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/01.20.21.4859.01

 Input Form

 Integrate[z^n E^(b Sqrt[z] + d z + e) Sinh[a Sqrt[z] + p z + q]^m Cosh[c Sqrt[z] + f z + g]^v, z] == I^m 2^(-1 - m - 2 n - v) d^(-2 - 2 n) E^(-(b^2/(4 d)) + e) Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2]) Sum[(-1)^(-h + i) 4^i b^(-h - i + 2 n) (b + 2 d Sqrt[z])^(h + i) (-((b + 2 d Sqrt[z])^2/d))^((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] (b (b + 2 d Sqrt[z]) Gamma[(1/2) (1 + h + i), -((b + 2 d Sqrt[z])^2/(4 d))] + 2 d Sqrt[-((b + 2 d Sqrt[z])^2/d)] Gamma[(1/2) (2 + h + i), -((b + 2 d Sqrt[z])^2/(4 d))]), {i, 0, n}, {h, 0, i}] + I^m 2^(-1 - 2 n - m - v) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(-1)^k Binomial[m, k] (E^(e - (b + a (2 k - m))^2/(4 (d + (2 k - m) p)) + (I m Pi)/2 + (2 k - m) q) (d + (2 k - m) p)^(-2 - 2 n) Sum[(-1)^(-h + i) 4^i (b + a (2 k - m))^(-h - i + 2 n) (b + a (2 k - m) + 2 (d + (2 k - m) p) Sqrt[z])^(h + i) (-((b + a (2 k - m) + 2 (d + (2 k - m) p) Sqrt[z])^2/ (d + (2 k - m) p)))^((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] ((b + a (2 k - m)) (b + a (2 k - m) + 2 (d + (2 k - m) p) Sqrt[z]) Gamma[(1/2) (1 + h + i), -((b + a (2 k - m) + 2 (d + (2 k - m) p) Sqrt[z])^2/(4 (d + (2 k - m) p)))] + 2 (d + (2 k - m) p) Sqrt[-((b + a (2 k - m) + 2 (d + (2 k - m) p) Sqrt[z])^2/(d + (2 k - m) p))] Gamma[(1/2) (2 + h + i), -((b + a (2 k - m) + 2 (d + (2 k - m) p) Sqrt[z])^2/(4 (d + (2 k - m) p)))]), {i, 0, n}, {h, 0, i}] + E^(e - (b + a (-2 k + m))^2/(4 (d + (-2 k + m) p)) - (I m Pi)/2 + (-2 k + m) q) (d + (-2 k + m) p)^(-2 - 2 n) Sum[(-1)^(-h + i) 4^i (b + a (-2 k + m))^(-h - i + 2 n) (b + a (-2 k + m) + 2 (d + (-2 k + m) p) Sqrt[z])^(h + i) (-((b + a (-2 k + m) + 2 (d + (-2 k + m) p) Sqrt[z])^2/ (d + (-2 k + m) p)))^((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] ((b + a (-2 k + m)) (b + a (-2 k + m) + 2 (d + (-2 k + m) p) Sqrt[z]) Gamma[(1/2) (1 + h + i), -((b + a (-2 k + m) + 2 (d + (-2 k + m) p) Sqrt[z])^2/(4 (d + (-2 k + m) p)))] + 2 (d + (-2 k + m) p) Sqrt[-((b + a (-2 k + m) + 2 (d + (-2 k + m) p) Sqrt[z])^2/(d + (-2 k + m) p))] Gamma[(1/2) (2 + h + i), -((b + a (-2 k + m) + 2 (d + (-2 k + m) p) Sqrt[z])^2/(4 (d + (-2 k + m) p)))]), {i, 0, n}, {h, 0, i}]), {k, 0, Floor[(1/2) (-1 + m)]}] + I^m 2^(-1 - 2 n - m - v) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[Binomial[v, s] (E^(e - (b + c (2 s - v))^2/(4 (d + f (2 s - v))) + g (2 s - v)) (d + f (2 s - v))^(-2 - 2 n) Sum[(-1)^(-h + i) 4^i (b + c (2 s - v))^(-h - i + 2 n) (b + c (2 s - v) + 2 (d + f (2 s - v)) Sqrt[z])^(h + i) (-((b + c (2 s - v) + 2 (d + f (2 s - v)) Sqrt[z])^2/ (d + f (2 s - v))))^((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] ((b + c (2 s - v)) (b + c (2 s - v) + 2 (d + f (2 s - v)) Sqrt[z]) Gamma[(1/2) (1 + h + i), -((b + c (2 s - v) + 2 (d + f (2 s - v)) Sqrt[z])^2/(4 (d + f (2 s - v))))] + 2 (d + f (2 s - v)) Sqrt[-((b + c (2 s - v) + 2 (d + f (2 s - v)) Sqrt[z])^2/(d + f (2 s - v)))] Gamma[(1/2) (2 + h + i), -((b + c (2 s - v) + 2 (d + f (2 s - v)) Sqrt[z])^2/(4 (d + f (2 s - v))))]), {i, 0, n}, {h, 0, i}] + E^(e + g (-2 s + v) - (b + c (-2 s + v))^2/(4 (d + f (-2 s + v)))) (d + f (-2 s + v))^(-2 - 2 n) Sum[(-1)^(-h + i) 4^i (b + c (-2 s + v))^(-h - i + 2 n) (b + c (-2 s + v) + 2 (d + f (-2 s + v)) Sqrt[z])^(h + i) (-((b + c (-2 s + v) + 2 (d + f (-2 s + v)) Sqrt[z])^2/ (d + f (-2 s + v))))^((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] ((b + c (-2 s + v)) (b + c (-2 s + v) + 2 (d + f (-2 s + v)) Sqrt[z]) Gamma[(1/2) (1 + h + i), -((b + c (-2 s + v) + 2 (d + f (-2 s + v)) Sqrt[z])^2/(4 (d + f (-2 s + v))))] + 2 (d + f (-2 s + v)) Sqrt[-((b + c (-2 s + v) + 2 (d + f (-2 s + v)) Sqrt[z])^2/(d + f (-2 s + v)))] Gamma[(1/2) (2 + h + i), -((b + c (-2 s + v) + 2 (d + f (-2 s + v)) Sqrt[z])^2/(4 (d + f (-2 s + v))))]), {i, 0, n}, {h, 0, i}]), {s, 0, Floor[(1/2) (-1 + v)]}] + I^m 2^(-1 - 2 n - m - v) Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] (E^(e + (I m Pi)/2 + (2 k - m) q - (b + a (2 k - m) + c (2 s - v))^2/ (4 (d + (2 k - m) p + f (2 s - v))) + g (2 s - v)) (d + (2 k - m) p + f (2 s - v))^(-2 - 2 n) Sum[(-1)^(-h + i) 4^i (b + a (2 k - m) + c (2 s - v))^ (-h - i + 2 n) (b + a (2 k - m) + c (2 s - v) + 2 (d + (2 k - m) p + f (2 s - v)) Sqrt[z])^(h + i) (-((b + a (2 k - m) + c (2 s - v) + 2 (d + (2 k - m) p + f (2 s - v)) Sqrt[z])^2/(d + (2 k - m) p + f (2 s - v))))^ ((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] ((b + a (2 k - m) + c (2 s - v)) (b + a (2 k - m) + c (2 s - v) + 2 (d + (2 k - m) p + f (2 s - v)) Sqrt[z]) Gamma[(1/2) (1 + h + i), -((b + a (2 k - m) + c (2 s - v) + 2 (d + (2 k - m) p + f (2 s - v)) Sqrt[z])^2/ (4 (d + (2 k - m) p + f (2 s - v))))] + 2 (d + (2 k - m) p + f (2 s - v)) Sqrt[-((b + a (2 k - m) + c (2 s - v) + 2 (d + (2 k - m) p + f (2 s - v)) Sqrt[z])^2/ (d + (2 k - m) p + f (2 s - v)))] Gamma[(1/2) (2 + h + i), -((b + a (2 k - m) + c (2 s - v) + 2 (d + (2 k - m) p + f (2 s - v)) Sqrt[z])^2/(4 (d + (2 k - m) p + f (2 s - v))))]), {i, 0, n}, {h, 0, i}] + E^(e - (I m Pi)/2 + (-2 k + m) q - (b + a (-2 k + m) + c (2 s - v))^ 2/(4 (d + (-2 k + m) p + f (2 s - v))) + g (2 s - v)) (d + (-2 k + m) p + f (2 s - v))^(-2 - 2 n) Sum[(-1)^(-h + i) 4^i (b + a (-2 k + m) + c (2 s - v))^ (-h - i + 2 n) (b + a (-2 k + m) + c (2 s - v) + 2 (d + (-2 k + m) p + f (2 s - v)) Sqrt[z])^(h + i) (-((b + a (-2 k + m) + c (2 s - v) + 2 (d + (-2 k + m) p + f (2 s - v)) Sqrt[z])^2/(d + (-2 k + m) p + f (2 s - v))))^ ((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] ((b + a (-2 k + m) + c (2 s - v)) (b + a (-2 k + m) + c (2 s - v) + 2 (d + (-2 k + m) p + f (2 s - v)) Sqrt[z]) Gamma[(1/2) (1 + h + i), -((b + a (-2 k + m) + c (2 s - v) + 2 (d + (-2 k + m) p + f (2 s - v)) Sqrt[z])^2/ (4 (d + (-2 k + m) p + f (2 s - v))))] + 2 (d + (-2 k + m) p + f (2 s - v)) Sqrt[-((b + a (-2 k + m) + c (2 s - v) + 2 (d + (-2 k + m) p + f (2 s - v)) Sqrt[z])^ 2/(d + (-2 k + m) p + f (2 s - v)))] Gamma[(1/2) (2 + h + i), -((b + a (-2 k + m) + c (2 s - v) + 2 (d + (-2 k + m) p + f (2 s - v)) Sqrt[z])^2/ (4 (d + (-2 k + m) p + f (2 s - v))))]), {i, 0, n}, {h, 0, i}] + E^(e + (I m Pi)/2 + (2 k - m) q + g (-2 s + v) - (b + a (2 k - m) + c (-2 s + v))^2/(4 (d + (2 k - m) p + f (-2 s + v)))) (d + (2 k - m) p + f (-2 s + v))^(-2 - 2 n) Sum[(-1)^(-h + i) 4^i (b + a (2 k - m) + c (-2 s + v))^ (-h - i + 2 n) (b + a (2 k - m) + c (-2 s + v) + 2 (d + (2 k - m) p + f (-2 s + v)) Sqrt[z])^(h + i) (-((b + a (2 k - m) + c (-2 s + v) + 2 (d + (2 k - m) p + f (-2 s + v)) Sqrt[z])^2/(d + (2 k - m) p + f (-2 s + v))))^((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] ((b + a (2 k - m) + c (-2 s + v)) (b + a (2 k - m) + c (-2 s + v) + 2 (d + (2 k - m) p + f (-2 s + v)) Sqrt[z]) Gamma[(1/2) (1 + h + i), -((b + a (2 k - m) + c (-2 s + v) + 2 (d + (2 k - m) p + f (-2 s + v)) Sqrt[z])^2/(4 (d + (2 k - m) p + f (-2 s + v))))] + 2 (d + (2 k - m) p + f (-2 s + v)) Sqrt[-((b + a (2 k - m) + c (-2 s + v) + 2 (d + (2 k - m) p + f (-2 s + v)) Sqrt[z])^2/(d + (2 k - m) p + f (-2 s + v)))] Gamma[(1/2) (2 + h + i), -((b + a (2 k - m) + c (-2 s + v) + 2 (d + (2 k - m) p + f (-2 s + v)) Sqrt[z])^2/(4 (d + (2 k - m) p + f (-2 s + v))))]), {i, 0, n}, {h, 0, i}] + E^(e - (I m Pi)/2 + (-2 k + m) q + g (-2 s + v) - (b + a (-2 k + m) + c (-2 s + v))^2/(4 (d + (-2 k + m) p + f (-2 s + v)))) (d + (-2 k + m) p + f (-2 s + v))^(-2 - 2 n) Sum[(-1)^(-h + i) 4^i (b + a (-2 k + m) + c (-2 s + v))^ (-h - i + 2 n) (b + a (-2 k + m) + c (-2 s + v) + 2 (d + (-2 k + m) p + f (-2 s + v)) Sqrt[z])^(h + i) (-((b + a (-2 k + m) + c (-2 s + v) + 2 (d + (-2 k + m) p + f (-2 s + v)) Sqrt[z])^2/(d + (-2 k + m) p + f (-2 s + v))))^((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] ((b + a (-2 k + m) + c (-2 s + v)) (b + a (-2 k + m) + c (-2 s + v) + 2 (d + (-2 k + m) p + f (-2 s + v)) Sqrt[z]) Gamma[(1/2) (1 + h + i), -((b + a (-2 k + m) + c (-2 s + v) + 2 (d + (-2 k + m) p + f (-2 s + v)) Sqrt[z])^2/(4 (d + (-2 k + m) p + f (-2 s + v))))] + 2 (d + (-2 k + m) p + f (-2 s + v)) Sqrt[-((b + a (-2 k + m) + c (-2 s + v) + 2 (d + (-2 k + m) p + f (-2 s + v)) Sqrt[z])^2/(d + (-2 k + m) p + f (-2 s + v)))] Gamma[(1/2) (2 + h + i), -((b + a (-2 k + m) + c (-2 s + v) + 2 (d + (-2 k + m) p + f (-2 s + v)) Sqrt[z])^2/(4 (d + (-2 k + m) p + f (-2 s + v))))]), {i, 0, n}, {h, 0, i}]), {s, 0, Floor[(1/2) (-1 + v)]}], {k, 0, Floor[(1/2) (-1 + m)]}] /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m > 0 && Element[v, Integers] && v > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List["d", " ", "z"]], "+", "e"]]], SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List[RowBox[List["a", " ", SqrtBox["z"]]], "+", RowBox[List["p", " ", "z"]], "+", "q"]], "]"]], "m"], SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c", " ", SqrtBox["z"]]], "+", RowBox[List["f", " ", "z"]], "+", "g"]], "]"]], "v"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["\[ImaginaryI]", "m"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m", "-", RowBox[List["2", " ", "n"]], "-", "v"]]], " ", SuperscriptBox["d", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", 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 MathML Form

 z n z b + d z + e sinh m ( z a + p z + q ) cosh v ( z c + f z + g ) z m 2 - m - 2 n - v - 1 d - 2 n - 2 e - b 2 4 d ( 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]]]]] ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - v mod 2 \$CellContext`v 2 ) i = 0 n h = 0 i ( - 1 ) i - h 4 i b - h - i + 2 n ( b + 2 d z ) h + i ( - ( b + 2 d z ) 2 d ) 1 2 ( - h - i - 1 ) ( i h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( b ( b + 2 d z ) Γ ( 1 2 ( h + i + 1 ) , - ( b + 2 d z ) 2 4 d ) + 2 - ( b + 2 d z ) 2 d d Γ ( 1 2 ( h + i + 2 ) , - ( b + 2 d z ) 2 4 d ) ) + m 2 - m - 2 n - v - 1 ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - v mod 2 \$CellContext`v 2 ) k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - ( b + a ( 2 k - m ) ) 2 4 ( d + ( 2 k - m ) p ) + e + ( 2 k - m ) q + m π 2 ( d + ( 2 k - m ) p ) - 2 n - 2 i = 0 n h = 0 i ( - 1 ) i - h 4 i ( b + a ( 2 k - m ) ) - h - i + 2 n ( b + a ( 2 k - m ) + 2 ( d + ( 2 k - m ) p ) z ) h + i ( - ( b + a ( 2 k - m ) + 2 ( d + ( 2 k - m ) p ) z ) 2 d + ( 2 k - m ) p ) 1 2 ( - h - i - 1 ) ( i h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( b + a ( 2 k - m ) ) ( b + a ( 2 k - m ) + 2 ( d + ( 2 k - m ) p ) z ) Γ ( 1 2 ( h + i + 1 ) , - ( b + a ( 2 k - m ) + 2 ( d + ( 2 k - m ) p ) z ) 2 4 ( d + ( 2 k - m ) p ) ) + 2 - ( b + a ( 2 k - m ) + 2 ( d + ( 2 k - m ) p ) z ) 2 d + ( 2 k - m ) p ( d + ( 2 k - m ) p ) Γ ( 1 2 ( h + i + 2 ) , - ( b + a ( 2 k - m ) + 2 ( d + ( 2 k - m ) p ) z ) 2 4 ( d + ( 2 k - m ) p ) ) ) + - ( b + a ( m - 2 k ) ) 2 4 ( d + ( m - 2 k ) p ) + e + ( m - 2 k ) q - m π 2 ( d + ( m - 2 k ) p ) - 2 n - 2 i = 0 n h = 0 i ( - 1 ) i - h 4 i ( b + a ( m - 2 k ) ) - h - i + 2 n ( b + a ( m - 2 k ) + 2 ( d + ( m - 2 k ) p ) z ) h + i ( - ( b + a ( m - 2 k ) + 2 ( d + ( m - 2 k ) p ) z ) 2 d + ( m - 2 k ) p ) 1 2 ( - h - i - 1 ) ( i h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( b + a ( m - 2 k ) ) ( b + a ( m - 2 k ) + 2 ( d + ( m - 2 k ) p ) z ) Γ ( 1 2 ( h + i + 1 ) , - ( b + a ( m - 2 k ) + 2 ( d + ( m - 2 k ) p ) z ) 2 4 ( d + ( m - 2 k ) p ) ) + 2 - ( b + a ( m - 2 k ) + 2 ( d + ( m - 2 k ) p ) z ) 2 d + ( m - 2 k ) p ( d + ( m - 2 k ) p ) Γ ( 1 2 ( h + i + 2 ) , - ( b + a ( m - 2 k ) + 2 ( d + ( m - 2 k ) p ) z ) 2 4 ( d + ( m - 2 k ) p ) ) ) ) + m 2 - m - 2 n - v - 1 ( 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]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - ( b + c ( 2 s - v ) ) 2 4 ( d + f ( 2 s - v ) ) + e + g ( 2 s - v ) ( d + f ( 2 s - v ) ) - 2 n - 2 i = 0 n h = 0 i ( - 1 ) i - h 4 i ( b + c ( 2 s - v ) ) - h - i + 2 n ( b + c ( 2 s - v ) + 2 ( d + f ( 2 s - v ) ) z ) h + i ( - ( b + c ( 2 s - v ) + 2 ( d + f ( 2 s - v ) ) z ) 2 d + f ( 2 s - v ) ) 1 2 ( - h - i - 1 ) ( i h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( b + c ( 2 s - v ) ) ( b + c ( 2 s - v ) + 2 ( d + f ( 2 s - v ) ) z ) Γ ( 1 2 ( h + i + 1 ) , - ( b + c ( 2 s - v ) + 2 ( d + f ( 2 s - v ) ) z ) 2 4 ( d + f ( 2 s - v ) ) ) + 2 - ( b + c ( 2 s - v ) + 2 ( d + f ( 2 s - v ) ) z ) 2 d + f ( 2 s - v ) ( d + f ( 2 s - v ) ) Γ ( 1 2 ( h + i + 2 ) , - ( b + c ( 2 s - v ) + 2 ( d + f ( 2 s - v ) ) z ) 2 4 ( d + f ( 2 s - v ) ) ) ) + - ( b + c ( v - 2 s ) ) 2 4 ( d + f ( v - 2 s ) ) + e + g ( v - 2 s ) ( d + f ( v - 2 s ) ) - 2 n - 2 i = 0 n h = 0 i ( - 1 ) i - h 4 i ( b + c ( v - 2 s ) ) - h - i + 2 n ( b + c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z ) h + i ( - ( b + c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z ) 2 d + f ( v - 2 s ) ) 1 2 ( - h - i - 1 ) ( i h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( b + c ( v - 2 s ) ) ( b + c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z ) Γ ( 1 2 ( h + i + 1 ) , - ( b + c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z ) 2 4 ( d + f ( v - 2 s ) ) ) + 2 - ( b + c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z ) 2 d + f ( v - 2 s ) ( d + f ( v - 2 s ) ) Γ ( 1 2 ( h + i + 2 ) , - ( b + c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z ) 2 4 ( d + f ( v - 2 s ) ) ) ) ) + m 2 - m - 2 n - v - 1 k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - ( b + a ( 2 k - m ) + c ( 2 s - v ) ) 2 4 ( d + ( 2 k - m ) p + f ( 2 s - v ) ) + e + ( 2 k - m ) q + g ( 2 s - v ) + m π 2 ( d + ( 2 k - m ) p + f ( 2 s - v ) ) - 2 n - 2 i = 0 n h = 0 i ( - 1 ) i - h 4 i ( b + a ( 2 k - m ) + c ( 2 s - v ) ) - h - i + 2 n ( b + a ( 2 k - m ) + c ( 2 s - v ) + 2 ( d + ( 2 k - m ) p + f ( 2 s - v ) ) z ) h + i ( - ( b + a ( 2 k - m ) + c ( 2 s - v ) + 2 ( d + ( 2 k - m ) p + f ( 2 s - v ) ) z ) 2 / ( d + ( 2 k - m ) p + f ( 2 s - v ) ) ) 1 2 ( - h - i - 1 ) ( i h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( b + a ( 2 k - m ) + c ( 2 s - v ) ) ( b + a ( 2 k - m ) + c ( 2 s - v ) + 2 ( d + ( 2 k - m ) p + f ( 2 s - v ) ) z ) Γ ( 1 2 ( h + i + 1 ) , - ( b + a ( 2 k - m ) + c ( 2 s - v ) + 2 ( d + ( 2 k - m ) p + f ( 2 s - v ) ) z ) 2 / ( 4 ( d + ( 2 k - m ) p + f ( 2 s - v ) ) ) ) + 2 ( d + ( 2 k - m ) p + f ( 2 s - v ) ) Γ ( 1 2 ( h + i + 2 ) , - ( b + a ( 2 k - m ) + c ( 2 s - v ) + 2 ( d + ( 2 k - m ) p + f ( 2 s - v ) ) z ) 2 / ( 4 ( d + ( 2 k - m ) p + f ( 2 s - v ) ) ) ) ( - ( b + a ( 2 k - m ) + c ( 2 s - v ) + 2 ( d + ( 2 k - m ) p + f ( 2 s - v ) ) z ) 2 / ( d + ( 2 k - m ) p + f ( 2 s - v ) ) ) ) + - ( b + a ( m - 2 k ) + c ( 2 s - v ) ) 2 4 ( d + ( m - 2 k ) p + f ( 2 s - v ) ) + e + ( m - 2 k ) q + g ( 2 s - v ) - m π 2 ( d + ( m - 2 k ) p + f ( 2 s - v ) ) - 2 n - 2 i = 0 n h = 0 i ( - 1 ) i - h 4 i ( b + a ( m - 2 k ) + c ( 2 s - v ) ) - h - i + 2 n ( b + a ( m - 2 k ) + c ( 2 s - v ) + 2 ( d + ( m - 2 k ) p + f ( 2 s - v ) ) z ) h + i ( - ( b + a ( m - 2 k ) + c ( 2 s - v ) + 2 ( d + ( m - 2 k ) p + f ( 2 s - v ) ) z ) 2 / ( d + ( m - 2 k ) p + f ( 2 s - v ) ) ) 1 2 ( - h - i - 1 ) ( i h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( b + a ( m - 2 k ) + c ( 2 s - v ) ) ( b + a ( m - 2 k ) + c ( 2 s - v ) + 2 ( d + ( m - 2 k ) p + f ( 2 s - v ) ) z ) Γ ( 1 2 ( h + i + 1 ) , - ( b + a ( m - 2 k ) + c ( 2 s - v ) + 2 ( d + ( m - 2 k ) p + f ( 2 s - v ) ) z ) 2 / ( 4 ( d + ( m - 2 k ) p + f ( 2 s - v ) ) ) ) + 2 ( d + ( m - 2 k ) p + f ( 2 s - v ) ) Γ ( 1 2 ( h + i + 2 ) , - ( b + a ( m - 2 k ) + c ( 2 s - v ) + 2 ( d + ( m - 2 k ) p + f ( 2 s - v ) ) z ) 2 / ( 4 ( d + ( m - 2 k ) p + f ( 2 s - v ) ) ) ) ( - ( b + a ( m - 2 k ) + c ( 2 s - v ) + 2 ( d + ( m - 2 k ) p + f ( 2 s - v ) ) z ) 2 / ( d + ( m - 2 k ) p + f ( 2 s - v ) ) ) ) + - ( b + a ( 2 k - m ) + c ( v - 2 s ) ) 2 4 ( d + ( 2 k - m ) p + f ( v - 2 s ) ) + e + ( 2 k - m ) q + g ( v - 2 s ) + m π 2 ( d + ( 2 k - m ) p + f ( v - 2 s ) ) - 2 n - 2 i = 0 n h = 0 i ( - 1 ) i - h 4 i ( b + a ( 2 k - m ) + c ( v - 2 s ) ) - h - i + 2 n ( b + a ( 2 k - m ) + c ( v - 2 s ) + 2 ( d + ( 2 k - m ) p + f ( v - 2 s ) ) z ) h + i ( - ( b + a ( 2 k - m ) + c ( v - 2 s ) + 2 ( d + ( 2 k - m ) p + f ( v - 2 s ) ) z ) 2 / ( d + ( 2 k - m ) p + f ( v - 2 s ) ) ) 1 2 ( - h - i - 1 ) ( i h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( b + a ( 2 k - m ) + c ( v - 2 s ) ) ( b + a ( 2 k - m ) + c ( v - 2 s ) + 2 ( d + ( 2 k - m ) p + f ( v - 2 s ) ) z ) Γ ( 1 2 ( h + i + 1 ) , - ( b + a ( 2 k - m ) + c ( v - 2 s ) + 2 ( d + ( 2 k - m ) p + f ( v - 2 s ) ) z ) 2 / ( 4 ( d + ( 2 k - m ) p + f ( v - 2 s ) ) ) ) + 2 ( d + ( 2 k - m ) p + f ( v - 2 s ) ) Γ ( 1 2 ( h + i + 2 ) , - ( b + a ( 2 k - m ) + c ( v - 2 s ) + 2 ( d + ( 2 k - m ) p + f ( v - 2 s ) ) z ) 2 / ( 4 ( d + ( 2 k - m ) p + f ( v - 2 s ) ) ) ) ( - ( b + a ( 2 k - m ) + c ( v - 2 s ) + 2 ( d + ( 2 k - m ) p + f ( v - 2 s ) ) z ) 2 / ( d + ( 2 k - m ) p + f ( v - 2 s ) ) ) ) + - ( b + a ( m - 2 k ) + c ( v - 2 s ) ) 2 4 ( d + ( m - 2 k ) p + f ( v - 2 s ) ) + e + ( m - 2 k ) q + g ( v - 2 s ) - m π 2 ( d + ( m - 2 k ) p + f ( v - 2 s ) ) - 2 n - 2 i = 0 n h = 0 i ( - 1 ) i - h 4 i ( b + a ( m - 2 k ) + c ( v - 2 s ) ) - h - i + 2 n ( b + a ( m - 2 k ) + c ( v - 2 s ) + 2 ( d + ( m - 2 k ) p + f ( v - 2 s ) ) z ) h + i ( - ( b + a ( m - 2 k ) + c ( v - 2 s ) + 2 ( d + ( m - 2 k ) p + f ( v - 2 s ) ) z ) 2 / ( d + ( m - 2 k ) p + f ( v - 2 s ) ) ) 1 2 ( - h - i - 1 ) ( i h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( b + a ( m - 2 k ) + c ( v - 2 s ) ) ( b + a ( m - 2 k ) + c ( v - 2 s ) + 2 ( d + ( m - 2 k ) p + f ( v - 2 s ) ) z ) Γ ( 1 2 ( h + i + 1 ) , - ( b + a ( m - 2 k ) + c ( v - 2 s ) + 2 ( d + ( m - 2 k ) p + f ( v - 2 s ) ) z ) 2 / ( 4 ( d + ( m - 2 k ) p + f ( v - 2 s ) ) ) ) + 2 ( d + ( m - 2 k ) p + f ( v - 2 s ) ) Γ ( 1 2 ( h + i + 2 ) , - ( b + a ( m - 2 k ) + c ( v - 2 s ) + 2 ( d + ( m - 2 k ) p + f ( v - 2 s ) ) z ) 2 / ( 4 ( d + ( m - 2 k ) p + f ( v - 2 s ) ) ) ) ( - ( b + a ( m - 2 k ) + c ( v - 2 s ) + 2 ( d + ( m - 2 k ) p + f ( v - 2 s ) ) z ) 2 / ( d + ( m - 2 k ) p + f ( v - 2 s ) ) ) ) ) /; n m + v + Condition z z n z 1 2 b d z e z 1 2 a p z q m z 1 2 c f z g v m 2 -1 m -1 2 n -1 v -1 d -2 n -2 e -1 b 2 4 d -1 Binomial m m 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 h 0 i i 0 n -1 i -1 h 4 i b -1 h -1 i 2 n b 2 d z 1 2 h i -1 b 2 d z 1 2 2 d -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i b b 2 d z 1 2 Gamma 1 2 h i 1 -1 b 2 d z 1 2 2 4 d -1 2 -1 b 2 d z 1 2 2 d -1 1 2 d Gamma 1 2 h i 2 -1 b 2 d z 1 2 2 4 d -1 m 2 -1 m -1 2 n -1 v -1 Binomial v v 2 -1 1 -1 \$CellContext`v 2 k 0 m -1 2 -1 -1 k Binomial m k -1 b a 2 k -1 m 2 4 d 2 k -1 m p -1 e 2 k -1 m q m 2 -1 d 2 k -1 m p -2 n -2 h 0 i i 0 n -1 i -1 h 4 i b a 2 k -1 m -1 h -1 i 2 n b a 2 k -1 m 2 d 2 k -1 m p z 1 2 h i -1 b a 2 k -1 m 2 d 2 k -1 m p z 1 2 2 d 2 k -1 m p -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i b a 2 k -1 m b a 2 k -1 m 2 d 2 k -1 m p z 1 2 Gamma 1 2 h i 1 -1 b a 2 k -1 m 2 d 2 k -1 m p z 1 2 2 4 d 2 k -1 m p -1 2 -1 b a 2 k -1 m 2 d 2 k -1 m p z 1 2 2 d 2 k -1 m p -1 1 2 d 2 k -1 m p Gamma 1 2 h i 2 -1 b a 2 k -1 m 2 d 2 k -1 m p z 1 2 2 4 d 2 k -1 m p -1 -1 b a m -1 2 k 2 4 d m -1 2 k p -1 e m -1 2 k q -1 m 2 -1 d m -1 2 k p -2 n -2 h 0 i i 0 n -1 i -1 h 4 i b a m -1 2 k -1 h -1 i 2 n b a m -1 2 k 2 d m -1 2 k p z 1 2 h i -1 b a m -1 2 k 2 d m -1 2 k p z 1 2 2 d m -1 2 k p -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i b a m -1 2 k b a m -1 2 k 2 d m -1 2 k p z 1 2 Gamma 1 2 h i 1 -1 b a m -1 2 k 2 d m -1 2 k p z 1 2 2 4 d m -1 2 k p -1 2 -1 b a m -1 2 k 2 d m -1 2 k p z 1 2 2 d m -1 2 k p -1 1 2 d m -1 2 k p Gamma 1 2 h i 2 -1 b a m -1 2 k 2 d m -1 2 k p z 1 2 2 4 d m -1 2 k p -1 m 2 -1 m -1 2 n -1 v -1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 s 0 v -1 2 -1 Binomial v s -1 b c 2 s -1 v 2 4 d f 2 s -1 v -1 e g 2 s -1 v d f 2 s -1 v -2 n -2 h 0 i i 0 n -1 i -1 h 4 i b c 2 s -1 v -1 h -1 i 2 n b c 2 s -1 v 2 d f 2 s -1 v z 1 2 h i -1 b c 2 s -1 v 2 d f 2 s -1 v z 1 2 2 d f 2 s -1 v -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i b c 2 s -1 v b c 2 s -1 v 2 d f 2 s -1 v z 1 2 Gamma 1 2 h i 1 -1 b c 2 s -1 v 2 d f 2 s -1 v z 1 2 2 4 d f 2 s -1 v -1 2 -1 b c 2 s -1 v 2 d f 2 s -1 v z 1 2 2 d f 2 s -1 v -1 1 2 d f 2 s -1 v Gamma 1 2 h i 2 -1 b c 2 s -1 v 2 d f 2 s -1 v z 1 2 2 4 d f 2 s -1 v -1 -1 b c v -1 2 s 2 4 d f v -1 2 s -1 e g v -1 2 s d f v -1 2 s -2 n -2 h 0 i i 0 n -1 i -1 h 4 i b c v -1 2 s -1 h -1 i 2 n b c v -1 2 s 2 d f v -1 2 s z 1 2 h i -1 b c v -1 2 s 2 d f v -1 2 s z 1 2 2 d f v -1 2 s -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i b c v -1 2 s b c v -1 2 s 2 d f v -1 2 s z 1 2 Gamma 1 2 h i 1 -1 b c v -1 2 s 2 d f v -1 2 s z 1 2 2 4 d f v -1 2 s -1 2 -1 b c v -1 2 s 2 d f v -1 2 s z 1 2 2 d f v -1 2 s -1 1 2 d f v -1 2 s Gamma 1 2 h i 2 -1 b c v -1 2 s 2 d f v -1 2 s z 1 2 2 4 d f v -1 2 s -1 m 2 -1 m -1 2 n -1 v -1 k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 Binomial v s -1 b a 2 k -1 m c 2 s -1 v 2 4 d 2 k -1 m p f 2 s -1 v -1 e 2 k -1 m q g 2 s -1 v m 2 -1 d 2 k -1 m p f 2 s -1 v -2 n -2 h 0 i i 0 n -1 i -1 h 4 i b a 2 k -1 m c 2 s -1 v -1 h -1 i 2 n b a 2 k -1 m c 2 s -1 v 2 d 2 k -1 m p f 2 s -1 v z 1 2 h i -1 b a 2 k -1 m c 2 s -1 v 2 d 2 k -1 m p f 2 s -1 v z 1 2 2 d 2 k -1 m p f 2 s -1 v -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i b a 2 k -1 m c 2 s -1 v b a 2 k -1 m c 2 s -1 v 2 d 2 k -1 m p f 2 s -1 v z 1 2 Gamma 1 2 h i 1 -1 b a 2 k -1 m c 2 s -1 v 2 d 2 k -1 m p f 2 s -1 v z 1 2 2 4 d 2 k -1 m p f 2 s -1 v -1 2 d 2 k -1 m p f 2 s -1 v Gamma 1 2 h i 2 -1 b a 2 k -1 m c 2 s -1 v 2 d 2 k -1 m p f 2 s -1 v z 1 2 2 4 d 2 k -1 m p f 2 s -1 v -1 -1 b a 2 k -1 m c 2 s -1 v 2 d 2 k -1 m p f 2 s -1 v z 1 2 2 d 2 k -1 m p f 2 s -1 v -1 -1 b a m -1 2 k c 2 s -1 v 2 4 d m -1 2 k p f 2 s -1 v -1 e m -1 2 k q g 2 s -1 v -1 m 2 -1 d m -1 2 k p f 2 s -1 v -2 n -2 h 0 i i 0 n -1 i -1 h 4 i b a m -1 2 k c 2 s -1 v -1 h -1 i 2 n b a m -1 2 k c 2 s -1 v 2 d m -1 2 k p f 2 s -1 v z 1 2 h i -1 b a m -1 2 k c 2 s -1 v 2 d m -1 2 k p f 2 s -1 v z 1 2 2 d m -1 2 k p f 2 s -1 v -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i b a m -1 2 k c 2 s -1 v b a m -1 2 k c 2 s -1 v 2 d m -1 2 k p f 2 s -1 v z 1 2 Gamma 1 2 h i 1 -1 b a m -1 2 k c 2 s -1 v 2 d m -1 2 k p f 2 s -1 v z 1 2 2 4 d m -1 2 k p f 2 s -1 v -1 2 d m -1 2 k p f 2 s -1 v Gamma 1 2 h i 2 -1 b a m -1 2 k c 2 s -1 v 2 d m -1 2 k p f 2 s -1 v z 1 2 2 4 d m -1 2 k p f 2 s -1 v -1 -1 b a m -1 2 k c 2 s -1 v 2 d m -1 2 k p f 2 s -1 v z 1 2 2 d m -1 2 k p f 2 s -1 v -1 -1 b a 2 k -1 m c v -1 2 s 2 4 d 2 k -1 m p f v -1 2 s -1 e 2 k -1 m q g v -1 2 s m 2 -1 d 2 k -1 m p f v -1 2 s -2 n -2 h 0 i i 0 n -1 i -1 h 4 i b a 2 k -1 m c v -1 2 s -1 h -1 i 2 n b a 2 k -1 m c v -1 2 s 2 d 2 k -1 m p f v -1 2 s z 1 2 h i -1 b a 2 k -1 m c v -1 2 s 2 d 2 k -1 m p f v -1 2 s z 1 2 2 d 2 k -1 m p f v -1 2 s -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i b a 2 k -1 m c v -1 2 s b a 2 k -1 m c v -1 2 s 2 d 2 k -1 m p f v -1 2 s z 1 2 Gamma 1 2 h i 1 -1 b a 2 k -1 m c v -1 2 s 2 d 2 k -1 m p f v -1 2 s z 1 2 2 4 d 2 k -1 m p f v -1 2 s -1 2 d 2 k -1 m p f v -1 2 s Gamma 1 2 h i 2 -1 b a 2 k -1 m c v -1 2 s 2 d 2 k -1 m p f v -1 2 s z 1 2 2 4 d 2 k -1 m p f v -1 2 s -1 -1 b a 2 k -1 m c v -1 2 s 2 d 2 k -1 m p f v -1 2 s z 1 2 2 d 2 k -1 m p f v -1 2 s -1 -1 b a m -1 2 k c v -1 2 s 2 4 d m -1 2 k p f v -1 2 s -1 e m -1 2 k q g v -1 2 s -1 m 2 -1 d m -1 2 k p f v -1 2 s -2 n -2 h 0 i i 0 n -1 i -1 h 4 i b a m -1 2 k c v -1 2 s -1 h -1 i 2 n b a m -1 2 k c v -1 2 s 2 d m -1 2 k p f v -1 2 s z 1 2 h i -1 b a m -1 2 k c v -1 2 s 2 d m -1 2 k p f v -1 2 s z 1 2 2 d m -1 2 k p f v -1 2 s -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i b a m -1 2 k c v -1 2 s b a m -1 2 k c v -1 2 s 2 d m -1 2 k p f v -1 2 s z 1 2 Gamma 1 2 h i 1 -1 b a m -1 2 k c v -1 2 s 2 d m -1 2 k p f v -1 2 s z 1 2 2 4 d m -1 2 k p f v -1 2 s -1 2 d m -1 2 k p f v -1 2 s Gamma 1 2 h i 2 -1 b a m -1 2 k c v -1 2 s 2 d m -1 2 k p f v -1 2 s z 1 2 2 4 d m -1 2 k p f v -1 2 s -1 -1 b a m -1 2 k c v -1 2 s 2 d m -1 2 k p f v -1 2 s z 1 2 2 d m -1 2 k p f v -1 2 s -1 n m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18