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/01.20.21.3721.01

 Input Form

 Integrate[z^n Cos[b Sqrt[z] + e]^m Cosh[c Sqrt[z] + f z + g]^v, z] == (1/(1 + n)) (2^(-m - v) z^(1 + n) Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2])) + ((-1)^n 2^(1 - m - v) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(Binomial[m, k] (E^(I e (-2 k + m)) Gamma[2 (1 + n), (-I) b (-2 k + m) Sqrt[z]] + Gamma[2 (1 + n), I b (-2 k + m) Sqrt[z]]/E^(I e (-2 k + m))))/(-2 k + m)^(2 (1 + n)), {k, 0, Floor[(1/2) (-1 + m)]}])/b^(2 (1 + n)) + 2^(-1 - m - 2 n - v) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[(Binomial[v, k] (E^((c^2 (-2 k + v))/(4 f) - g (-2 k + v)) Sum[(-1)^(-h + j) 4^j ((-c) (-2 k + v))^(-h - j + 2 n) ((-c) (-2 k + v) - 2 f (-2 k + v) Sqrt[z])^(h + j) (((-c) (-2 k + v) - 2 f (-2 k + v) Sqrt[z])^2/(f (-2 k + v)))^ ((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((-c) (-2 k + v) ((-c) (-2 k + v) - 2 f (-2 k + v) Sqrt[z]) Gamma[(1/2) (1 + h + j), ((-c) (-2 k + v) - 2 f (-2 k + v) Sqrt[z])^2/(4 f (-2 k + v))] - 2 f (-2 k + v) Sqrt[((-c) (-2 k + v) - 2 f (-2 k + v) Sqrt[z])^2/(f (-2 k + v))] Gamma[(1/2) (2 + h + j), ((-c) (-2 k + v) - 2 f (-2 k + v) Sqrt[z])^2/(4 f (-2 k + v))]), {j, 0, n}, {h, 0, j}] + E^(-((c^2 (-2 k + v))/(4 f)) + g (-2 k + v)) Sum[(-1)^(-h + j) 4^j (c (-2 k + v))^(-h - j + 2 n) (c (-2 k + v) + 2 f (-2 k + v) Sqrt[z])^(h + j) (-((c (-2 k + v) + 2 f (-2 k + v) Sqrt[z])^2/(f (-2 k + v))))^ ((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] (c (-2 k + v) (c (-2 k + v) + 2 f (-2 k + v) Sqrt[z]) Gamma[(1/2) (1 + h + j), -((c (-2 k + v) + 2 f (-2 k + v) Sqrt[z])^2/(4 f (-2 k + v)))] + 2 f (-2 k + v) Sqrt[-((c (-2 k + v) + 2 f (-2 k + v) Sqrt[z])^2/ (f (-2 k + v)))] Gamma[(1/2) (2 + h + j), -((c (-2 k + v) + 2 f (-2 k + v) Sqrt[z])^2/(4 f (-2 k + v)))]), {j, 0, n}, {h, 0, j}]))/(f (-2 k + v))^(2 (1 + n)), {k, 0, Floor[(1/2) (-1 + v)]}] + 2^(-1 - m - 2 n - v) Sum[Binomial[m, k] Sum[(Binomial[v, s] (E^((-I) e (-2 k + m) - g (-2 s + v) + ((-I) b (-2 k + m) - c (-2 s + v))^2/(4 f (-2 s + v))) Sum[(-1)^(-h + j) 4^j ((-I) b (-2 k + m) - c (-2 s + v))^(-h - j + 2 n) ((-I) b (-2 k + m) - c (-2 s + v) - 2 f (-2 s + v) Sqrt[z])^ (h + j) ((1/(f (-2 s + v))) ((-I) b (-2 k + m) - c (-2 s + v) - 2 f (-2 s + v) Sqrt[z])^2)^((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] (((-I) b (-2 k + m) - c (-2 s + v)) ((-I) b (-2 k + m) - c (-2 s + v) - 2 f (-2 s + v) Sqrt[z]) Gamma[(1/2) (1 + h + j), ((-I) b (-2 k + m) - c (-2 s + v) - 2 f (-2 s + v) Sqrt[z])^2/ (4 f (-2 s + v))] - 2 f (-2 s + v) Sqrt[(1/(f (-2 s + v))) ((-I) b (-2 k + m) - c (-2 s + v) - 2 f (-2 s + v) Sqrt[z])^ 2] Gamma[(1/2) (2 + h + j), ((-I) b (-2 k + m) - c (-2 s + v) - 2 f (-2 s + v) Sqrt[z])^2/(4 f (-2 s + v))]), {j, 0, n}, {h, 0, j}] + E^(I e (-2 k + m) - g (-2 s + v) + (I b (-2 k + m) - c (-2 s + v))^ 2/(4 f (-2 s + v))) Sum[(-1)^(-h + j) 4^j (I b (-2 k + m) - c (-2 s + v))^(-h - j + 2 n) (I b (-2 k + m) - c (-2 s + v) - 2 f (-2 s + v) Sqrt[z])^(h + j) ((1/(f (-2 s + v))) (I b (-2 k + m) - c (-2 s + v) - 2 f (-2 s + v) Sqrt[z])^2)^((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((I b (-2 k + m) - c (-2 s + v)) ( I b (-2 k + m) - c (-2 s + v) - 2 f (-2 s + v) Sqrt[z]) Gamma[ (1/2) (1 + h + j), (I b (-2 k + m) - c (-2 s + v) - 2 f (-2 s + v) Sqrt[z])^2/(4 f (-2 s + v))] - 2 f (-2 s + v) Sqrt[(1/(f (-2 s + v))) (I b (-2 k + m) - c (-2 s + v) - 2 f (-2 s + v) Sqrt[z])^2] Gamma[ (1/2) (2 + h + j), (I b (-2 k + m) - c (-2 s + v) - 2 f (-2 s + v) Sqrt[z])^2/(4 f (-2 s + v))]), {j, 0, n}, {h, 0, j}] + E^((-I) e (-2 k + m) + g (-2 s + v) - ((-I) b (-2 k + m) + c (-2 s + v))^2/(4 f (-2 s + v))) Sum[(-1)^(-h + j) 4^j ((-I) b (-2 k + m) + c (-2 s + v))^ (-h - j + 2 n) ((-I) b (-2 k + m) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])^(h + j) (-((1/(f (-2 s + v))) ((-I) b (-2 k + m) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])^ 2))^((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] (((-I) b (-2 k + m) + c (-2 s + v)) ((-I) b (-2 k + m) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z]) Gamma[ (1/2) (1 + h + j), -(((-I) b (-2 k + m) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])^2/(4 f (-2 s + v)))] + 2 f (-2 s + v) Sqrt[-(((-I) b (-2 k + m) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])^2/(f (-2 s + v)))] Gamma[ (1/2) (2 + h + j), -(((-I) b (-2 k + m) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])^2/(4 f (-2 s + v)))]), {j, 0, n}, {h, 0, j}] + E^(I e (-2 k + m) + g (-2 s + v) - (I b (-2 k + m) + c (-2 s + v))^2/(4 f (-2 s + v))) Sum[(-1)^(-h + j) 4^j (I b (-2 k + m) + c (-2 s + v))^ (-h - j + 2 n) (I b (-2 k + m) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])^(h + j) (-((1/(f (-2 s + v))) (I b (-2 k + m) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])^2))^((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((I b (-2 k + m) + c (-2 s + v)) (I b (-2 k + m) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z]) Gamma[(1/2) (1 + h + j), -((I b (-2 k + m) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])^2/(4 f (-2 s + v)))] + 2 f (-2 s + v) Sqrt[-((I b (-2 k + m) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])^2/(f (-2 s + v)))] Gamma[(1/2) (2 + h + j), -((I b (-2 k + m) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])^2/(4 f (-2 s + v)))]), {j, 0, n}, {h, 0, j}]))/(f (-2 s + v))^(2 (1 + n)), {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

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 MathML Form

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