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/01.20.21.4596.01

 Input Form

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

 Rule Form

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"2"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], RowBox[List["n", "+", "1"]]], "+", RowBox[List[SuperscriptBox["\[ImaginaryI]", "m"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m", "-", "v", "-", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["c", "2"], " ", 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 Date Added to functions.wolfram.com (modification date)

 2002-12-18