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/01.20.21.1365.01

 Input Form

 Integrate[z^n E^(p z) Sin[b Sqrt[z]] Cosh[c Sqrt[z]], z] == (-I) 2^(-3 - 2 n) p^(-2 - 2 n) ((-E^(-(((-I) b - c)^2/(4 p)))) Sum[(-1)^(-h + j) 4^j ((-I) b - c)^(-h - j + 2 n) ((-I) b - c + 2 p Sqrt[z])^(h + j) (-(((-I) b - c + 2 p Sqrt[z])^2/p))^ ((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] (((-I) b - c) ((-I) b - c + 2 p Sqrt[z]) Gamma[(1/2) (1 + h + j), -(((-I) b - c + 2 p Sqrt[z])^2/(4 p))] + 2 p Sqrt[-(((-I) b - c + 2 p Sqrt[z])^2/p)] Gamma[(1/2) (2 + h + j), -(((-I) b - c + 2 p Sqrt[z])^2/(4 p))]), {j, 0, n}, {h, 0, j}] + Sum[(-1)^(-h + j) 4^j (I b - c)^(-h - j + 2 n) (I b - c + 2 p Sqrt[z])^ (h + j) (-((I b - c + 2 p Sqrt[z])^2/p))^((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((I b - c) (I b - c + 2 p Sqrt[z]) Gamma[(1/2) (1 + h + j), -((I b - c + 2 p Sqrt[z])^2/(4 p))] + 2 p Sqrt[-((I b - c + 2 p Sqrt[z])^2/p)] Gamma[(1/2) (2 + h + j), -((I b - c + 2 p Sqrt[z])^2/(4 p))]), {j, 0, n}, {h, 0, j}]/ E^((I b - c)^2/(4 p)) - Sum[(-1)^(-h + j) 4^j ((-I) b + c)^(-h - j + 2 n) ((-I) b + c + 2 p Sqrt[z])^(h + j) (-(((-I) b + c + 2 p Sqrt[z])^2/p))^ ((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] (((-I) b + c) ((-I) b + c + 2 p Sqrt[z]) Gamma[(1/2) (1 + h + j), -(((-I) b + c + 2 p Sqrt[z])^2/(4 p))] + 2 p Sqrt[-(((-I) b + c + 2 p Sqrt[z])^2/p)] Gamma[(1/2) (2 + h + j), -(((-I) b + c + 2 p Sqrt[z])^2/(4 p))]), {j, 0, n}, {h, 0, j}]/ E^(((-I) b + c)^2/(4 p)) + Sum[(-1)^(-h + j) 4^j (I b + c)^(-h - j + 2 n) (I b + c + 2 p Sqrt[z])^ (h + j) (-((I b + c + 2 p Sqrt[z])^2/p))^((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((I b + c) (I b + c + 2 p Sqrt[z]) Gamma[(1/2) (1 + h + j), -((I b + c + 2 p Sqrt[z])^2/(4 p))] + 2 p Sqrt[-((I b + c + 2 p Sqrt[z])^2/p)] Gamma[(1/2) (2 + h + j), -((I b + c + 2 p Sqrt[z])^2/(4 p))]), {j, 0, n}, {h, 0, j}]/ E^((I b + c)^2/(4 p))) /; Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 z n p z sin ( b z ) cosh ( c z ) z - 2 - 2 n - 3 p - 2 n - 2 ( - - ( - c - b ) 2 4 p j = 0 n h = 0 j ( - 1 ) j - h 4 j ( - c - b ) - h - j + 2 n ( - c - b + 2 p z ) h + j ( - ( - c - 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]]]]] ( ( - c - b ) ( - c - b + 2 p z ) Γ ( 1 2 ( h + j + 1 ) , - ( - c - b + 2 p z ) 2 4 p ) + 2 - ( - c - b + 2 p z ) 2 p p Γ ( 1 2 ( h + j + 2 ) , - ( - c - b + 2 p z ) 2 4 p ) ) + - ( b - c ) 2 4 p j = 0 n h = 0 j ( - 1 ) j - h 4 j ( b - c ) - h - j + 2 n ( - c + b + 2 p z ) h + j ( - ( - c + 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 - c ) ( - c + b + 2 p z ) Γ ( 1 2 ( h + j + 1 ) , - ( - c + b + 2 p z ) 2 4 p ) + 2 - ( - c + b + 2 p z ) 2 p p Γ ( 1 2 ( h + j + 2 ) , - ( - c + b + 2 p z ) 2 4 p ) ) - - ( c - b ) 2 4 p j = 0 n h = 0 j ( - 1 ) j - h 4 j ( c - b ) - h - j + 2 n ( c - b + 2 p z ) h + j ( - ( c - 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]]]]] ( ( c - b ) ( c - b + 2 p z ) Γ ( 1 2 ( h + j + 1 ) , - ( c - b + 2 p z ) 2 4 p ) + 2 - ( c - b + 2 p z ) 2 p p Γ ( 1 2 ( h + j + 2 ) , - ( c - b + 2 p z ) 2 4 p ) ) + - ( c + b ) 2 4 p j = 0 n h = 0 j ( - 1 ) j - h 4 j ( c + b ) - h - j + 2 n ( c + b + 2 p z ) h + j ( - ( c + 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]]]]] ( ( c + b ) ( c + b + 2 p z ) Γ ( 1 2 ( h + j + 1 ) , - ( c + b + 2 p z ) 2 4 p ) + 2 - ( c + b + 2 p z ) 2 p p Γ ( 1 2 ( h + j + 2 ) , - ( c + b + 2 p z ) 2 4 p ) ) ) /; n Condition z z n p z b z 1 2 c z 1 2 -1 2 -2 n -3 p -2 n -2 -1 -1 -1 c -1 b 2 4 p -1 h 0 j j 0 n -1 j -1 h 4 j -1 c -1 b -1 h -1 j 2 n -1 c -1 b 2 p z 1 2 h j -1 -1 c -1 b 2 p z 1 2 2 p -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j -1 c -1 b -1 c -1 b 2 p z 1 2 Gamma 1 2 h j 1 -1 -1 c -1 b 2 p z 1 2 2 4 p -1 2 -1 -1 c -1 b 2 p z 1 2 2 p -1 1 2 p Gamma 1 2 h j 2 -1 -1 c -1 b 2 p z 1 2 2 4 p -1 -1 b -1 c 2 4 p -1 h 0 j j 0 n -1 j -1 h 4 j b -1 c -1 h -1 j 2 n -1 c b 2 p z 1 2 h j -1 -1 c b 2 p z 1 2 2 p -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j b -1 c -1 c b 2 p z 1 2 Gamma 1 2 h j 1 -1 -1 c b 2 p z 1 2 2 4 p -1 2 -1 -1 c b 2 p z 1 2 2 p -1 1 2 p Gamma 1 2 h j 2 -1 -1 c b 2 p z 1 2 2 4 p -1 -1 -1 c -1 b 2 4 p -1 h 0 j j 0 n -1 j -1 h 4 j c -1 b -1 h -1 j 2 n c -1 b 2 p z 1 2 h j -1 c -1 b 2 p z 1 2 2 p -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j c -1 b c -1 b 2 p z 1 2 Gamma 1 2 h j 1 -1 c -1 b 2 p z 1 2 2 4 p -1 2 -1 c -1 b 2 p z 1 2 2 p -1 1 2 p Gamma 1 2 h j 2 -1 c -1 b 2 p z 1 2 2 4 p -1 -1 c b 2 4 p -1 h 0 j j 0 n -1 j -1 h 4 j c b -1 h -1 j 2 n c b 2 p z 1 2 h j -1 c b 2 p z 1 2 2 p -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j c b c b 2 p z 1 2 Gamma 1 2 h j 1 -1 c b 2 p z 1 2 2 4 p -1 2 -1 c b 2 p z 1 2 2 p -1 1 2 p Gamma 1 2 h j 2 -1 c b 2 p z 1 2 2 4 p -1 n [/itex]

 Rule Form

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

 2002-12-18