Complementary error function: Integration (formula 06.27.21.0049)

Erfc

$Mathematica Notation$

Gamma, Beta, Erf

Erfc[z]

Integration

Indefinite integration

Involving one direct function and elementary functions

Involving trigonometric functions and a power function

Involving cos and power

http://functions.wolfram.com/06.27.21.0049.01

Input Form

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

Standard Form

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

MathML Form

Rule Form

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

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

2001-10-29