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/01.24.21.0276.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18