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 Csch

 http://functions.wolfram.com/01.23.21.0230.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], RowBox[List["Cosh", "[", RowBox[List["a", "+", RowBox[List["b", " ", "z"]]]], "]"]], RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], " ", RowBox[List["n", "!"]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", "+", RowBox[List[RowBox[List["(", RowBox[List["p", "+", "b"]], ")"]], " ", "z"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "j"]], ")"]], "!"]], SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", "p", "+", "b"]], ")"]], RowBox[List["j", "+", "1"]]]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["j", "+", "1"]]], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", RowBox[List["j", "+", "1"]]]]]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List[RowBox[List["(", RowBox[List["p", "-", "b"]], ")"]], " ", "z"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "j"]], ")"]], "!"]], SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", "p", "-", "b"]], ")"]], RowBox[List["j", "+", "1"]]]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", RowBox[List["j", "+", "1"]]], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["b", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", RowBox[List["j", "+", "1"]]]]]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "\[Equal]", SubscriptBox["a", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["n", "+", "1"]]], "\[Equal]", FractionBox[RowBox[List["b", "+", "p", "+", "c"]], RowBox[List["2", " ", "c"]]]]], "\[And]", RowBox[List[SubscriptBox["b", "1"], "\[Equal]", SubscriptBox["b", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["b", RowBox[List["n", "+", "1"]]], "\[Equal]", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", "p", "+", "c"]], RowBox[List["2", " ", "c"]]]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[RowBox[List["p", "+", "b"]], "\[NotEqual]", RowBox[List[RowBox[List["-", "c"]], " ", "\[Nu]"]]]], "\[And]", RowBox[List[RowBox[List["p", "-", "b"]], "\[NotEqual]", RowBox[List[RowBox[List["-", "c"]], " ", "\[Nu]"]]]]]]]]]]

 MathML Form

 z n p z cosh ( a + b z ) csch ( c z ) z - c z n ! ( a + ( p + b ) z j = 0 n ( - 1 ) j z n - j ( b + p + c ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( c + p + b 2 c , , c + p + b 2 c , 1 ; c + p + b 2 c + 1 , , c + p + b 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", "+", "b"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["c", "+", "p", "+", "b"]], 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", "+", "b"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["c", "+", "p", "+", "b"]], 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] + - a + ( p - b ) z j = 0 n ( - 1 ) j z n - j ( - b + p + c ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( c + p - b 2 c , , c + p - b 2 c , 1 ; c + p - b 2 c + 1 , , c + p - b 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", "-", "b"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["c", "+", "p", "-", "b"]], 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", "-", "b"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["c", "+", "p", "-", "b"]], 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 Condition z z n p z a b z c z -1 c z n a p b z j 0 n -1 j z n -1 j b p c -1 j -1 n -1 j -1 HypergeometricPFQ c p b 2 c -1 c p b 2 c -1 1 c p b 2 c -1 1 c p b 2 c -1 1 2 c z -1 a p -1 b z j 0 n -1 j z n -1 j -1 b p c -1 j -1 n -1 j -1 HypergeometricPFQ c p -1 b 2 c -1 c p -1 b 2 c -1 1 c p -1 b 2 c -1 1 c p -1 b 2 c -1 1 2 c z n [/itex]

 Rule Form

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

 2002-12-18