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; }

 Erfi

 http://functions.wolfram.com/06.28.21.0041.01

 Input Form

 Integrate[z^(\[Alpha] - 1) Cos[b z] Erfi[a z], z] == ((I a z^\[Alpha])/(b Sqrt[Pi])) Sum[((((-1)^k a^(2 k))/((1 + 2 k) k!)) (Gamma[1 + 2 k + \[Alpha], I b z]/ (I b z)^\[Alpha] - Gamma[1 + 2 k + \[Alpha], (-I) b z]/ ((-I) b z)^\[Alpha]))/b^(2 k), {k, 0, Infinity}]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["Cos", "[", RowBox[List["b", " ", "z"]], "]"]], RowBox[List["Erfi", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", SuperscriptBox["z", "\[Alpha]"]]], RowBox[List["b", SqrtBox["\[Pi]"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SuperscriptBox["a", RowBox[List["2", "k"]]], " "]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], RowBox[List["k", "!"]]]]], SuperscriptBox["b", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b", " ", "z"]], ")"]], RowBox[List["-", "\[Alpha]"]]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", "k"]], "+", "\[Alpha]"]], ",", RowBox[List["\[ImaginaryI]", " ", "b", " ", "z"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", "z"]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", "k"]], "+", "\[Alpha]"]], ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", "z"]]]], "]"]]]]]], ")"]]]]]]]]]]]]

 MathML Form

 z α - 1 cos ( b z ) erfi ( a z ) z a z α b π k = 0 ( - 1 ) k a 2 k b - 2 k ( 2 k + 1 ) k ! ( ( b z ) - α Γ ( 2 k + α + 1 , b z ) - ( - b z ) - α Γ ( 2 k + α + 1 , - b z ) ) z z α -1 b z Erfi a z a z α b 1 2 -1 k 0 -1 k a 2 k b -2 k 2 k 1 k -1 b z -1 α Gamma 2 k α 1 b z -1 -1 b z -1 α Gamma 2 k α 1 -1 b z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["Cos", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", RowBox[List["Erfi", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "a", " ", SuperscriptBox["z", "\[Alpha]"]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["a", RowBox[List["2", " ", "k"]]]]], ")"]], " ", SuperscriptBox["b", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b", " ", "z"]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", "k"]], "+", "\[Alpha]"]], ",", RowBox[List["\[ImaginaryI]", " ", "b", " ", "z"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", "z"]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", "k"]], "+", "\[Alpha]"]], ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", "z"]]]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]], RowBox[List["b", " ", SqrtBox["\[Pi]"]]]]]]]]

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

 2001-10-29