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/01.07.21.2534.01

 Input Form

 Integrate[z^n Sin[b Sqrt[z] + d z + e]^m Cos[c Sqrt[z]]^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 z^(1 + n) Binomial[m, m/2] (-1 + Mod[m, 2]) Sum[Binomial[v, s] (Gamma[2 (1 + n), (-I) c (2 s - v) Sqrt[z]]/((-I) c (2 s - v) Sqrt[z])^ (2 (1 + n)) + Gamma[2 (1 + n), I c (2 s - v) Sqrt[z]]/ (I c (2 s - v) Sqrt[z])^(2 (1 + n))), {s, 0, Floor[(1/2) (-1 + v)]}] - (Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[((-1)^(u - 1) (d^2 (m - 2 u)^2)^(-1 - 2 n) Binomial[m, u] (E^(2 I e (m - 2 u)) ((-I) d (m - 2 u))^(2 n) Sum[(-(-1)^(-h + k)) 4^k (m - 2 u) (I b (m - 2 u))^(-h - k + 2 n) (I (m - 2 u) (b + 2 d Sqrt[z]))^(h + k) (-((I (m - 2 u) (b + 2 d Sqrt[z])^2)/d))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (b (m - 2 u) (b + 2 d Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I (m - 2 u) (b + 2 d Sqrt[z])^2)/ (4 d))] - 2 I d Sqrt[-((I (m - 2 u) (b + 2 d Sqrt[z])^2)/ d)] Gamma[(1/2) (2 + h + k), -((I (m - 2 u) (b + 2 d Sqrt[z])^2)/(4 d))]), {k, 0, n}, {h, 0, k}] + (-1)^m E^((I b^2 (m - 2 u))/(2 d)) (I d (m - 2 u))^(2 n) Sum[(-(-1)^(-h + k)) 4^k (m - 2 u) ((-I) b (m - 2 u))^ (-h - k + 2 n) ((-I) (m - 2 u) (b + 2 d Sqrt[z]))^(h + k) ((I (m - 2 u) (b + 2 d Sqrt[z])^2)/d)^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (b (m - 2 u) (b + 2 d Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (m - 2 u) (b + 2 d Sqrt[z])^2)/ (4 d)] + 2 I d Sqrt[(I (m - 2 u) (b + 2 d Sqrt[z])^2)/d] Gamma[(1/2) (2 + h + k), (I (m - 2 u) (b + 2 d Sqrt[z])^2)/ (4 d)]), {k, 0, n}, {h, 0, k}]))/ E^((I (b^2 + 4 d e) (m - 2 u))/(4 d)), {u, 0, Floor[(1/2) (-1 + m)]}])/(I^m 4^n) + Sum[(-1)^u Binomial[m, u] Sum[(-E^(-((I (b^2 (m - 2 u)^2 + 4 d e (m - 2 u)^2 + 2 b c (m - 2 u) (2 s - v) + c^2 (-2 s + v)^2))/(4 d (m - 2 u))))) (d^2 (m - 2 u)^2)^(-1 - 2 n) Binomial[v, s] (E^(2 I e (m - 2 u)) ((-I) d (m - 2 u))^(2 n) 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 d (m - 2 u) Sqrt[z]))^(h + k) (-((1/(d (m - 2 u))) (I (b (m - 2 u) + c (2 s - v) + 2 d (m - 2 u) 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 d (m - 2 u) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I (b (m - 2 u) + c (2 s - v) + 2 d (m - 2 u) Sqrt[z])^2)/(4 d (m - 2 u)))] + 2 I d (m - 2 u) Sqrt[-((1/(d (m - 2 u))) (I (b (m - 2 u) + c (2 s - v) + 2 d (m - 2 u) Sqrt[z])^2))] Gamma[ (1/2) (2 + h + k), -((I (b (m - 2 u) + c (2 s - v) + 2 d (m - 2 u) Sqrt[z])^2)/(4 d (m - 2 u)))]), {k, 0, n}, {h, 0, k}] + (-1)^m E^((I (b m + 2 c s - 2 b u - c v)^2)/ (2 d (m - 2 u))) (I d (m - 2 u))^(2 n) 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 d (m - 2 u) Sqrt[z]))^(h + k) ((I (b (m - 2 u) + c (2 s - v) + 2 d (m - 2 u) Sqrt[z])^2)/(d (m - 2 u)))^((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 d (m - 2 u) Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (b (m - 2 u) + c (2 s - v) + 2 d (m - 2 u) Sqrt[z])^2)/(4 d (m - 2 u))] - 2 I d (m - 2 u) Sqrt[(1/(d (m - 2 u))) (I (b (m - 2 u) + c (2 s - v) + 2 d (m - 2 u) Sqrt[z])^2)] Gamma[ (1/2) (2 + h + k), (I (b (m - 2 u) + c (2 s - v) + 2 d (m - 2 u) Sqrt[z])^2)/(4 d (m - 2 u))]), {k, 0, n}, {h, 0, k}] + E^(I (2 e (m - 2 u) + (b c (2 s - v))/d)) ((-I) d (m - 2 u))^(2 n) 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 d (m - 2 u) Sqrt[z]))^(h + k) (-((1/(d (m - 2 u))) (I (b (m - 2 u) + c (-2 s + v) + 2 d (m - 2 u) 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 d (m - 2 u) Sqrt[z]) Gamma[ (1/2) (1 + h + k), -((I (b (m - 2 u) + c (-2 s + v) + 2 d (m - 2 u) Sqrt[z])^2)/(4 d (m - 2 u)))] + 2 I d (m - 2 u) Sqrt[-((1/(d (m - 2 u))) (I (b (m - 2 u) + c (-2 s + v) + 2 d (m - 2 u) Sqrt[z])^2))] Gamma[ (1/2) (2 + h + k), -((I (b (m - 2 u) + c (-2 s + v) + 2 d (m - 2 u) Sqrt[z])^2)/(4 d (m - 2 u)))]), {k, 0, n}, {h, 0, k}] + (-1)^m E^((I (b^2 (m - 2 u)^2 + c^2 (-2 s + v)^2))/ (2 d (m - 2 u))) (I d (m - 2 u))^(2 n) 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 d (m - 2 u) Sqrt[z]))^(h + k) ((1/(d (m - 2 u))) (I (b (m - 2 u) + c (-2 s + v) + 2 d (m - 2 u) 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 d (m - 2 u) Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (b (m - 2 u) + c (-2 s + v) + 2 d (m - 2 u) Sqrt[z])^2)/ (4 d (m - 2 u))] - 2 I d (m - 2 u) Sqrt[(1/(d (m - 2 u))) (I (b (m - 2 u) + c (-2 s + v) + 2 d (m - 2 u) Sqrt[z])^2)] Gamma[(1/2) (2 + h + k), (I (b (m - 2 u) + c (-2 s + v) + 2 d (m - 2 u) Sqrt[z])^2)/(4 d (m - 2 u))]), {k, 0, n}, {h, 0, k}]), {s, 0, Floor[(1/2) (-1 + v)]}], {u, 0, Floor[(1/2) (-1 + m)]}]/(I^m 4^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 ( z b + d z + e ) cos v ( c z ) z 2 - m - v - 1 ( 2 z n + 1 ( m mod 2 \$CellContext`m 2 - 1 ) ( v mod 2 \$CellContext`v 2 - 1 ) 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]]]]] + 4 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]]]]] ( m mod 2 \$CellContext`m 2 - 1 ) 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]]]]] ( Γ ( 2 ( n + 1 ) , - c ( 2 s - v ) z ) ( - c ( 2 s - v ) z ) - 2 ( n + 1 ) + ( c ( 2 s - v ) z ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , c ( 2 s - v ) z ) ) - - m 4 - n ( 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]]]]] ( v mod 2 \$CellContext`v 2 - 1 ) u = 0 m - 1 2 ( - 1 ) u - 1 - ( b 2 + 4 d e ) ( m - 2 u ) 4 d ( d 2 ( m - 2 u ) 2 ) - 2 n - 1 ( m u ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["u", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 e ( m - 2 u ) ( k = 0 n h = 0 k - ( - 1 ) k - h 4 k ( m - 2 u ) ( b ( m - 2 u ) ) - h - k + 2 n ( ( m - 2 u ) ( b + 2 d z ) ) h + k ( - ( m - 2 u ) ( b + 2 d z ) 2 d ) 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 ) ( b + 2 d z ) Γ ( 1 2 ( h + k + 1 ) , - ( m - 2 u ) ( b + 2 d z ) 2 4 d ) - 2 d - ( m - 2 u ) ( b + 2 d z ) 2 d Γ ( 1 2 ( h + k + 2 ) , - ( m - 2 u ) ( b + 2 d z ) 2 4 d ) ) ) ( - d ( m - 2 u ) ) 2 n + ( - 1 ) m b 2 ( m - 2 u ) 2 d ( d ( m - 2 u ) ) 2 n k = 0 n h = 0 k - ( - 1 ) k - h 4 k ( m - 2 u ) ( - b ( m - 2 u ) ) - h - k + 2 n ( - ( m - 2 u ) ( b + 2 d z ) ) h + k ( ( m - 2 u ) ( b + 2 d z ) 2 d ) 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 ) ( b + 2 d z ) Γ ( 1 2 ( h + k + 1 ) , ( m - 2 u ) ( b + 2 d z ) 2 4 d ) + 2 ( m - 2 u ) ( b + 2 d z ) 2 d d Γ ( 1 2 ( h + k + 2 ) , ( m - 2 u ) ( b + 2 d z ) 2 4 d ) ) ) + - m 4 - n 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 - - ( b 2 ( m - 2 u ) 2 + 4 d e ( m - 2 u ) 2 + 2 b c ( 2 s - v ) ( m - 2 u ) + c 2 ( v - 2 s ) 2 ) 4 d ( m - 2 u ) ( d 2 ( m - 2 u ) 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]]]]] ( 2 e ( m - 2 u ) ( 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 ) + 2 d z ( m - 2 u ) + c ( 2 s - v ) ) ) h + k ( - ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( 2 s - v ) ) 2 d ( m - 2 u ) ) 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]]]]] ( 2 d ( m - 2 u ) - ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( 2 s - v ) ) 2 d ( m - 2 u ) Γ ( 1 2 ( h + k + 2 ) , - ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( 2 s - v ) ) 2 4 d ( m - 2 u ) ) - ( b m + 2 c s - 2 b u - c v ) ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( 2 s - v ) ) Γ ( 1 2 ( h + k + 1 ) , - ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( 2 s - v ) ) 2 4 d ( m - 2 u ) ) ) ) ( - d ( m - 2 u ) ) 2 n + ( 2 e ( m - 2 u ) + b c ( 2 s - v ) d ) ( 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 ) + 2 d z ( m - 2 u ) + c ( v - 2 s ) ) ) h + k ( - ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( v - 2 s ) ) 2 d ( m - 2 u ) ) 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]]]]] ( 2 d ( m - 2 u ) - ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( v - 2 s ) ) 2 d ( m - 2 u ) Γ ( 1 2 ( h + k + 2 ) , - ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( v - 2 s ) ) 2 4 d ( m - 2 u ) ) - ( b m - 2 c s - 2 b u + c v ) ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( v - 2 s ) ) Γ ( 1 2 ( h + k + 1 ) , - ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( v - 2 s ) ) 2 4 d ( m - 2 u ) ) ) ) ( - d ( m - 2 u ) ) 2 n + ( - 1 ) m ( b m + 2 c s - 2 b u - c v ) 2 2 d ( m - 2 u ) ( d ( m - 2 u ) ) 2 n 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 ) + 2 d z ( m - 2 u ) + c ( 2 s - v ) ) ) h + k ( ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( 2 s - v ) ) 2 d ( m - 2 u ) ) 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 ) + 2 d z ( m - 2 u ) + c ( 2 s - v ) ) Γ ( 1 2 ( h + k + 1 ) , ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( 2 s - v ) ) 2 4 d ( m - 2 u ) ) - 2 d ( m - 2 u ) ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( 2 s - v ) ) 2 d ( m - 2 u ) Γ ( 1 2 ( h + k + 2 ) , ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( 2 s - v ) ) 2 4 d ( m - 2 u ) ) ) + ( - 1 ) m ( b 2 ( m - 2 u ) 2 + c 2 ( v - 2 s ) 2 ) 2 d ( m - 2 u ) ( d ( m - 2 u ) ) 2 n 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 ) + 2 d z ( m - 2 u ) + c ( v - 2 s ) ) ) h + k ( ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( v - 2 s ) ) 2 d ( m - 2 u ) ) 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 ) + 2 d z ( m - 2 u ) + c ( v - 2 s ) ) Γ ( 1 2 ( h + k + 1 ) , ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( v - 2 s ) ) 2 4 d ( m - 2 u ) ) - 2 d ( m - 2 u ) ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( v - 2 s ) ) 2 d ( m - 2 u ) Γ ( 1 2 ( h + k + 2 ) , ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( v - 2 s ) ) 2 4 d ( m - 2 u ) ) ) ) ) /; n m + v + Condition z z n z 1 2 b d z e m c z 1 2 v 2 -1 m -1 v -1 2 z n 1 \$CellContext`m 2 -1 \$CellContext`v 2 -1 n 1 -1 Binomial m m 2 -1 Binomial v v 2 -1 4 z n 1 Binomial m m 2 -1 \$CellContext`m 2 -1 s 0 v -1 2 -1 Binomial v s Gamma 2 n 1 -1 c 2 s -1 v z 1 2 -1 c 2 s -1 v z 1 2 -2 n 1 c 2 s -1 v z 1 2 -2 n 1 Gamma 2 n 1 c 2 s -1 v z 1 2 -1 -1 m 4 -1 n Binomial v v 2 -1 \$CellContext`v 2 -1 u 0 m -1 2 -1 -1 u -1 -1 b 2 4 d e m -1 2 u 4 d -1 d 2 m -1 2 u 2 -2 n -1 Binomial m u 2 e m -1 2 u h 0 k k 0 n -1 -1 k -1 h 4 k m -1 2 u b m -1 2 u -1 h -1 k 2 n m -1 2 u b 2 d z 1 2 h k -1 m -1 2 u b 2 d z 1 2 2 d -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b m -1 2 u b 2 d z 1 2 Gamma 1 2 h k 1 -1 m -1 2 u b 2 d z 1 2 2 4 d -1 -1 2 d -1 m -1 2 u b 2 d z 1 2 2 d -1 1 2 Gamma 1 2 h k 2 -1 m -1 2 u b 2 d z 1 2 2 4 d -1 -1 d m -1 2 u 2 n -1 m b 2 m -1 2 u 2 d -1 d m -1 2 u 2 n h 0 k k 0 n -1 -1 k -1 h 4 k m -1 2 u -1 b m -1 2 u -1 h -1 k 2 n -1 m -1 2 u b 2 d z 1 2 h k m -1 2 u b 2 d z 1 2 2 d -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b m -1 2 u b 2 d z 1 2 Gamma 1 2 h k 1 m -1 2 u b 2 d z 1 2 2 4 d -1 2 m -1 2 u b 2 d z 1 2 2 d -1 1 2 d Gamma 1 2 h k 2 m -1 2 u b 2 d z 1 2 2 4 d -1 -1 m 4 -1 n u 0 m -1 2 -1 -1 u Binomial m u s 0 v -1 2 -1 -1 -1 b 2 m -1 2 u 2 4 d e m -1 2 u 2 2 b c 2 s -1 v m -1 2 u c 2 v -1 2 s 2 4 d m -1 2 u -1 d 2 m -1 2 u 2 -2 n -1 Binomial v s 2 e m -1 2 u 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 2 d z 1 2 m -1 2 u c 2 s -1 v h k -1 b m -1 2 u 2 d z 1 2 m -1 2 u c 2 s -1 v 2 d m -1 2 u -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k 2 d m -1 2 u -1 b m -1 2 u 2 d z 1 2 m -1 2 u c 2 s -1 v 2 d m -1 2 u -1 1 2 Gamma 1 2 h k 2 -1 b m -1 2 u 2 d z 1 2 m -1 2 u c 2 s -1 v 2 4 d m -1 2 u -1 -1 b m 2 c s -1 2 b u -1 c v b m -1 2 u 2 d z 1 2 m -1 2 u c 2 s -1 v Gamma 1 2 h k 1 -1 b m -1 2 u 2 d z 1 2 m -1 2 u c 2 s -1 v 2 4 d m -1 2 u -1 -1 d m -1 2 u 2 n 2 e m -1 2 u b c 2 s -1 v d -1 h 0 k k 0 n -1 k -1 h 4 k b m -1 2 c s -1 2 b u c v -1 h -1 k 2 n b m -1 2 u 2 d z 1 2 m -1 2 u c v -1 2 s h k -1 b m -1 2 u 2 d z 1 2 m -1 2 u c v -1 2 s 2 d m -1 2 u -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k 2 d m -1 2 u -1 b m -1 2 u 2 d z 1 2 m -1 2 u c v -1 2 s 2 d m -1 2 u -1 1 2 Gamma 1 2 h k 2 -1 b m -1 2 u 2 d z 1 2 m -1 2 u c v -1 2 s 2 4 d m -1 2 u -1 -1 b m -1 2 c s -1 2 b u c v b m -1 2 u 2 d z 1 2 m -1 2 u c v -1 2 s Gamma 1 2 h k 1 -1 b m -1 2 u 2 d z 1 2 m -1 2 u c v -1 2 s 2 4 d m -1 2 u -1 -1 d m -1 2 u 2 n -1 m b m 2 c s -1 2 b u -1 c v 2 2 d m -1 2 u -1 d m -1 2 u 2 n 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 2 d z 1 2 m -1 2 u c 2 s -1 v h k b m -1 2 u 2 d z 1 2 m -1 2 u c 2 s -1 v 2 d m -1 2 u -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k -1 b m 2 c s -1 2 b u -1 c v b m -1 2 u 2 d z 1 2 m -1 2 u c 2 s -1 v Gamma 1 2 h k 1 b m -1 2 u 2 d z 1 2 m -1 2 u c 2 s -1 v 2 4 d m -1 2 u -1 -1 2 d m -1 2 u b m -1 2 u 2 d z 1 2 m -1 2 u c 2 s -1 v 2 d m -1 2 u -1 1 2 Gamma 1 2 h k 2 b m -1 2 u 2 d z 1 2 m -1 2 u c 2 s -1 v 2 4 d m -1 2 u -1 -1 m b 2 m -1 2 u 2 c 2 v -1 2 s 2 2 d m -1 2 u -1 d m -1 2 u 2 n h 0 k k 0 n -1 k -1 h 4 k -1 b m -1 2 c s -1 2 b u c v -1 h -1 k 2 n -1 b m -1 2 u 2 d z 1 2 m -1 2 u c v -1 2 s h k b m -1 2 u 2 d z 1 2 m -1 2 u c v -1 2 s 2 d m -1 2 u -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k -1 b m -1 2 c s -1 2 b u c v b m -1 2 u 2 d z 1 2 m -1 2 u c v -1 2 s Gamma 1 2 h k 1 b m -1 2 u 2 d z 1 2 m -1 2 u c v -1 2 s 2 4 d m -1 2 u -1 -1 2 d m -1 2 u b m -1 2 u 2 d z 1 2 m -1 2 u c v -1 2 s 2 d m -1 2 u -1 1 2 Gamma 1 2 h k 2 b m -1 2 u 2 d z 1 2 m -1 2 u c v -1 2 s 2 4 d m -1 2 u -1 n m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18