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/01.07.21.2599.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", SuperscriptBox["z", "2"]]]], RowBox[List["Sin", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]], SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "]"]], "v"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "v"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "p"]]], " ", "z"]], "]"]], SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "p"]]]], "-", FractionBox[RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "p"]]], " ", "z"]], "]"]], SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "p"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "v"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "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", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "p", "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]], " ", "z"]], "]"]], SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "p", "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]], "-", FractionBox[RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "p", "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]], " ", "z"]], "]"]], SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "p", "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]], "+", FractionBox[RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "p", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]], " ", "z"]], "]"]], SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "p", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]], "-", FractionBox[RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "p", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]], " ", "z"]], "]"]], SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "p", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

 p z 2 sin ( b z 2 ) cos v ( c z 2 ) z 2 - v - 2 ( erfi ( - b + p z ) - b + p - erfi ( b + p z ) b + p ) π ( 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 ) + 2 - v - 2 π 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]]]]] ( erfi ( - b + p + c ( v - 2 s ) z ) - b + p + c ( v - 2 s ) + erfi ( - b + p - c ( v - 2 s ) z ) - b + p - c ( v - 2 s ) - erfi ( b + p + c ( v - 2 s ) z ) b + p + c ( v - 2 s ) - erfi ( b + p - c ( v - 2 s ) z ) b + p - c ( v - 2 s ) ) /; v + Condition z p z 2 b z 2 c z 2 v 2 -1 v -2 Erfi -1 b p 1 2 z -1 b p 1 2 -1 -1 Erfi b p 1 2 z b p 1 2 -1 1 2 Binomial v v 2 -1 1 -1 \$CellContext`v 2 2 -1 v -2 1 2 s 0 v -1 2 -1 Binomial v s Erfi -1 b p c v -1 2 s 1 2 z -1 b p c v -1 2 s 1 2 -1 Erfi -1 b p -1 c v -1 2 s 1 2 z -1 b p -1 c v -1 2 s 1 2 -1 -1 Erfi b p c v -1 2 s 1 2 z b p c v -1 2 s 1 2 -1 -1 Erfi b p -1 c v -1 2 s 1 2 z b p -1 c v -1 2 s 1 2 -1 v SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p_", " ", SuperscriptBox["z_", "2"]]]], " ", RowBox[List["Sin", "[", RowBox[List["b_", " ", SuperscriptBox["z_", "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c_", " ", SuperscriptBox["z_", "2"]]], "]"]], "v_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "v"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "p"]]], " ", "z"]], "]"]], SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "p"]]]], "-", FractionBox[RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "p"]]], " ", "z"]], "]"]], SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "p"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "v"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "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", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "p", "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]], " ", "z"]], "]"]], SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "p", "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]], "-", FractionBox[RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "p", "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]], " ", "z"]], "]"]], SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "p", "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]], "+", FractionBox[RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "p", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]], " ", "z"]], "]"]], SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "p", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]], "-", FractionBox[RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "p", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]], " ", "z"]], "]"]], SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "p", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "&&", RowBox[List["v", ">", "0"]]]]]]]]]]

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

 2002-12-18