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/06.28.21.0063.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2001-10-29