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 Coth

 http://functions.wolfram.com/01.22.21.0199.01

 Input Form

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

 Standard Form

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 MathML Form

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