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; }

 Sech

 http://functions.wolfram.com/01.24.21.0173.01

 Input Form

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

 Standard Form

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"0"]], "\[And]", RowBox[List["u", "\[Element]", "Integers"]], "\[And]", RowBox[List["u", ">", "0"]]]]]]]]

 MathML Form

 z n tanh u ( c z ) sech ( c z ) z 2 u c ( u + 1 ) z ( u u 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["u", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] n ! ( 1 - u mod 2 \$CellContext`u 2 ) j = 0 n ( ( - 1 ) j z n - j ( c ( u + 1 ) ) - j - 1 ) ( n - j ) ! j + 2 F j + 1 ( u + 1 2 , , u + 1 2 , u + 1 ; u + 1 2 + 1 , , u + 1 2 + 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["u", "+", "1"]], "2"], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["u", "+", "1"]], "2"], HypergeometricPFQ], ",", TagBox[RowBox[List["u", "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["u", "+", "1"]], "2"], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["u", "+", "1"]], "2"], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] + 2 n ! c ( u + 1 ) z k = 0 u - 1 2 ( - 1 ) k ( u k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( ( - 1 ) u ( - c ( u - 2 k ) ) z j = 0 n ( ( - 1 ) j z n - j ( c ( 2 k + 1 ) ) - j - 1 ) ( n - j ) ! j + 2 F j + 1 ( 1 2 ( 2 k + 1 ) , , 1 2 ( 2 k + 1 ) , u + 1 ; 1 2 ( 2 k + 1 ) + 1 , , 1 2 ( 2 k + 1 ) + 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[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]]]], HypergeometricPFQ], ",", TagBox[RowBox[List["u", "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] + ( c ( u - 2 k ) ) z j = 0 n ( ( - 1 ) j z n - j ( c ( - 2 k + 2 u + 1 ) ) - j - 1 ) ( n - j ) ! j + 2 F j + 1 ( 1 2 ( - 2 k + 2 u + 1 ) , , 1 2 ( - 2 k + 2 u + 1 ) , u + 1 ; 1 2 ( - 2 k + 2 u + 1 ) + 1 , , 1 2 ( - 2 k + 2 u + 1 ) + 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[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", RowBox[List["2", " ", "u"]], "+", "1"]], ")"]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", RowBox[List["2", " ", "u"]], "+", "1"]], ")"]]]], HypergeometricPFQ], ",", TagBox[RowBox[List["u", "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", RowBox[List["2", " ", "u"]], "+", "1"]], ")"]]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", RowBox[List["2", " ", "u"]], "+", "1"]], ")"]]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) /; n u + Condition z z n c z u c z 2 u c u 1 z Binomial u u 2 -1 n 1 -1 \$CellContext`u 2 j 0 n -1 j z n -1 j c u 1 -1 j -1 n -1 j -1 HypergeometricPFQ u 1 2 -1 u 1 2 -1 u 1 u 1 2 -1 1 u 1 2 -1 1 -1 2 c z 2 n c u 1 z k 0 u -1 2 -1 -1 k Binomial u k -1 u -1 c u -1 2 k z j 0 n -1 j z n -1 j c 2 k 1 -1 j -1 n -1 j -1 HypergeometricPFQ 1 2 2 k 1 1 2 2 k 1 u 1 1 2 2 k 1 1 1 2 2 k 1 1 -1 2 c z c u -1 2 k z j 0 n -1 j z n -1 j c -2 k 2 u 1 -1 j -1 n -1 j -1 HypergeometricPFQ 1 2 -2 k 2 u 1 1 2 -2 k 2 u 1 u 1 1 2 -2 k 2 u 1 1 1 2 -2 k 2 u 1 1 -1 2 c z n u SuperPlus [/itex]

 Rule Form

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

 2002-12-18