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 Csch

 http://functions.wolfram.com/01.23.21.0240.01

 Input Form

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

 Standard Form

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 MathML Form

 z n p z tanh u ( c z ) csch ( c z ) z 2 u - 1 ( p + c u ) z ( u - 1 u - 1 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["u", "-", "1"]], Identity]], List[TagBox[FractionBox[RowBox[List["u", "-", "1"]], "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] n ! ( 1 - ( u - 1 ) mod 2 -1 \$CellContext`u 2 ) j = 0 n ( - 1 ) j ( p + c u ) - j - 1 z n - j ( n - j ) ! j + 2 F j + 1 ( c u + p 2 c , , c u + p 2 c , u ; c u + p 2 c + 1 , , c u + 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", " ", "u"]], "+", "p"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List[RowBox[List["c", " ", "u"]], "+", "p"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["u", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["c", " ", "u"]], "+", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["c", " ", "u"]], "+", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] + 2 c u z n ! k = 0 u - 2 2 ( - 1 ) k ( u - 1 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["u", "-", "1"]], Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( p + c ( - 2 k + u - 1 ) ) z j = 0 n ( - 1 ) j ( p + c ( - 2 k + 2 u - 1 ) ) - j - 1 z n - j ( n - j ) ! j + 2 F j + 1 ( p + c ( - 2 k + 2 u - 1 ) 2 c , , p + c ( - 2 k + 2 u - 1 ) 2 c , u ; p + c ( - 2 k + 2 u - 1 ) 2 c + 1 , , p + c ( - 2 k + 2 u - 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[RowBox[List["-", "2"]], " ", "k"]], "+", RowBox[List["2", " ", "u"]], "-", "1"]], ")"]]]]]], 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"]], " ", "k"]], "+", RowBox[List["2", " ", "u"]], "-", "1"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["u", 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"]], " ", "k"]], "+", RowBox[List["2", " ", "u"]], "-", "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[RowBox[List["-", "2"]], " ", "k"]], "+", RowBox[List["2", " ", "u"]], "-", "1"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] - ( - 1 ) u ( p - c ( - 2 k + u - 1 ) ) z j = 0 n ( - 1 ) j ( ( 2 k + 1 ) c + p ) - j - 1 z n - j ( n - j ) ! j + 2 F j + 1 ( ( 2 k + 1 ) c + p 2 c , , ( 2 k + 1 ) c + p 2 c , u ; ( 2 k + 1 ) c + p 2 c + 1 , , ( 2 k + 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[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["u", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", 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 2 u -1 p c u z Binomial u -1 u -1 2 -1 n 1 -1 -1 \$CellContext`u 2 j 0 n -1 j p c u -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ c u p 2 c -1 c u p 2 c -1 u c u p 2 c -1 1 c u p 2 c -1 1 -1 2 c z 2 c u z n k 0 u -2 2 -1 -1 k Binomial u -1 k p c -2 k u -1 z j 0 n -1 j p c -2 k 2 u -1 -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ p c -2 k 2 u -1 2 c -1 p c -2 k 2 u -1 2 c -1 u p c -2 k 2 u -1 2 c -1 1 p c -2 k 2 u -1 2 c -1 1 -1 2 c z -1 -1 u p -1 c -2 k u -1 z j 0 n -1 j 2 k 1 c p -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ 2 k 1 c p 2 c -1 2 k 1 c p 2 c -1 u 2 k 1 c p 2 c -1 1 2 k 1 c p 2 c -1 1 -1 2 c z n u SuperPlus [/itex]

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

 2002-12-18