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/01.22.21.0160.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18