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; }

 Csch

 http://functions.wolfram.com/01.23.21.0268.01

 Input Form

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

 Standard Form

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RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "u"]], ")"]]]]]], RowBox[List["2", " ", "c"]]]]], "\[And]", RowBox[List[SubscriptBox["g", "1"], "\[Equal]", SubscriptBox["g", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["g", RowBox[List["n", "+", "1"]]], "\[Equal]", FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "u"]], ")"]]]]]], RowBox[List["2", " ", "c"]]]]], "\[And]", RowBox[List[SubscriptBox["h", "1"], "\[Equal]", SubscriptBox["h", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["h", RowBox[List["n", "+", "1"]]], "\[Equal]", FractionBox[RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["b", " ", 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 MathML Form

 z n sin m ( a z ) sinh u ( b z ) csch ( c z ) z - u 2 - m - u + 1 c z ( 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]]]]] ( u u 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity]], List[TagBox[FractionBox["u", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] n ! ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - u mod 2 \$CellContext`u 2 ) j = 0 n ( - 1 ) j z n - j c - j - 1 ( n - j ) ! j + 2 F j + 1 ( 1 2 , , 1 2 , 1 ; 3 2 , , 3 2 ; 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["1", "2"], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox["1", "2"], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "2"], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox["3", "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]]]]], HypergeometricPFQ] + m + u 2 - m - u + 1 c z ( u u 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity]], List[TagBox[FractionBox["u", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] n ! ( u mod 2 \$CellContext`u 2 - 1 ) s = 0 m - 1 2 ( - 1 ) s ( m s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - 1 ) m ( a ( m - 2 s ) ) z j = 0 n ( - 1 ) j z n - j ( a ( m - 2 s ) + c ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( a ( m - 2 s ) + c 2 c , , a ( m - 2 s ) + c 2 c , 1 ; a ( m - 2 s ) + 3 c 2 c , , a ( m - 2 s ) + 3 c 2 c ; 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["a", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", "c"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List[RowBox[List["a", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", "c"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List[RowBox[List["a", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["3", "c"]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List[RowBox[List["a", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["3", "c"]]]], RowBox[List["2", " ", "c"]]], 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]]]]], HypergeometricPFQ] + ( - a ( m - 2 s ) ) z j = 0 n ( - 1 ) j z n - j ( - a ( m - 2 s ) + c ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( - a ( m - 2 s ) + c 2 c , , - a ( m - 2 s ) + c 2 c , 1 ; - a ( m - 2 s ) + 3 c 2 c , , - a ( m - 2 s ) + 3 c 2 c ; 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[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", "c"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", "c"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["3", "c"]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["3", "c"]]]], RowBox[List["2", " ", "c"]]], 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]]]]], HypergeometricPFQ] ) + 2 - m - u + 1 c z ( 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]]]]] n ! ( m mod 2 \$CellContext`m 2 - 1 ) k = 0 u - 1 2 ( - 1 ) k ( 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 z n - j ( b ( u - 2 k ) + c ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( b ( u - 2 k ) + c 2 c , , b ( u - 2 k ) + c 2 c , 1 ; b ( u - 2 k ) + 3 c 2 c , , b ( u - 2 k ) + 3 c 2 c ; 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["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", "c"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", "c"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["3", "c"]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["3", "c"]]]], RowBox[List["2", " ", "c"]]], 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]]]]], HypergeometricPFQ] + ( - 1 ) u ( - b ( u - 2 k ) ) z j = 0 n ( - 1 ) j z n - j ( - b ( u - 2 k ) + c ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( - b ( u - 2 k ) + c 2 c , , - b ( u - 2 k ) + c 2 c , 1 ; - b ( u - 2 k ) + 3 c 2 c , , - b ( u - 2 k ) + 3 c 2 c ; 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[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", "c"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", "c"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["3", "c"]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["3", "c"]]]], RowBox[List["2", " ", "c"]]], 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]]]]], HypergeometricPFQ] ) - 2 1 - m - u c z n ! k = 0 u - 1 2 s = 0 m - 1 2 ( - 1 ) k + s ( m s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( u k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( m π 2 + ( - a ( - 2 s + m ) + b ( - 2 k + u ) ) z j = 0 n ( - 1 ) j z n - j ( - a ( m - 2 s ) + b ( - 2 k + u ) + c ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( - a ( m - 2 s ) + b ( - 2 k + u ) + c 2 c , , - a ( m - 2 s ) + b ( - 2 k + u ) + c 2 c , 1 ; - a ( m - 2 s ) + b ( - 2 k + u ) + 3 c 2 c , , - a ( m - 2 s ) + b ( - 2 k + u ) + 3 c 2 c ; 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[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "u"]], ")"]]]], "+", "c"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "u"]], ")"]]]], "+", "c"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "u"]], ")"]]]], "+", RowBox[List["3", "c"]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "u"]], ")"]]]], "+", RowBox[List["3", "c"]]]], RowBox[List["2", " ", "c"]]], 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]]]]], HypergeometricPFQ] + ( - 1 ) u - 1 2 m π + ( a ( - 2 s + m ) - b ( - 2 k + u ) ) z j = 0 n ( - 1 ) j z n - j ( a ( m - 2 s ) - b ( - 2 k + u ) + c ) - j - 1 ( n - j ) ! 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j + 2 F j + 1 ( a ( m - 2 s ) + b ( - 2 k + u ) + c 2 c , , a ( m - 2 s ) + b ( - 2 k + u ) + c 2 c , 1 ; a ( m - 2 s ) + b ( - 2 k + u ) + 3 c 2 c , , a ( m - 2 s ) + b ( - 2 k + u ) + 3 c 2 c ; 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", " ", "s"]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "u"]], ")"]]]], "+", "c"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "u"]], ")"]]]], "+", "c"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "u"]], ")"]]]], "+", RowBox[List["3", "c"]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "u"]], ")"]]]], "+", RowBox[List["3", "c"]]]], RowBox[List["2", " ", "c"]]], 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]]]]], HypergeometricPFQ] ) /; n m + u + Condition z z n a z m b z u c z -1 u 2 -1 m -1 u 1 c z Binomial m m 2 -1 Binomial u u 2 -1 n 1 -1 \$CellContext`m 2 1 -1 \$CellContext`u 2 j 0 n -1 j z n -1 j c -1 j -1 n -1 j -1 HypergeometricPFQ 1 2 1 2 1 3 2 3 2 2 c z m u 2 -1 m -1 u 1 c z Binomial u u 2 -1 n \$CellContext`u 2 -1 s 0 m -1 2 -1 -1 s Binomial m s -1 m a m -1 2 s z j 0 n -1 j z n -1 j a m -1 2 s c -1 j -1 n -1 j -1 HypergeometricPFQ a m -1 2 s c 2 c -1 a m -1 2 s c 2 c -1 1 a m -1 2 s 3 c 2 c -1 a m -1 2 s 3 c 2 c -1 2 c z -1 a m -1 2 s z j 0 n -1 j z n -1 j -1 a m -1 2 s c -1 j -1 n -1 j -1 HypergeometricPFQ -1 a m -1 2 s c 2 c -1 -1 a m -1 2 s c 2 c -1 1 -1 a m -1 2 s 3 c 2 c -1 -1 a m -1 2 s 3 c 2 c -1 2 c z 2 -1 m -1 u 1 c z Binomial m m 2 -1 n \$CellContext`m 2 -1 k 0 u -1 2 -1 -1 k Binomial u k b u -1 2 k z j 0 n -1 j z n -1 j b u -1 2 k c -1 j -1 n -1 j -1 HypergeometricPFQ b u -1 2 k c 2 c -1 b u -1 2 k c 2 c -1 1 b u -1 2 k 3 c 2 c -1 b u -1 2 k 3 c 2 c -1 2 c z -1 u -1 b u -1 2 k z j 0 n -1 j z n -1 j -1 b u -1 2 k c -1 j -1 n -1 j -1 HypergeometricPFQ -1 b u -1 2 k c 2 c -1 -1 b u -1 2 k c 2 c -1 1 -1 b u -1 2 k 3 c 2 c -1 -1 b u -1 2 k 3 c 2 c -1 2 c z -1 2 1 -1 m -1 u c z n s 0 m -1 2 -1 k 0 u -1 2 -1 -1 k s Binomial m s Binomial u k m 2 -1 -1 a -2 s m b -2 k u z j 0 n -1 j z n -1 j -1 a m -1 2 s b -2 k u c -1 j -1 n -1 j -1 HypergeometricPFQ -1 a m -1 2 s b -2 k u c 2 c -1 -1 a m -1 2 s b -2 k u c 2 c -1 1 -1 a m -1 2 s b -2 k u 3 c 2 c -1 -1 a m -1 2 s b -2 k u 3 c 2 c -1 2 c z -1 u -1 1 2 m a -2 s m -1 b -2 k u z j 0 n -1 j z n -1 j a m -1 2 s -1 b -2 k u c -1 j -1 n -1 j -1 HypergeometricPFQ a m -1 2 s -1 b -2 k u c 2 c -1 a m -1 2 s -1 b -2 k u c 2 c -1 1 a m -1 2 s -1 b -2 k u 3 c 2 c -1 a m -1 2 s -1 b -2 k u 3 c 2 c -1 2 c z -1 u m 2 -1 -1 a -2 s m -1 b -2 k u z j 0 n -1 j z n -1 j -1 a m -1 2 s -1 b -2 k u c -1 j -1 n -1 j -1 HypergeometricPFQ -1 a m -1 2 s -1 b -2 k u c 2 c -1 -1 a m -1 2 s -1 b -2 k u c 2 c -1 1 -1 a m -1 2 s -1 b -2 k u 3 c 2 c -1 -1 a m -1 2 s -1 b -2 k u 3 c 2 c -1 2 c z -1 1 2 m a -2 s m b -2 k u z j 0 n -1 j z n -1 j a m -1 2 s b -2 k u c -1 j -1 n -1 j -1 HypergeometricPFQ a m -1 2 s b -2 k u c 2 c -1 a m -1 2 s b -2 k u c 2 c -1 1 a m -1 2 s b -2 k u 3 c 2 c -1 a m -1 2 s b -2 k u 3 c 2 c -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

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