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/01.24.21.0240.01

 Input Form

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

 Standard Form

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 MathML Form

 z n p z tanh u ( c z ) sech ( c z ) z u 2 ( p + 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 ( p + c ( u + 1 ) ) - j - 1 ) ( n - j ) ! j + 2 F j + 1 ( p + c ( u + 1 ) 2 c , , p + c ( u + 1 ) 2 c , u + 1 ; p + c ( u + 1 ) 2 c + 1 , , p + c ( u + 1 ) 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["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["u", "+", "1"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["u", "+", "1"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox[RowBox[List["u", "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["u", "+", "1"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["u", "+", "1"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "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 ( p - c ( u - 2 k ) ) z j = 0 n ( ( - 1 ) j z n - j ( c ( 2 k + 1 ) + p ) - j - 1 ) ( n - j ) ! j + 2 F j + 1 ( c ( 2 k + 1 ) + p 2 c , , c ( 2 k + 1 ) + p 2 c , u + 1 ; c ( 2 k + 1 ) + p 2 c + 1 , , c ( 2 k + 1 ) + p 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["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]]]], "+", "p"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]]]], "+", "p"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox[RowBox[List["u", "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]]]], "+", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]]]], "+", "p"]], RowBox[List["2", " ", "c"]]], "+", "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] + ( p + c ( u - 2 k ) ) z j = 0 n ( ( - 1 ) j z n - j ( c ( - 2 k + 2 u + 1 ) + p ) - j - 1 ) ( n - j ) ! j + 2 F j + 1 ( p + c ( - 2 k + 2 u + 1 ) 2 c , , p + c ( - 2 k + 2 u + 1 ) 2 c , u + 1 ; p + c ( - 2 k + 2 u + 1 ) 2 c + 1 , , p + c ( - 2 k + 2 u + 1 ) 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["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", RowBox[List["2", " ", "u"]], "+", "1"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", RowBox[List["2", " ", "u"]], "+", "1"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox[RowBox[List["u", "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", RowBox[List["2", " ", "u"]], "+", "1"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", RowBox[List["2", " ", "u"]], "+", "1"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "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 p z c z u c z u 2 p c u 1 z Binomial u u 2 -1 n 1 -1 \$CellContext`u 2 j 0 n -1 j z n -1 j p c u 1 -1 j -1 n -1 j -1 HypergeometricPFQ p c u 1 2 c -1 p c u 1 2 c -1 u 1 p c u 1 2 c -1 1 p c u 1 2 c -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 p -1 c u -1 2 k z j 0 n -1 j z n -1 j c 2 k 1 p -1 j -1 n -1 j -1 HypergeometricPFQ c 2 k 1 p 2 c -1 c 2 k 1 p 2 c -1 u 1 c 2 k 1 p 2 c -1 1 c 2 k 1 p 2 c -1 1 -1 2 c z p c u -1 2 k z j 0 n -1 j z n -1 j c -2 k 2 u 1 p -1 j -1 n -1 j -1 HypergeometricPFQ p c -2 k 2 u 1 2 c -1 p c -2 k 2 u 1 2 c -1 u 1 p c -2 k 2 u 1 2 c -1 1 p c -2 k 2 u 1 2 c -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