html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Coth

 http://functions.wolfram.com/01.22.21.0214.01

 Input Form

 Integrate[z^n Sin[a z] Sinh[b z] Coth[c z], z] == (-(1/4)) I n! (E^((I a - b) z) Sum[(1/(-j + n)!) ((-1)^j (I a - b)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[v, 1], \[Ellipsis], Subscript[v, 1 + j], 1}, {1 + Subscript[v, 1], \[Ellipsis], 1 + Subscript[v, 1 + j]}, E^(2 c z)]), {j, 0, n}] + E^(((-I) a + b) z) Sum[(1/(-j + n)!) ((-1)^j ((-I) a + b)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[w, 1], \[Ellipsis], Subscript[w, 1 + j], 1}, {1 + Subscript[w, 1], \[Ellipsis], 1 + Subscript[w, 1 + j]}, E^(2 c z)]), {j, 0, n}] + E^((I a - b + 2 c) z) Sum[(1/(-j + n)!) ((-1)^j (I a - b + 2 c)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[x, 1], \[Ellipsis], Subscript[x, 1 + j], 1}, {1 + Subscript[x, 1], \[Ellipsis], 1 + Subscript[x, 1 + j]}, E^(2 c z)]), {j, 0, n}] + E^(((-I) a + b + 2 c) z) Sum[(1/(-j + n)!) ((-1)^j ((-I) a + b + 2 c)^(-1 - j) z^(-j + n) HypergeometricPFQ[ {Subscript[y, 1], \[Ellipsis], Subscript[y, 1 + j], 1}, {1 + Subscript[y, 1], \[Ellipsis], 1 + Subscript[y, 1 + j]}, E^(2 c z)]), {j, 0, n}] - E^(((-I) a - b) z) Sum[(1/(-j + n)!) ((-1)^j ((-I) a - b)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[z, 1], \[Ellipsis], Subscript[z, 1 + j], 1}, {1 + Subscript[z, 1], \[Ellipsis], 1 + Subscript[z, 1 + j]}, E^(2 c z)]), {j, 0, n}] - E^((I a + b) z) Sum[(1/(-j + n)!) ((-1)^j (I a + b)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[\[Alpha], 1], \[Ellipsis], Subscript[\[Alpha], 1 + j], 1}, {1 + Subscript[\[Alpha], 1], \[Ellipsis], 1 + Subscript[\[Alpha], 1 + j]}, E^(2 c z)]), {j, 0, n}] - E^(((-I) a - b + 2 c) z) Sum[(1/(-j + n)!) ((-1)^j ((-I) a - b + 2 c)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[\[Beta], 1], \[Ellipsis], Subscript[\[Beta], 1 + j], 1}, {1 + Subscript[\[Beta], 1], \[Ellipsis], 1 + Subscript[\[Beta], 1 + j]}, E^(2 c z)]), {j, 0, n}] - E^((I a + b + 2 c) z) Sum[(1/(-j + n)!) ((-1)^j (I a + b + 2 c)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[\[Gamma], 1], \[Ellipsis], Subscript[\[Gamma], 1 + j], 1}, {1 + Subscript[\[Gamma], 1], \[Ellipsis], 1 + Subscript[\[Gamma], 1 + j]}, E^(2 c z)]), {j, 0, n}]) /; Subscript[v, 1] == Subscript[v, 2] == \[Ellipsis] == Subscript[v, n + 1] == (I a - b)/(2 c) && Subscript[w, 1] == Subscript[w, 2] == \[Ellipsis] == Subscript[w, n + 1] == ((-I) a + b)/(2 c) && Subscript[x, 1] == Subscript[x, 2] == \[Ellipsis] == Subscript[x, n + 1] == (I a - b + 2 c)/(2 c) && Subscript[y, 1] == Subscript[y, 2] == \[Ellipsis] == Subscript[y, n + 1] == ((-I) a + b + 2 c)/(2 c) && Subscript[z, 1] == Subscript[z, 2] == \[Ellipsis] == Subscript[z, n + 1] == ((-I) a - b)/(2 c) && Subscript[\[Alpha], 1] == Subscript[\[Alpha], 2] == \[Ellipsis] == Subscript[\[Alpha], n + 1] == (I a + b)/(2 c) && Subscript[\[Beta], 1] == Subscript[\[Beta], 2] == \[Ellipsis] == Subscript[\[Beta], n + 1] == ((-I) a - b + 2 c)/(2 c) && Subscript[\[Gamma], 1] == Subscript[\[Gamma], 2] == \[Ellipsis] == Subscript[\[Gamma], n + 1] == (I a + b + 2 c)/(2 c) && Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 z n sin ( a z ) sinh ( b z ) coth ( c z ) z - 1 4 n ! 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j + 2 F j + 1 ( b + a 2 c , , b + a 2 c , 1 ; b + a 2 c + 1 , , b + a 2 c + 1 ; 2 c z ) 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["b", "+", RowBox[List["\[ImaginaryI]", " ", "a", " "]]]], RowBox[List["2", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "a", " "]]]], RowBox[List["2", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "a", " "]]]], RowBox[List["2", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "a", " "]]]], 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]]]]] - ( - b + 2 c - a ) z j = 0 n ( - 1 ) j ( - b + 2 c - a ) - j - 1 z n - j ( n - j ) ! 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j + 2 F j + 1 ( b + 2 c + a 2 c , , b + 2 c + a 2 c , 1 ; b + 2 c + a 2 c + 1 , , b + 2 c + a 2 c + 1 ; 2 c z ) 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["b", "+", RowBox[List["2", " ", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " "]]]], RowBox[List["2", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " "]]]], RowBox[List["2", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " "]]]], RowBox[List["2", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " "]]]], 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]]]]] ) /; n Condition z z n a z b z c z -1 1 4 n a -1 b z j 0 n -1 j a -1 b -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ a -1 b 2 c -1 a -1 b 2 c -1 1 a -1 b 2 c -1 1 a -1 b 2 c -1 1 2 c z b -1 a z j 0 n -1 j b -1 a -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ b -1 a 2 c -1 b -1 a 2 c -1 1 b -1 a 2 c -1 1 b -1 a 2 c -1 1 2 c z -1 b 2 c a z j 0 n -1 j -1 b 2 c a -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ -1 b 2 c a 2 c -1 -1 b 2 c a 2 c -1 1 -1 b 2 c a 2 c -1 1 -1 b 2 c a 2 c -1 1 2 c z b 2 c -1 a z j 0 n -1 j b 2 c -1 a -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ b 2 c -1 a 2 c -1 b 2 c -1 a 2 c -1 1 b 2 c -1 a 2 c -1 1 b 2 c -1 a 2 c -1 1 2 c z -1 -1 b -1 a z j 0 n -1 j -1 b -1 a -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ -1 b -1 a 2 c -1 -1 b -1 a 2 c -1 1 -1 b -1 a 2 c -1 1 -1 b -1 a 2 c -1 1 2 c z -1 b a z j 0 n -1 j b a -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ b a 2 c -1 b a 2 c -1 1 b a 2 c -1 1 b a 2 c -1 1 2 c z -1 -1 b 2 c -1 a z j 0 n -1 j -1 b 2 c -1 a -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ -1 b 2 c -1 a 2 c -1 -1 b 2 c -1 a 2 c -1 1 -1 b 2 c -1 a 2 c -1 1 -1 b 2 c -1 a 2 c -1 1 2 c z -1 b 2 c a z j 0 n -1 j b 2 c a -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ b 2 c a 2 c -1 b 2 c a 2 c -1 1 b 2 c a 2 c -1 1 b 2 c a 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