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/01.20.21.3923.01

 Input Form

 Integrate[z^n E^(p z) Sin[b Sqrt[z]] Cosh[c Sqrt[z]]^v, z] == I 2^(-2 - 2 n - v) p^(-2 - 2 n) Binomial[v, v/2] (1 - Mod[v, 2]) (E^(b^2/(4 p)) Sum[(-1)^(-h + j) 4^j ((-I) b)^(-h - j + 2 n) ((-I) b + 2 p Sqrt[z])^(h + j) (-(((-I) b + 2 p Sqrt[z])^2/p))^ ((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((-I) b ((-I) b + 2 p Sqrt[z]) Gamma[(1/2) (1 + h + j), -(((-I) b + 2 p Sqrt[z])^2/(4 p))] + 2 p Sqrt[-(((-I) b + 2 p Sqrt[z])^2/p)] Gamma[(1/2) (2 + h + j), -(((-I) b + 2 p Sqrt[z])^2/(4 p))]), {j, 0, n}, {h, 0, j}] - E^(b^2/(4 p)) Sum[(-1)^(-h + j) 4^j (I b)^(-h - j + 2 n) (I b + 2 p Sqrt[z])^(h + j) (-((I b + 2 p Sqrt[z])^2/p))^ ((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] (I b (I b + 2 p Sqrt[z]) Gamma[(1/2) (1 + h + j), -((I b + 2 p Sqrt[z])^2/(4 p))] + 2 p Sqrt[-((I b + 2 p Sqrt[z])^2/p)] Gamma[(1/2) (2 + h + j), -((I b + 2 p Sqrt[z])^2/(4 p))]), {j, 0, n}, {h, 0, j}]) - I 2^(-2 - 2 n - v) p^(-2 - 2 n) Sum[Binomial[v, s] ((-E^(-(((-I) b - c (2 s - v))^2/(4 p)))) Sum[(-1)^(-h + j) 4^j ((-I) b - c (2 s - v))^(-h - j + 2 n) ((-I) b - c (2 s - v) + 2 p Sqrt[z])^(h + j) (-(((-I) b - c (2 s - v) + 2 p Sqrt[z])^2/p))^((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] (((-I) b - c (2 s - v)) ((-I) b - c (2 s - v) + 2 p Sqrt[z]) Gamma[(1/2) (1 + h + j), -(((-I) b - c (2 s - v) + 2 p Sqrt[z])^2/(4 p))] + 2 p Sqrt[-(((-I) b - c (2 s - v) + 2 p Sqrt[z])^2/p)] Gamma[(1/2) (2 + h + j), -(((-I) b - c (2 s - v) + 2 p Sqrt[z])^ 2/(4 p))]), {j, 0, n}, {h, 0, j}] + Sum[(-1)^(-h + j) 4^j (I b - c (2 s - v))^(-h - j + 2 n) (I b - c (2 s - v) + 2 p Sqrt[z])^(h + j) (-((I b - c (2 s - v) + 2 p Sqrt[z])^2/p))^((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((I b - c (2 s - v)) (I b - c (2 s - v) + 2 p Sqrt[z]) Gamma[(1/2) (1 + h + j), -((I b - c (2 s - v) + 2 p Sqrt[z])^2/(4 p))] + 2 p Sqrt[-((I b - c (2 s - v) + 2 p Sqrt[z])^2/p)] Gamma[(1/2) (2 + h + j), -((I b - c (2 s - v) + 2 p Sqrt[z])^2/(4 p))]), {j, 0, n}, {h, 0, j}]/E^((I b - c (2 s - v))^2/ (4 p)) - Sum[(-1)^(-h + j) 4^j ((-I) b - c (-2 s + v))^ (-h - j + 2 n) ((-I) b - c (-2 s + v) + 2 p Sqrt[z])^(h + j) (-(((-I) b - c (-2 s + v) + 2 p Sqrt[z])^2/p))^((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] (((-I) b - c (-2 s + v)) ((-I) b - c (-2 s + v) + 2 p Sqrt[z]) Gamma[(1/2) (1 + h + j), -(((-I) b - c (-2 s + v) + 2 p Sqrt[z])^2/(4 p))] + 2 p Sqrt[-(((-I) b - c (-2 s + v) + 2 p Sqrt[z])^2/p)] Gamma[(1/2) (2 + h + j), -(((-I) b - c (-2 s + v) + 2 p Sqrt[z])^ 2/(4 p))]), {j, 0, n}, {h, 0, j}]/ E^(((-I) b - c (-2 s + v))^2/(4 p)) + Sum[(-1)^(-h + j) 4^j (I b - c (-2 s + v))^(-h - j + 2 n) (I b - c (-2 s + v) + 2 p Sqrt[z])^(h + j) (-((I b - c (-2 s + v) + 2 p Sqrt[z])^2/p))^((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((I b - c (-2 s + v)) (I b - c (-2 s + v) + 2 p Sqrt[z]) Gamma[(1/2) (1 + h + j), -((I b - c (-2 s + v) + 2 p Sqrt[z])^2/(4 p))] + 2 p Sqrt[-((I b - c (-2 s + v) + 2 p Sqrt[z])^2/p)] Gamma[(1/2) (2 + h + j), -((I b - c (-2 s + v) + 2 p Sqrt[z])^2/( 4 p))]), {j, 0, n}, {h, 0, j}]/ E^((I b - c (-2 s + v))^2/(4 p))), {s, 0, Floor[(1/2) (-1 + v)]}] /; Element[n, Integers] && n >= 0 && Element[v, Integers] && v > 0

 Standard Form

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 MathML Form

 z n p z sin ( b z ) cosh v ( c z ) z 2 - 2 n - v - 2 p - 2 n - 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 - v mod 2 \$CellContext`v 2 ) ( b 2 4 p j = 0 n h = 0 j ( - 1 ) j - h 4 j ( - b ) - h - j + 2 n ( - b + 2 p z ) h + j ( - ( - b + 2 p z ) 2 p ) 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 p - ( - b + 2 p z ) 2 p Γ ( 1 2 ( h + j + 2 ) , - ( - b + 2 p z ) 2 4 p ) - b ( - b + 2 p z ) Γ ( 1 2 ( h + j + 1 ) , - ( - b + 2 p z ) 2 4 p ) ) - b 2 4 p j = 0 n h = 0 j ( - 1 ) j - h 4 j ( b ) - h - j + 2 n ( b + 2 p z ) h + j ( - ( b + 2 p z ) 2 p ) 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 ( b + 2 p z ) Γ ( 1 2 ( h + j + 1 ) , - ( b + 2 p z ) 2 4 p ) + 2 - ( b + 2 p z ) 2 p p Γ ( 1 2 ( h + j + 2 ) , - ( b + 2 p z ) 2 4 p ) ) ) - 2 - 2 n - v - 2 p - 2 n - 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]]]]] ( - - ( - b - c ( 2 s - v ) ) 2 4 p j = 0 n h = 0 j ( - 1 ) j - h 4 j ( - b - c ( 2 s - v ) ) - h - j + 2 n ( - b - c ( 2 s - v ) + 2 p z ) h + j ( - ( - b - c ( 2 s - v ) + 2 p z ) 2 p ) 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 - c ( 2 s - v ) ) ( - b - c ( 2 s - v ) + 2 p z ) Γ ( 1 2 ( h + j + 1 ) , - ( - b - c ( 2 s - v ) + 2 p z ) 2 4 p ) + 2 - ( - b - c ( 2 s - v ) + 2 p z ) 2 p p Γ ( 1 2 ( h + j + 2 ) , - ( - b - c ( 2 s - v ) + 2 p z ) 2 4 p ) ) + - ( b - c ( 2 s - v ) ) 2 4 p j = 0 n h = 0 j ( - 1 ) j - h 4 j ( b - c ( 2 s - v ) ) - h - j + 2 n ( b - c ( 2 s - v ) + 2 p z ) h + j ( - ( b - c ( 2 s - v ) + 2 p z ) 2 p ) 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 - c ( 2 s - v ) ) ( b - c ( 2 s - v ) + 2 p z ) Γ ( 1 2 ( h + j + 1 ) , - ( b - c ( 2 s - v ) + 2 p z ) 2 4 p ) + 2 - ( b - c ( 2 s - v ) + 2 p z ) 2 p p Γ ( 1 2 ( h + j + 2 ) , - ( b - c ( 2 s - v ) + 2 p z ) 2 4 p ) ) - - ( - b - c ( v - 2 s ) ) 2 4 p j = 0 n h = 0 j ( - 1 ) j - h 4 j ( - b - c ( v - 2 s ) ) - h - j + 2 n ( - b - c ( v - 2 s ) + 2 p z ) h + j ( - ( - b - c ( v - 2 s ) + 2 p z ) 2 p ) 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 - c ( v - 2 s ) ) ( - b - c ( v - 2 s ) + 2 p z ) Γ ( 1 2 ( h + j + 1 ) , - ( - b - c ( v - 2 s ) + 2 p z ) 2 4 p ) + 2 - ( - b - c ( v - 2 s ) + 2 p z ) 2 p p Γ ( 1 2 ( h + j + 2 ) , - ( - b - c ( v - 2 s ) + 2 p z ) 2 4 p ) ) + - ( b - c ( v - 2 s ) ) 2 4 p j = 0 n h = 0 j ( - 1 ) j - h 4 j ( b - c ( v - 2 s ) ) - h - j + 2 n ( b - c ( v - 2 s ) + 2 p z ) h + j ( - ( b - c ( v - 2 s ) + 2 p z ) 2 p ) 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 - c ( v - 2 s ) ) ( b - c ( v - 2 s ) + 2 p z ) Γ ( 1 2 ( h + j + 1 ) , - ( b - c ( v - 2 s ) + 2 p z ) 2 4 p ) + 2 - ( b - c ( v - 2 s ) + 2 p z ) 2 p p Γ ( 1 2 ( h + j + 2 ) , - ( b - c ( v - 2 s ) + 2 p z ) 2 4 p ) ) ) /; n v + Condition z z n p z b z 1 2 c z 1 2 v 2 -2 n -1 v -2 p -2 n -2 Binomial v v 2 -1 1 -1 \$CellContext`v 2 b 2 4 p -1 h 0 j j 0 n -1 j -1 h 4 j -1 b -1 h -1 j 2 n -1 b 2 p z 1 2 h j -1 -1 b 2 p z 1 2 2 p -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j 2 p -1 -1 b 2 p z 1 2 2 p -1 1 2 Gamma 1 2 h j 2 -1 -1 b 2 p z 1 2 2 4 p -1 -1 b -1 b 2 p z 1 2 Gamma 1 2 h j 1 -1 -1 b 2 p z 1 2 2 4 p -1 -1 b 2 4 p -1 h 0 j j 0 n -1 j -1 h 4 j b -1 h -1 j 2 n b 2 p z 1 2 h j -1 b 2 p z 1 2 2 p -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j b b 2 p z 1 2 Gamma 1 2 h j 1 -1 b 2 p z 1 2 2 4 p -1 2 -1 b 2 p z 1 2 2 p -1 1 2 p Gamma 1 2 h j 2 -1 b 2 p z 1 2 2 4 p -1 -1 2 -2 n -1 v -2 p -2 n -2 s 0 v -1 2 -1 Binomial v s -1 -1 -1 b -1 c 2 s -1 v 2 4 p -1 h 0 j j 0 n -1 j -1 h 4 j -1 b -1 c 2 s -1 v -1 h -1 j 2 n -1 b -1 c 2 s -1 v 2 p z 1 2 h j -1 -1 b -1 c 2 s -1 v 2 p z 1 2 2 p -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j -1 b -1 c 2 s -1 v -1 b -1 c 2 s -1 v 2 p z 1 2 Gamma 1 2 h j 1 -1 -1 b -1 c 2 s -1 v 2 p z 1 2 2 4 p -1 2 -1 -1 b -1 c 2 s -1 v 2 p z 1 2 2 p -1 1 2 p Gamma 1 2 h j 2 -1 -1 b -1 c 2 s -1 v 2 p z 1 2 2 4 p -1 -1 b -1 c 2 s -1 v 2 4 p -1 h 0 j j 0 n -1 j -1 h 4 j b -1 c 2 s -1 v -1 h -1 j 2 n b -1 c 2 s -1 v 2 p z 1 2 h j -1 b -1 c 2 s -1 v 2 p z 1 2 2 p -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j b -1 c 2 s -1 v b -1 c 2 s -1 v 2 p z 1 2 Gamma 1 2 h j 1 -1 b -1 c 2 s -1 v 2 p z 1 2 2 4 p -1 2 -1 b -1 c 2 s -1 v 2 p z 1 2 2 p -1 1 2 p Gamma 1 2 h j 2 -1 b -1 c 2 s -1 v 2 p z 1 2 2 4 p -1 -1 -1 -1 b -1 c v -1 2 s 2 4 p -1 h 0 j j 0 n -1 j -1 h 4 j -1 b -1 c v -1 2 s -1 h -1 j 2 n -1 b -1 c v -1 2 s 2 p z 1 2 h j -1 -1 b -1 c v -1 2 s 2 p z 1 2 2 p -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j -1 b -1 c v -1 2 s -1 b -1 c v -1 2 s 2 p z 1 2 Gamma 1 2 h j 1 -1 -1 b -1 c v -1 2 s 2 p z 1 2 2 4 p -1 2 -1 -1 b -1 c v -1 2 s 2 p z 1 2 2 p -1 1 2 p Gamma 1 2 h j 2 -1 -1 b -1 c v -1 2 s 2 p z 1 2 2 4 p -1 -1 b -1 c v -1 2 s 2 4 p -1 h 0 j j 0 n -1 j -1 h 4 j b -1 c v -1 2 s -1 h -1 j 2 n b -1 c v -1 2 s 2 p z 1 2 h j -1 b -1 c v -1 2 s 2 p z 1 2 2 p -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j b -1 c v -1 2 s b -1 c v -1 2 s 2 p z 1 2 Gamma 1 2 h j 1 -1 b -1 c v -1 2 s 2 p z 1 2 2 4 p -1 2 -1 b -1 c v -1 2 s 2 p z 1 2 2 p -1 1 2 p Gamma 1 2 h j 2 -1 b -1 c v -1 2 s 2 p z 1 2 2 4 p -1 n v SuperPlus [/itex]

 Rule Form

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

 2002-12-18