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; }

 Cos

 http://functions.wolfram.com/01.07.21.2566.01

 Input Form

 Integrate[z^n Sin[b Sqrt[z]]^m Cos[c Sqrt[z] + f z + g]^v, z] == 2^(-1 - m - v) ((1/(1 + n)) (2 z^(1 + n) Binomial[m, m/2] Binomial[v, v/2] (-1 + Mod[m, 2]) (-1 + Mod[v, 2])) + (4 Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(-1)^(k + n) (-2 k + m)^(-2 n - 2) Binomial[m, k] ((-1)^m Gamma[2 (1 + n), (-I) b (2 k - m) Sqrt[z]] + Gamma[2 (1 + n), I b (2 k - m) Sqrt[z]]), {k, 0, Floor[(1/2) (-1 + m)]}])/(I^m b^(2 (1 + n))) + (1/c) ((I Binomial[m, m/2] (-1 + Mod[m, 2]) Sum[((f^2 (-2 s + v)^2)^(-1 - 2 n) Binomial[v, s] (((-I) f (2 s - v))^(2 n) Sum[(-1)^(-h + k) 4^k (I c (2 s - v))^( 1 - h - k + 2 n) (I (2 s - v) (c + 2 f Sqrt[z]))^(h + k) (-((I (2 s - v) (c + 2 f Sqrt[z])^2)/f))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (c (2 s - v) (c + 2 f Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I (2 s - v) (c + 2 f Sqrt[z])^2)/ (4 f))] - 2 I f Sqrt[-((I (2 s - v) (c + 2 f Sqrt[z])^2)/ f)] Gamma[(1/2) (2 + h + k), -((I (2 s - v) (c + 2 f Sqrt[z])^2)/(4 f))]), {k, 0, n}, {h, 0, k}] - E^((I (c^2 - 4 f g) (2 s - v))/(2 f)) (I f (2 s - v))^(2 n) Sum[(-1)^(-h + k) 4^k ((-I) c (2 s - v))^(1 - h - k + 2 n) ((-I) (2 s - v) (c + 2 f Sqrt[z]))^(h + k) ((I (2 s - v) (c + 2 f Sqrt[z])^2)/f)^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (c (2 s - v) (c + 2 f Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (2 s - v) (c + 2 f Sqrt[z])^2)/ (4 f)] + 2 I f Sqrt[(I (2 s - v) (c + 2 f Sqrt[z])^2)/f] Gamma[(1/2) (2 + h + k), (I (2 s - v) (c + 2 f Sqrt[z])^2)/ (4 f)]), {k, 0, n}, {h, 0, k}]))/ E^((I (c^2 - 4 f g) (2 s - v))/(4 f)), {s, 0, Floor[(1/2) (-1 + v)]}])/4^n) - ((-1)^n Sum[(-1)^u Binomial[m, u] Sum[(Binomial[v, s] ((Sum[(-1)^(-h + k) 4^k (I (b (m - 2 u) + c (-2 s + v)))^(-h - k + 2 n) (I (b (m - 2 u) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z]))^(h + k) (-((I (b (m - 2 u) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])^ 2)/(f (-2 s + v))))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (((-b) (m - 2 u) - c (-2 s + v)) (b (m - 2 u) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I (b (m - 2 u) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])^2)/(4 f (-2 s + v)))] + 2 I f (-2 s + v) Sqrt[-((I (b (m - 2 u) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])^2)/(f (-2 s + v)))] Gamma[(1/2) (2 + h + k), -((I (b (m - 2 u) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])^2)/(4 f (-2 s + v)))]), {k, 0, n}, {h, 0, k}] + (-1)^m E^((1/2) I (8 g s - 4 g v + (b (m - 2 u) + c (-2 s + v))^2/(f (-2 s + v)))) Sum[ (-1)^(-h + k) 4^k ((-I) (b (m - 2 u) + c (-2 s + v)))^ (-h - k + 2 n) ((-I) (b (m - 2 u) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z]))^(h + k) ((I (b (m - 2 u) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])^2)/ (f (-2 s + v)))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (((-b) (m - 2 u) - c (-2 s + v)) (b (m - 2 u) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (b (m - 2 u) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])^2)/(4 f (-2 s + v))] - 2 I f (-2 s + v) Sqrt[(I (b (m - 2 u) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])^2)/(f (-2 s + v))] Gamma[(1/2) (2 + h + k), (I (b (m - 2 u) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])^2)/(4 f (-2 s + v))]), {k, 0, n}, {h, 0, k}])/E^((I (b (m - 2 u) + c (-2 s + v))^2)/( 4 f (-2 s + v))) + E^(I (g (4 s - 2 v) - (b m + 2 c s - 2 b u - c v)^2/(8 f s - 4 f v))) Sum[(-1)^(-h + k) 4^k (I (b m + 2 c s - 2 b u - c v))^ (-h - k + 2 n) (I (b (m - 2 u) + c (2 s - v) + 2 (2 f s - f v) Sqrt[z]))^(h + k) (-((1/(2 f s - f v)) (I (b (m - 2 u) + c (2 s - v) + 2 (2 f s - f v) Sqrt[z])^ 2)))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ( ((-b) m - 2 c s + 2 b u + c v) (b (m - 2 u) + c (2 s - v) + 2 (2 f s - f v) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I (b (m - 2 u) + c (2 s - v) + 2 (2 f s - f v) Sqrt[z])^ 2)/(8 f s - 4 f v))] + 2 I (2 f s - f v) Sqrt[-((1/(2 f s - f v)) (I (b (m - 2 u) + c (2 s - v) + 2 (2 f s - f v) Sqrt[z])^2))] Gamma[(1/2) (2 + h + k), -((I (b (m - 2 u) + c (2 s - v) + 2 (2 f s - f v) Sqrt[z])^ 2)/(8 f s - 4 f v))]), {k, 0, n}, {h, 0, k}] + (-1)^m E^((I (b m + 2 c s - 2 b u - c v)^2)/(8 f s - 4 f v)) Sum[(-1)^(-h + k) 4^k ((-I) (b m + 2 c s - 2 b u - c v))^ (-h - k + 2 n) ((-I) (b (m - 2 u) + c (2 s - v) + 2 (2 f s - f v) Sqrt[z]))^(h + k) ((1/(2 f s - f v)) (I (b (m - 2 u) + c (2 s - v) + 2 (2 f s - f v) Sqrt[z])^2))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ( ((-b) m - 2 c s + 2 b u + c v) (b (m - 2 u) + c (2 s - v) + 2 (2 f s - f v) Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (b (m - 2 u) + c (2 s - v) + 2 (2 f s - f v) Sqrt[z])^2)/ (8 f s - 4 f v)] - 2 I (2 f s - f v) Sqrt[ (1/(2 f s - f v)) (I (b (m - 2 u) + c (2 s - v) + 2 (2 f s - f v) Sqrt[z])^2)] Gamma[(1/2) (2 + h + k), (I (b (m - 2 u) + c (2 s - v) + 2 (2 f s - f v) Sqrt[z])^2)/ (8 f s - 4 f v)]), {k, 0, n}, {h, 0, k}]))/ (E^(I (2 g s - g v)) (-2 s + v)^(2 (1 + n))), {s, 0, Floor[(1/2) (-1 + v)]}], {u, 0, Floor[(1/2) (-1 + m)]}])/ (I^m 4^n f^(2 (1 + n)))) /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m > 0 && Element[v, Integers] && v > 0

 Standard Form

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 MathML Form

 z n sin m ( b z ) cos v ( z c + f z + g ) z 2 - m - v - 1 ( 2 z n + 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]]]]] ( 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]]]]] ( m mod 2 \$CellContext`m 2 - 1 ) ( v mod 2 \$CellContext`v 2 - 1 ) n + 1 + 4 - m b - 2 ( n + 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 + n ( m - 2 k ) - 2 n - 2 ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - 1 ) m Γ ( 2 ( n + 1 ) , - b ( 2 k - m ) z ) + Γ ( 2 ( n + 1 ) , b ( 2 k - m ) z ) ) + 1 c ( 4 - n ( 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]]]]] ( m mod 2 \$CellContext`m 2 - 1 ) s = 0 v - 1 2 - ( c 2 - 4 f g ) ( 2 s - v ) 4 f ( f 2 ( v - 2 s ) 2 ) - 2 n - 1 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - f ( 2 s - v ) ) 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k ( c ( 2 s - v ) ) - h - k + 2 n + 1 ( ( 2 s - v ) ( c + 2 f z ) ) h + k ( - ( 2 s - v ) ( c + 2 f z ) 2 f ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( c ( 2 s - v ) ( c + 2 f z ) Γ ( 1 2 ( h + k + 1 ) , - ( 2 s - v ) ( c + 2 f z ) 2 4 f ) - 2 f - ( 2 s - v ) ( c + 2 f z ) 2 f Γ ( 1 2 ( h + k + 2 ) , - ( 2 s - v ) ( c + 2 f z ) 2 4 f ) ) - ( c 2 - 4 f g ) ( 2 s - v ) 2 f ( f ( 2 s - v ) ) 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - c ( 2 s - v ) ) - h - k + 2 n + 1 ( - ( 2 s - v ) ( c + 2 f z ) ) h + k ( ( 2 s - v ) ( c + 2 f z ) 2 f ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( c ( 2 s - v ) ( c + 2 f z ) Γ ( 1 2 ( h + k + 1 ) , ( 2 s - v ) ( c + 2 f z ) 2 4 f ) + 2 ( 2 s - v ) ( c + 2 f z ) 2 f f Γ ( 1 2 ( h + k + 2 ) , ( 2 s - v ) ( c + 2 f z ) 2 4 f ) ) ) ) - - m ( - 1 ) n 4 - n f - 2 ( n + 1 ) u = 0 m - 1 2 ( - 1 ) u ( m u ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["u", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] s = 0 v - 1 2 - ( 2 g s - g v ) ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( v - 2 s ) - 2 ( n + 1 ) ( - ( b ( m - 2 u ) + c ( v - 2 s ) ) 2 4 f ( v - 2 s ) ( ( - 1 ) m 1 2 ( ( b ( m - 2 u ) + c ( v - 2 s ) ) 2 f ( v - 2 s ) + 8 g s - 4 g v ) k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - ( b ( m - 2 u ) + c ( v - 2 s ) ) ) - h - k + 2 n ( - ( b ( m - 2 u ) + c ( v - 2 s ) + 2 f ( v - 2 s ) z ) ) h + k ( ( b ( m - 2 u ) + c ( v - 2 s ) + 2 f ( v - 2 s ) z ) 2 f ( v - 2 s ) ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - b ( m - 2 u ) - c ( v - 2 s ) ) ( b ( m - 2 u ) + c ( v - 2 s ) + 2 f ( v - 2 s ) z ) Γ ( 1 2 ( h + k + 1 ) , ( b ( m - 2 u ) + c ( v - 2 s ) + 2 f ( v - 2 s ) z ) 2 4 f ( v - 2 s ) ) - 2 f ( v - 2 s ) ( b ( m - 2 u ) + c ( v - 2 s ) + 2 f ( v - 2 s ) z ) 2 f ( v - 2 s ) Γ ( 1 2 ( h + k + 2 ) , ( b ( m - 2 u ) + c ( v - 2 s ) + 2 f ( v - 2 s ) z ) 2 4 f ( v - 2 s ) ) ) + k = 0 n h = 0 k ( - 1 ) k - h 4 k ( ( b ( m - 2 u ) + c ( v - 2 s ) ) ) - h - k + 2 n ( ( b ( m - 2 u ) + c ( v - 2 s ) + 2 f ( v - 2 s ) z ) ) h + k ( - ( b ( m - 2 u ) + c ( v - 2 s ) + 2 f ( v - 2 s ) z ) 2 f ( v - 2 s ) ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - b ( m - 2 u ) - c ( v - 2 s ) ) ( b ( m - 2 u ) + c ( v - 2 s ) + 2 f ( v - 2 s ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( b ( m - 2 u ) + c ( v - 2 s ) + 2 f ( v - 2 s ) z ) 2 4 f ( v - 2 s ) ) + 2 f ( v - 2 s ) Γ ( 1 2 ( h + k + 2 ) , - ( b ( m - 2 u ) + c ( v - 2 s ) + 2 f ( v - 2 s ) z ) 2 4 f ( v - 2 s ) ) - ( b ( m - 2 u ) + c ( v - 2 s ) + 2 f ( v - 2 s ) z ) 2 f ( v - 2 s ) ) ) + ( - 1 ) m ( b m + 2 c s - 2 b u - c v ) 2 8 f s - 4 f v k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - ( b m + 2 c s - 2 b u - c v ) ) - h - k + 2 n ( - ( b ( m - 2 u ) + c ( 2 s - v ) + 2 ( 2 f s - f v ) z ) ) h + k ( ( b ( m - 2 u ) + c ( 2 s - v ) + 2 ( 2 f s - f v ) z ) 2 2 f s - f v ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - b m - 2 c s + 2 b u + c v ) ( b ( m - 2 u ) + c ( 2 s - v ) + 2 ( 2 f s - f v ) z ) Γ ( 1 2 ( h + k + 1 ) , ( b ( m - 2 u ) + c ( 2 s - v ) + 2 ( 2 f s - f v ) z ) 2 8 f s - 4 f v ) - 2 ( 2 f s - f v ) ( b ( m - 2 u ) + c ( 2 s - v ) + 2 ( 2 f s - f v ) z ) 2 2 f s - f v Γ ( 1 2 ( h + k + 2 ) , ( b ( m - 2 u ) + c ( 2 s - v ) + 2 ( 2 f s - f v ) z ) 2 8 f s - 4 f v ) ) + ( g ( 4 s - 2 v ) - ( b m + 2 c s - 2 b u - c v ) 2 8 f s - 4 f v ) k = 0 n h = 0 k ( - 1 ) k - h 4 k ( ( b m + 2 c s - 2 b u - c v ) ) - h - k + 2 n ( ( b ( m - 2 u ) + c ( 2 s - v ) + 2 ( 2 f s - f v ) z ) ) h + k ( - ( b ( m - 2 u ) + c ( 2 s - v ) + 2 ( 2 f s - f v ) z ) 2 2 f s - f v ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - b m - 2 c s + 2 b u + c v ) ( b ( m - 2 u ) + c ( 2 s - v ) + 2 ( 2 f s - f v ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( b ( m - 2 u ) + c ( 2 s - v ) + 2 ( 2 f s - f v ) z ) 2 8 f s - 4 f v ) + 2 ( 2 f s - f v ) Γ ( 1 2 ( h + k + 2 ) , - ( b ( m - 2 u ) + c ( 2 s - v ) + 2 ( 2 f s - f v ) z ) 2 8 f s - 4 f v ) - ( b ( m - 2 u ) + c ( 2 s - v ) + 2 ( 2 f s - f v ) z ) 2 2 f s - f v ) ) ) /; n m + v + Condition z z n b z 1 2 m z 1 2 c f z g v 2 -1 m -1 v -1 2 z n 1 Binomial m m 2 -1 Binomial v v 2 -1 \$CellContext`m 2 -1 \$CellContext`v 2 -1 n 1 -1 4 -1 m b -2 n 1 Binomial v v 2 -1 1 -1 \$CellContext`v 2 k 0 m -1 2 -1 -1 k n m -1 2 k -2 n -2 Binomial m k -1 m Gamma 2 n 1 -1 b 2 k -1 m z 1 2 Gamma 2 n 1 b 2 k -1 m z 1 2 1 c -1 4 -1 n Binomial m m 2 -1 \$CellContext`m 2 -1 s 0 v -1 2 -1 -1 c 2 -1 4 f g 2 s -1 v 4 f -1 f 2 v -1 2 s 2 -2 n -1 Binomial v s -1 f 2 s -1 v 2 n h 0 k k 0 n -1 k -1 h 4 k c 2 s -1 v -1 h -1 k 2 n 1 2 s -1 v c 2 f z 1 2 h k -1 2 s -1 v c 2 f z 1 2 2 f -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k c 2 s -1 v c 2 f z 1 2 Gamma 1 2 h k 1 -1 2 s -1 v c 2 f z 1 2 2 4 f -1 -1 2 f -1 2 s -1 v c 2 f z 1 2 2 f -1 1 2 Gamma 1 2 h k 2 -1 2 s -1 v c 2 f z 1 2 2 4 f -1 -1 c 2 -1 4 f g 2 s -1 v 2 f -1 f 2 s -1 v 2 n h 0 k k 0 n -1 k -1 h 4 k -1 c 2 s -1 v -1 h -1 k 2 n 1 -1 2 s -1 v c 2 f z 1 2 h k 2 s -1 v c 2 f z 1 2 2 f -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k c 2 s -1 v c 2 f z 1 2 Gamma 1 2 h k 1 2 s -1 v c 2 f z 1 2 2 4 f -1 2 2 s -1 v c 2 f z 1 2 2 f -1 1 2 f Gamma 1 2 h k 2 2 s -1 v c 2 f z 1 2 2 4 f -1 -1 -1 m -1 n 4 -1 n f -2 n 1 u 0 m -1 2 -1 -1 u Binomial m u s 0 v -1 2 -1 -1 2 g s -1 g v Binomial v s v -1 2 s -2 n 1 -1 b m -1 2 u c v -1 2 s 2 4 f v -1 2 s -1 -1 m 1 2 b m -1 2 u c v -1 2 s 2 f v -1 2 s -1 8 g s -1 4 g v h 0 k k 0 n -1 k -1 h 4 k -1 b m -1 2 u c v -1 2 s -1 h -1 k 2 n -1 b m -1 2 u c v -1 2 s 2 f v -1 2 s z 1 2 h k b m -1 2 u c v -1 2 s 2 f v -1 2 s z 1 2 2 f v -1 2 s -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k -1 b m -1 2 u -1 c v -1 2 s b m -1 2 u c v -1 2 s 2 f v -1 2 s z 1 2 Gamma 1 2 h k 1 b m -1 2 u c v -1 2 s 2 f v -1 2 s z 1 2 2 4 f v -1 2 s -1 -1 2 f v -1 2 s b m -1 2 u c v -1 2 s 2 f v -1 2 s z 1 2 2 f v -1 2 s -1 1 2 Gamma 1 2 h k 2 b m -1 2 u c v -1 2 s 2 f v -1 2 s z 1 2 2 4 f v -1 2 s -1 h 0 k k 0 n -1 k -1 h 4 k b m -1 2 u c v -1 2 s -1 h -1 k 2 n b m -1 2 u c v -1 2 s 2 f v -1 2 s z 1 2 h k -1 b m -1 2 u c v -1 2 s 2 f v -1 2 s z 1 2 2 f v -1 2 s -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k -1 b m -1 2 u -1 c v -1 2 s b m -1 2 u c v -1 2 s 2 f v -1 2 s z 1 2 Gamma 1 2 h k 1 -1 b m -1 2 u c v -1 2 s 2 f v -1 2 s z 1 2 2 4 f v -1 2 s -1 2 f v -1 2 s Gamma 1 2 h k 2 -1 b m -1 2 u c v -1 2 s 2 f v -1 2 s z 1 2 2 4 f v -1 2 s -1 -1 b m -1 2 u c v -1 2 s 2 f v -1 2 s z 1 2 2 f v -1 2 s -1 1 2 -1 m b m 2 c s -1 2 b u -1 c v 2 8 f s -1 4 f v -1 h 0 k k 0 n -1 k -1 h 4 k -1 b m 2 c s -1 2 b u -1 c v -1 h -1 k 2 n -1 b m -1 2 u c 2 s -1 v 2 2 f s -1 f v z 1 2 h k b m -1 2 u c 2 s -1 v 2 2 f s -1 f v z 1 2 2 2 f s -1 f v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k -1 b m -1 2 c s 2 b u c v b m -1 2 u c 2 s -1 v 2 2 f s -1 f v z 1 2 Gamma 1 2 h k 1 b m -1 2 u c 2 s -1 v 2 2 f s -1 f v z 1 2 2 8 f s -1 4 f v -1 -1 2 2 f s -1 f v b m -1 2 u c 2 s -1 v 2 2 f s -1 f v z 1 2 2 2 f s -1 f v -1 1 2 Gamma 1 2 h k 2 b m -1 2 u c 2 s -1 v 2 2 f s -1 f v z 1 2 2 8 f s -1 4 f v -1 g 4 s -1 2 v -1 b m 2 c s -1 2 b u -1 c v 2 8 f s -1 4 f v -1 h 0 k k 0 n -1 k -1 h 4 k b m 2 c s -1 2 b u -1 c v -1 h -1 k 2 n b m -1 2 u c 2 s -1 v 2 2 f s -1 f v z 1 2 h k -1 b m -1 2 u c 2 s -1 v 2 2 f s -1 f v z 1 2 2 2 f s -1 f v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k -1 b m -1 2 c s 2 b u c v b m -1 2 u c 2 s -1 v 2 2 f s -1 f v z 1 2 Gamma 1 2 h k 1 -1 b m -1 2 u c 2 s -1 v 2 2 f s -1 f v z 1 2 2 8 f s -1 4 f v -1 2 2 f s -1 f v Gamma 1 2 h k 2 -1 b m -1 2 u c 2 s -1 v 2 2 f s -1 f v z 1 2 2 8 f s -1 4 f v -1 -1 b m -1 2 u c 2 s -1 v 2 2 f s -1 f v z 1 2 2 2 f s -1 f v -1 1 2 n m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18