c u 1 p 2 c -1 1 u c u 1 p 2 c -1 1 c u 1 p 2 c -1 1 2 c z -1 2 1 -1 u c z 1 -1 2 c z u c z u n s 0 u -1 2 -1 Binomial u s p -1 c -2 s u z j 0 n 1 n -1 j -1 -1 j z n -1 j p c 2 s 1 -1 j -1 HypergeometricPFQ p c 2 s 1 2 c -1 p c 2 s 1 2 c -1 u 1 p c 2 s 1 2 c -1 1 p c 2 s 1 2 c -1 1 2 c z p c -2 s u z j 0 n 1 n -1 j -1 -1 j z n -1 j p c 2 u -1 2 s 1 -1 j -1 HypergeometricPFQ p c 2 u -1 2 s 1 2 c -1 p c 2 u -1 2 s 1 2 c -1 u 1 p c 2 u -1 2 s 1 2 c -1 1 p c 2 u -1 2 s 1 2 c -1 1 2 c z n u SuperPlus [/itex]

 Rule Form

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

 2002-12-18

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