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; }

 Cos

 http://functions.wolfram.com/01.07.21.2729.01

 Input Form

 Integrate[z^n E^(p z) Sin[b z^2] Cos[c z^2]^v, z] == (-(1/Sqrt[b^2])) ((I 2^(-2 - v) (Sqrt[(-I) b] E^((I p^2)/(2 b)) Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[2^(-n + q) (I b)^(-(1/2) - n) (-p)^(n - q) (p + 2 I b z)^(1 + q) ((I (p + 2 I b z)^2)/b)^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (p + 2 I b z)^2)/(4 b)], {q, 0, n}] - Sqrt[I b] Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[2^(-n + q) ((-I) b)^(-(1/2) - n) (-p)^(n - q) (p - 2 I b z)^(1 + q) ((I (I p + 2 b z)^2)/b)^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (I p + 2 b z)^2)/(4 b)], {q, 0, n}] + Sqrt[b^2] E^((I p^2)/(4 b)) Sum[(E^((I b p^2)/(2 (b + 2 c s - c v) (b - 2 c s + c v))) Binomial[v, s] ((-E^(-((I p^2)/(4 (b - 2 c s + c v))))) Sqrt[(-I) (b + 2 c s - c v)] Sqrt[(b - 2 c s + c v)^2] Sum[2^(-n + q) (-p)^(n - q) (I (b + 2 c s - c v))^(-(1/2) - n) (p + 2 I (b + 2 c s - c v) z)^(1 + q) ((I (p + 2 I (b + 2 c s - c v) z)^2)/(b + 2 c s - c v))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (p + 2 I (b + 2 c s - c v) z)^2)/(4 (b + 2 c s - c v))], {q, 0, n}] + (Sqrt[I (b + 2 c s - c v)] Sqrt[(b - 2 c s + c v)^ 2] Sum[2^(-n + q) (-p)^(n - q) ((-I) (b + 2 c s - c v))^ (-(1/2) - n) (p - 2 I (b + 2 c s - c v) z)^(1 + q) ((I (I p + 2 (b + 2 c s - c v) z)^2)/(b + 2 c s - c v))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (I p + 2 (b + 2 c s - c v) z)^2)/(4 (b + 2 c s - c v))], {q, 0, n}])/E^((I p^2 (3 b - 2 c s + c v))/(4 (b + 2 c s - c v) (b - 2 c s + c v))) + (Sqrt[(b + 2 c s - c v)^2] ((-E^((I p^2)/(2 (b - 2 c s + c v)))) Sqrt[(-I) (b - 2 c s + c v)] Sum[2^(-n + q) (-p)^(n - q) (I (b - 2 c s + c v))^ (-(1/2) - n) (p + 2 I (b - 2 c s + c v) z)^(1 + q) ((I (p + 2 I (b - 2 c s + c v) z)^2)/(b - 2 c s + c v))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (p + 2 I (b - 2 c s + c v) z)^2)/(4 (b - 2 c s + c v))], {q, 0, n}] + Sqrt[I (b - 2 c s + c v)] Sum[2^(-n + q) (-p)^(n - q) ((-I) (b - 2 c s + c v))^(-(1/2) - n) (p - 2 I (b - 2 c s + c v) z)^(1 + q) ((I (I p + 2 (b - 2 c s + c v) z)^2)/(b - 2 c s + c v))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (I p + 2 (b - 2 c s + c v) z)^2)/(4 (b - 2 c s + c v))], {q, 0, n}]))/E^((I p^2 (3 b + 2 c s - c v))/(4 (b + 2 c s - c v) (b - 2 c s + c v)))))/(Sqrt[(b + 2 c s - c v)^2] Sqrt[(b - 2 c s + c v)^2]), {s, 0, Floor[(1/2) (-1 + v)]}]))/ E^((I p^2)/(4 b))) /; Element[v, Integers] && v > 0 && Element[n, Integers] && n >= 0

 Standard Form

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" ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["b", "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List["b", "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]]]]]]], "]"]]]]]]]]]], ")"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List[SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], "2"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], "2"]]]], ")"]]]]]]]]]], ")"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 z n p z sin ( b z 2 ) cos v ( c z 2 ) z - 2 - v - 2 b 2 - p 2 4 b ( b 2 p 2 4 b s = 0 v - 1 2 ( b p 2 2 ( b + 2 c s - c v ) ( b - 2 c s + c v ) ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - p 2 ( 3 b + 2 c s - c v ) 4 ( b + 2 c s - c v ) ( b - 2 c s + c v ) ( b + 2 c s - c v ) 2 ( ( b - 2 c s + c v ) q = 0 n 2 q - n ( - p ) n - q ( - ( b - 2 c s + c v ) ) - n - 1 2 ( p - 2 ( b - 2 c s + c v ) z ) q + 1 ( ( p + 2 ( b - 2 c s + c v ) z ) 2 b - 2 c s + c v ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( p + 2 ( b - 2 c s + c v ) z ) 2 4 ( b - 2 c s + c v ) ) - p 2 2 ( b - 2 c s + c v ) - ( b - 2 c s + c v ) q = 0 n 2 q - n ( - p ) n - q ( ( b - 2 c s + c v ) ) - n - 1 2 ( p + 2 ( b - 2 c s + c v ) z ) q + 1 ( ( p + 2 ( b - 2 c s + c v ) z ) 2 b - 2 c s + c v ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( p + 2 ( b - 2 c s + c v ) z ) 2 4 ( b - 2 c s + c v ) ) ) + - p 2 ( 3 b - 2 c s + c v ) 4 ( b + 2 c s - c v ) ( b - 2 c s + c v ) ( b - 2 c s + c v ) 2 ( b + 2 c s - c v ) q = 0 n 2 q - n ( - p ) n - q ( - ( b + 2 c s - c v ) ) - n - 1 2 ( p - 2 ( b + 2 c s - c v ) z ) q + 1 ( ( p + 2 ( b + 2 c s - c v ) z ) 2 b + 2 c s - c v ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( p + 2 ( b + 2 c s - c v ) z ) 2 4 ( b + 2 c s - c v ) ) - - p 2 4 ( b - 2 c s + c v ) ( b - 2 c s + c v ) 2 - ( b + 2 c s - c v ) q = 0 n 2 q - n ( - p ) n - q ( ( b + 2 c s - c v ) ) - n - 1 2 ( p + 2 ( b + 2 c s - c v ) z ) q + 1 ( ( p + 2 ( b + 2 c s - c v ) z ) 2 b + 2 c s - c v ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( p + 2 ( b + 2 c s - c v ) z ) 2 4 ( b + 2 c s - c v ) ) ) ) / ( ( b + 2 c s - c v ) 2 ( b - 2 c s + c v ) 2 ) - b ( 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]]]]] ( v mod 2 \$CellContext`v 2 - 1 ) q = 0 n 2 q - n ( - b ) - n - 1 2 ( - p ) n - q ( p - 2 b z ) q + 1 ( ( p + 2 b z ) 2 b ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( p + 2 b z ) 2 4 b ) + - b p 2 2 b ( 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]]]]] ( v mod 2 \$CellContext`v 2 - 1 ) q = 0 n 2 q - n ( b ) - n - 1 2 ( - p ) n - q ( p + 2 b z ) q + 1 ( ( p + 2 b z ) 2 b ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( p + 2 b z ) 2 4 b ) ) /; v + n Condition z z n p z b z 2 c z 2 v -1 2 -1 v -2 b 2 1 2 -1 -1 p 2 4 b -1 b 2 1 2 p 2 4 b -1 s 0 v -1 2 -1 b p 2 2 b 2 c s -1 c v b -1 2 c s c v -1 Binomial v s -1 p 2 3 b 2 c s -1 c v 4 b 2 c s -1 c v b -1 2 c s c v -1 b 2 c s -1 c v 2 1 2 b -1 2 c s c v 1 2 q 0 n 2 q -1 n -1 p n -1 q -1 b -1 2 c s c v -1 n -1 1 2 p -1 2 b -1 2 c s c v z q 1 p 2 b -1 2 c s c v z 2 b -1 2 c s c v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 p 2 b -1 2 c s c v z 2 4 b -1 2 c s c v -1 -1 p 2 2 b -1 2 c s c v -1 -1 b -1 2 c s c v 1 2 q 0 n 2 q -1 n -1 p n -1 q b -1 2 c s c v -1 n -1 1 2 p 2 b -1 2 c s c v z q 1 p 2 b -1 2 c s c v z 2 b -1 2 c s c v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 p 2 b -1 2 c s c v z 2 4 b -1 2 c s c v -1 -1 p 2 3 b -1 2 c s c v 4 b 2 c s -1 c v b -1 2 c s c v -1 b -1 2 c s c v 2 1 2 b 2 c s -1 c v 1 2 q 0 n 2 q -1 n -1 p n -1 q -1 b 2 c s -1 c v -1 n -1 1 2 p -1 2 b 2 c s -1 c v z q 1 p 2 b 2 c s -1 c v z 2 b 2 c s -1 c v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 p 2 b 2 c s -1 c v z 2 4 b 2 c s -1 c v -1 -1 -1 p 2 4 b -1 2 c s c v -1 b -1 2 c s c v 2 1 2 -1 b 2 c s -1 c v 1 2 q 0 n 2 q -1 n -1 p n -1 q b 2 c s -1 c v -1 n -1 1 2 p 2 b 2 c s -1 c v z q 1 p 2 b 2 c s -1 c v z 2 b 2 c s -1 c v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 p 2 b 2 c s -1 c v z 2 4 b 2 c s -1 c v -1 b 2 c s -1 c v 2 1 2 b -1 2 c s c v 2 1 2 -1 -1 b 1 2 Binomial v v 2 -1 \$CellContext`v 2 -1 q 0 n 2 q -1 n -1 b -1 n -1 1 2 -1 p n -1 q p -1 2 b z q 1 p 2 b z 2 b -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 p 2 b z 2 4 b -1 -1 b 1 2 p 2 2 b -1 Binomial v v 2 -1 \$CellContext`v 2 -1 q 0 n 2 q -1 n b -1 n -1 1 2 -1 p n -1 q p 2 b z q 1 p 2 b z 2 b -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 p 2 b z 2 4 b -1 v SuperPlus n [/itex]

 Rule Form

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

 2002-12-18