Complementary error function: Integration (formula 06.27.21.0053)

Erfc

$Mathematica Notation$

Gamma, Beta, Erf

Erfc[z]

Integration

Indefinite integration

Involving one direct function and elementary functions

Involving exponential function and trigonometric functions

Involving exp and cos

http://functions.wolfram.com/06.27.21.0053.01

Input Form

Integrate[E^(b z^2) Cos[c z^2] Erfc[a z], z] == (1/(4 (b^2 + c^2))) (Sqrt[Pi] (Sqrt[b - I c] (b + I c) Erfi[Sqrt[b - I c] z] + (b - I c) Sqrt[b + I c] Erfi[Sqrt[b + I c] z])) - (1/(2 Sqrt[Pi] (b + I c))) Sum[(a^(2 k + 1)/((b + I c)^k ((1 + 2 k) k!))) Gamma[1 + k, (-(b + I c)) z^2], {k, 0, Infinity}] - (1/(2 Sqrt[Pi] (b - I c))) Sum[(a^(2 k + 1)/((b - I c)^k ((1 + 2 k) k!))) Gamma[1 + k, (-(b - I c)) z^2], {k, 0, Infinity}]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["Cos", "[", RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "]"]], " ", RowBox[List["Erfc", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["4", " ", RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "+", SuperscriptBox["c", "2"]]], ")"]]]]], RowBox[List["(", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SqrtBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], " ", "z"]], "]"]]]]]], ")"]]]], ")"]]]], "-", RowBox[List[FractionBox["1", RowBox[List["2", SqrtBox["\[Pi]"], RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], RowBox[List["-", "k"]]], SuperscriptBox["a", RowBox[List[RowBox[List["2", "k"]], "+", "1"]]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], RowBox[List["k", "!"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "k"]], ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]]]]]], "-", RowBox[List[FractionBox["1", RowBox[List["2", SqrtBox["\[Pi]"], RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], RowBox[List["-", "k"]]], SuperscriptBox["a", RowBox[List[RowBox[List["2", "k"]], "+", "1"]]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], RowBox[List["k", "!"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "k"]], ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]]]]]]]]]]]]

MathML Form

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["b_", " ", SuperscriptBox["z_", "2"]]]], " ", RowBox[List["Cos", "[", RowBox[List["c_", " ", SuperscriptBox["z_", "2"]]], "]"]], " ", RowBox[List["Erfc", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SqrtBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], " ", "z"]], "]"]]]]]], ")"]]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "+", SuperscriptBox["c", "2"]]], ")"]]]]], "-", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], RowBox[List["-", "k"]]], " ", SuperscriptBox["a", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "k"]], ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], " ", RowBox[List["k", "!"]]]]]]], RowBox[List["2", " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]]], "-", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], RowBox[List["-", "k"]]], " ", SuperscriptBox["a", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "k"]], ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], " ", RowBox[List["k", "!"]]]]]]], RowBox[List["2", " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]]]]]]]]]

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

2001-10-29