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; }

 Cosh

 http://functions.wolfram.com/01.20.21.4527.01

 Input Form

 Integrate[z^n Sinh[c z]^m Cosh[a z + b]^\[Nu], z] == ((-2^(-m)) Cosh[b + a z]^\[Nu] n! ((Binomial[m, m/2] (-1 + Mod[m, 2]) Sum[(1/(-j + n)!) ((-1)^j z^(-j + n) (a \[Nu])^(-1 - j) HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, 1 + j], -\[Nu]}, {1 + Subscript[a, 1], \[Ellipsis], 1 + Subscript[a, 1 + j]}, -E^(-2 (b + a z))]), {j, 0, n}])/I^m - Sum[(-1)^k E^(-2 c k z - c m z) Binomial[m, k] (E^(2 m c z) Sum[(1/(-j + n)!) ((-1)^j z^(-j + n) (-2 c k + c m + a \[Nu])^(-1 - j) HypergeometricPFQ[ {Subscript[b, 1], \[Ellipsis], Subscript[b, 1 + j], -\[Nu]}, {1 + Subscript[b, 1], \[Ellipsis], 1 + Subscript[b, 1 + j]}, -E^(-2 (b + a z))]), {j, 0, n}] + (-1)^m E^(4 k c z) Sum[(1/(-j + n)!) ((-1)^j z^(-j + n) (2 c k - c m + a \[Nu])^ (-1 - j) HypergeometricPFQ[{Subscript[c, 1], \[Ellipsis], Subscript[c, 1 + j], -\[Nu]}, {1 + Subscript[c, 1], \[Ellipsis], 1 + Subscript[c, 1 + j]}, -E^(-2 (b + a z))]), {j, 0, n}]), {k, 0, Floor[(1/2) (-1 + m)]}]))/ (1 + E^(-2 (b + a z)))^\[Nu] /; Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, n + 1] == -(\[Nu]/2) && Subscript[b, 1] == Subscript[b, 2] == \[Ellipsis] == Subscript[b, n + 1] == -((c (-2 k + m) + a \[Nu])/(2 a)) && Subscript[c, 1] == Subscript[c, 2] == \[Ellipsis] == Subscript[c, n + 1] == -(((-c) (-2 k + m) + a \[Nu])/(2 a)) && Element[n, Integers] && n >= 0 && Element[m, Integers] && m > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "m"], SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List[RowBox[List["a", " ", "z"]], "+", "b"]], "]"]], "\[Nu]"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", SuperscriptBox["2", RowBox[List["-", "m"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]]]]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], "\[Nu]"], " ", RowBox[List["n", "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["-", "m"]]], RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", 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 MathML Form

 z n sinh m ( c z ) cosh ν ( b + a z ) z - 2 - m ( 1 + - 2 ( b + a z ) ) - ν cosh ν ( b + a z ) n ! ( - m ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( m mod 2 \$CellContext`m 2 - 1 ) j = 0 n ( - 1 ) j z n - j ( a ν ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( - ν 2 , , - ν 2 , - ν ; 1 - ν 2 , , 1 - ν 2 ; - - 2 ( b + a 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[RowBox[List["-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ], ",", TagBox[RowBox[List["-", "\[Nu]"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] - k = 0 m - 1 2 ( - 1 ) k - 2 c k z - c m z ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - 1 ) m 4 k c z j = 0 n ( - 1 ) j z n - j ( 2 c k - c m + a ν ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( - a ν - c ( m - 2 k ) 2 a , , - a ν - c ( m - 2 k ) 2 a , - ν ; 1 - a ν - c ( m - 2 k ) 2 a , , 1 - a ν - c ( m - 2 k ) 2 a ; - - 2 ( b + a 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[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["a", " ", "\[Nu]"]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "a"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["a", " ", "\[Nu]"]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "a"]]]]], HypergeometricPFQ], ",", TagBox[RowBox[List["-", "\[Nu]"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["a", " ", "\[Nu]"]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "a"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["a", " ", "\[Nu]"]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "a"]]]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] + 2 m c z j = 0 n ( - 1 ) j z n - j ( - 2 c k + c m + a ν ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( - c ( m - 2 k ) + a ν 2 a , , - c ( m - 2 k ) + a ν 2 a , - ν ; 1 - c ( m - 2 k ) + a ν 2 a , , 1 - c ( m - 2 k ) + a ν 2 a ; - - 2 ( b + a 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[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]]]], HypergeometricPFQ], ",", TagBox[RowBox[List["-", "\[Nu]"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) ) /; n m + Condition z z n c z m b a z ν -1 2 -1 m 1 -2 b a z -1 ν b a z ν n -1 m Binomial m m 2 -1 \$CellContext`m 2 -1 j 0 n -1 j z n -1 j a ν -1 j -1 n -1 j -1 HypergeometricPFQ -1 ν 2 -1 -1 ν 2 -1 -1 ν 1 -1 ν 2 -1 1 -1 ν 2 -1 -1 -2 b a z -1 k 0 m -1 2 -1 -1 k -2 c k z -1 c m z Binomial m k -1 m 4 k c z j 0 n -1 j z n -1 j 2 c k -1 c m a ν -1 j -1 n -1 j -1 HypergeometricPFQ -1 a ν -1 c m -1 2 k 2 a -1 -1 a ν -1 c m -1 2 k 2 a -1 -1 ν 1 -1 a ν -1 c m -1 2 k 2 a -1 1 -1 a ν -1 c m -1 2 k 2 a -1 -1 -2 b a z 2 m c z j 0 n -1 j z n -1 j -2 c k c m a ν -1 j -1 n -1 j -1 HypergeometricPFQ -1 c m -1 2 k a ν 2 a -1 -1 c m -1 2 k a ν 2 a -1 -1 ν 1 -1 c m -1 2 k a ν 2 a -1 1 -1 c m -1 2 k a ν 2 a -1 -1 -2 b a z n m SuperPlus [/itex]

 Rule Form

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

 2002-12-18