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 Erf

 http://functions.wolfram.com/06.25.21.0040.01

 Input Form

 Integrate[(Sin[b z^2] Erf[a z])/z, z] == ((I 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}] - ((I 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["Sin", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]], " ", RowBox[List["Erf", "[", RowBox[List["a", " ", "z"]], "]"]]]], "z"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "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["\[ImaginaryI]", " ", "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

 sin ( b z 2 ) erf ( a z ) z z a z 2 - π b z 2 k = 0 ( b ) - k a 2 k Γ ( k + 1 2 , - b z 2 ) ( 2 k + 1 ) k ! - a z 2 π b z 2 k = 0 ( - b ) - k a 2 k Γ ( k + 1 2 , b z 2 ) ( 2 k + 1 ) k ! z b z 2 Erf a z z -1 a z 2 -1 b z 2 1 2 -1 k 0 b -1 k a 2 k Gamma k 1 2 -1 b z 2 2 k 1 k -1 -1 a z 2 b z 2 1 2 -1 k 0 -1 b -1 k a 2 k Gamma k 1 2 b 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[RowBox[List["Sin", "[", RowBox[List["b_", " ", 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["\[ImaginaryI]", " ", "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["\[ImaginaryI]", " ", "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