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 Csch

 http://functions.wolfram.com/01.23.21.0281.01

 Input Form

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

 Standard Form

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 MathML Form

 z n p z sin ( a z ) sinh ( b z ) csch ( c z ) z - 1 2 c z n ! ( - ( - a - b + p ) z j = 0 n ( - 1 ) j z n - j ( - a - b + p + c ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( c + p - a - b 2 c , , c + p - a - b 2 c , 1 ; c + p - a - b 2 c + 1 , , c + p - a - 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", "-", RowBox[List["\[ImaginaryI]", " ", "a"]], "-", "b"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["c", "+", "p", "-", RowBox[List["\[ImaginaryI]", " ", "a"]], "-", "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", "-", RowBox[List["\[ImaginaryI]", " ", "a"]], "-", "b"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["c", "+", "p", "-", RowBox[List["\[ImaginaryI]", " ", "a"]], "-", "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 + b + p ) z j = 0 n ( - 1 ) j z n - j ( b + a + p + c ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( c + p + a + b 2 c , , c + p + a + b 2 c , 1 ; c + p + a + b 2 c + 1 , , c + p + a + 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", "+", RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["c", "+", "p", "+", RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "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", "+", RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["c", "+", "p", "+", RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "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 - b + p ) z j = 0 n ( - 1 ) j z n - j ( a - b + p + c ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( a - b + p + c 2 c , , a - b + p + c 2 c , 1 ; a - b + p + c 2 c + 1 , , a - b + p + c 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["\[ImaginaryI]", " ", "a"]], "-", "b", "+", "p", "+", "c"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "-", "b", "+", "p", "+", "c"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "-", "b", "+", "p", "+", "c"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "-", "b", "+", "p", "+", "c"]], 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 + b + p ) z j = 0 n ( - 1 ) j z n - j ( - a + b + p + c ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( - a + b + p + c 2 c , , - a + b + p + c 2 c , 1 ; - a + b + p + c 2 c + 1 , , - a + b + p + c 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["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b", "+", "p", "+", "c"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b", "+", "p", "+", "c"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b", "+", "p", "+", "c"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b", "+", "p", "+", "c"]], 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 z b z c z -1 1 2 c z n -1 -1 a -1 b p z j 0 n -1 j z n -1 j -1 a -1 b p c -1 j -1 n -1 j -1 HypergeometricPFQ c p -1 a -1 b 2 c -1 c p -1 a -1 b 2 c -1 1 c p -1 a -1 b 2 c -1 1 c p -1 a -1 b 2 c -1 1 2 c z -1 a b p z j 0 n -1 j z n -1 j b a p c -1 j -1 n -1 j -1 HypergeometricPFQ c p a b 2 c -1 c p a b 2 c -1 1 c p a b 2 c -1 1 c p a b 2 c -1 1 2 c z a -1 b p z j 0 n -1 j z n -1 j a -1 b p c -1 j -1 n -1 j -1 HypergeometricPFQ a -1 b p c 2 c -1 a -1 b p c 2 c -1 1 a -1 b p c 2 c -1 1 a -1 b p c 2 c -1 1 2 c z -1 a b p z j 0 n -1 j z n -1 j -1 a b p c -1 j -1 n -1 j -1 HypergeometricPFQ -1 a b p c 2 c -1 -1 a b p c 2 c -1 1 -1 a b p c 2 c -1 1 -1 a b p c 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