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 Cosh

 http://functions.wolfram.com/01.20.21.3663.01

 Input Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "n_"], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List[RowBox[List["b_", " ", SqrtBox["z_"]]], "+", RowBox[List["d_", " ", "z_"]]]], "]"]], "m_"], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]], "v_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "m"]], "-", "v"]]], " ", SuperscriptBox["z", RowBox[List["1", "+", "n"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], RowBox[List["1", "+", "n"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "m"]], "-", "v"]]], " ", 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["k", "=", "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", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List[RowBox[List["-", "c"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", "z"]]]], "]"]]]]]], ")"]]]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m", "-", RowBox[List["2", " ", "n"]], "-", "v"]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", 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 Date Added to functions.wolfram.com (modification date)

 2002-12-18