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 Erf

 http://functions.wolfram.com/06.25.21.0070.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2001-10-29