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 Cos

 http://functions.wolfram.com/01.07.21.2560.01

 Input Form

 Integrate[z^n Sin[b Sqrt[z] + d z + e]^m Cos[c Sqrt[z] + f 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])) + (Binomial[v, v/2] (1 - Mod[v, 2]) Sum[((-1)^(n + u) Binomial[m, u] (E^(2 I e (m - 2 u)) 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)) 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)) (m - 2 u)^(2 (1 + n))), {u, 0, Floor[(1/2) (-1 + m)]}])/ (I^m 4^n d^(2 (1 + n))) + (1/c) ((I (-(1/4))^n Binomial[m, m/2] (-1 + Mod[m, 2]) Sum[((-2 s + v)^(-2 - 2 n) Binomial[v, s] (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 (2 s - v))/(2 f)) 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 (2 s - v))/(4 f)), {s, 0, Floor[(1/2) (-1 + v)]}])/ f^(2 (1 + n))) - Sum[(-1)^u Binomial[m, u] Sum[(Binomial[v, s] ((E^(I (2 e (m - 2 u) - (b m + 2 c s - 2 b u - c v)^2/(4 d m + 8 f s - 8 d u - 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 (d m + 2 f s - 2 d u - f v) Sqrt[z]))^(h + k) (-((I (b (m - 2 u) + c (2 s - v) + 2 (d m + 2 f s - 2 d u - f v) Sqrt[z])^2)/(d m + 2 f s - 2 d u - f v)))^ ((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 f s - 2 d u - f v) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I (b (m - 2 u) + c (2 s - v) + 2 (d m + 2 f s - 2 d u - f v) Sqrt[z])^2)/(4 d m + 8 f s - 8 d u - 4 f v))] + 2 I (d m + 2 f s - 2 d u - f v) Sqrt[-((I (b (m - 2 u) + c (2 s - v) + 2 (d m + 2 f s - 2 d u - f v) Sqrt[z])^2)/(d m + 2 f s - 2 d u - f v))] Gamma[(1/2) (2 + h + k), -((I (b (m - 2 u) + c (2 s - v) + 2 (d m + 2 f s - 2 d u - f v) Sqrt[z])^2)/(4 d m + 8 f s - 8 d u - 4 f v))]), {k, 0, n}, {h, 0, k}])/ (I (d m + 2 f s - 2 d u - f v))^(2 n)/(d m + 2 f s - 2 d u - f v)^2 + ((-1)^m E^((I (b m + 2 c s - 2 b u - c v)^2)/ (4 d m + 8 f s - 8 d u - 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 (d m + 2 f s - 2 d u - f v) Sqrt[z]))^(h + k) ((I (b (m - 2 u) + c (2 s - v) + 2 (d m + 2 f s - 2 d u - f v) Sqrt[z])^2)/(d m + 2 f s - 2 d u - f v))^((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 f s - 2 d u - f v) Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (b (m - 2 u) + c (2 s - v) + 2 (d m + 2 f s - 2 d u - f v) Sqrt[z])^2)/(4 d m + 8 f s - 8 d u - 4 f v)] - 2 I (d m + 2 f s - 2 d u - f v) Sqrt[(I (b (m - 2 u) + c (2 s - v) + 2 (d m + 2 f s - 2 d u - f v) Sqrt[z])^2)/(d m + 2 f s - 2 d u - f v)] Gamma[(1/2) (2 + h + k), (I (b (m - 2 u) + c (2 s - v) + 2 (d m + 2 f s - 2 d u - f v) Sqrt[z])^ 2)/(4 d m + 8 f s - 8 d u - 4 f v)]), {k, 0, n}, {h, 0, k}])/((-I) (d m + 2 f s - 2 d u - f v))^(2 n)/ (d m + 2 f s - 2 d u - f v)^2 + (((d (m - 2 u) + f (-2 s + v))^2)^ (-1 - 2 n) (E^(2 I e (m - 2 u)) ((-I) (d (m - 2 u) + f (-2 s + v)))^(2 n) 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 (d (m - 2 u) + f (-2 s + v)) Sqrt[z]))^(h + k) (-((I (b (m - 2 u) + c (-2 s + v) + 2 (d (m - 2 u) + f (-2 s + v)) Sqrt[z])^2)/(d (m - 2 u) + 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 (d (m - 2 u) + f (-2 s + v)) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I (b (m - 2 u) + c (-2 s + v) + 2 (d (m - 2 u) + f (-2 s + v)) Sqrt[z])^2)/(4 (d (m - 2 u) + f (-2 s + v))))] + 2 I (d (m - 2 u) + f (-2 s + v)) Sqrt[-((I (b (m - 2 u) + c (-2 s + v) + 2 (d (m - 2 u) + f (-2 s + v)) Sqrt[z])^2)/(d (m - 2 u) + f (-2 s + v)))] Gamma[(1/2) (2 + h + k), -((I (b (m - 2 u) + c (-2 s + v) + 2 (d (m - 2 u) + f (-2 s + v)) Sqrt[z])^2)/(4 (d (m - 2 u) + f (-2 s + v))))]), {k, 0, n}, {h, 0, k}] + (-1)^m E^((I (b (m - 2 u) + c (-2 s + v))^2)/(2 (d (m - 2 u) + f (-2 s + v)))) (I (d (m - 2 u) + f (-2 s + v)))^(2 n) 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 (d (m - 2 u) + f (-2 s + v)) Sqrt[z]))^(h + k) ((I (b (m - 2 u) + c (-2 s + v) + 2 (d (m - 2 u) + f (-2 s + v)) Sqrt[z])^2)/(d (m - 2 u) + 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 (d (m - 2 u) + f (-2 s + v)) Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (b (m - 2 u) + c (-2 s + v) + 2 (d (m - 2 u) + f (-2 s + v)) Sqrt[z])^2)/(4 (d (m - 2 u) + f (-2 s + v)))] - 2 I (d (m - 2 u) + f (-2 s + v)) Sqrt[(I (b (m - 2 u) + c (-2 s + v) + 2 (d (m - 2 u) + f (-2 s + v)) Sqrt[z])^2)/(d (m - 2 u) + f (-2 s + v))] Gamma[(1/2) (2 + h + k), (I (b (m - 2 u) + c (-2 s + v) + 2 (d (m - 2 u) + f (-2 s + v)) Sqrt[z])^2)/(4 (d (m - 2 u) + f (-2 s + v)))]), {k, 0, n}, {h, 0, k}]))/ E^((I (b (m - 2 u) + c (-2 s + v))^2)/(4 (d (m - 2 u) + f (-2 s + v))))))/E^(I (e m - 2 e u)), {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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RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

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