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 Erfc

 http://functions.wolfram.com/06.27.21.0095.01

 Input Form

 Integrate[E^(b z^2) Sinh[c z^2] Erfc[a z], z] == (Sqrt[Pi]/(4 (c^2 - b^2))) (Sqrt[b - c] (b + c) Erfi[Sqrt[b - c] z] + (-b + c) Sqrt[b + c] Erfi[Sqrt[b + c] z]) + (1/(2 Sqrt[Pi] (b - c))) Sum[(a^(2 k + 1)/((b - c)^k ((1 + 2 k) k!))) Gamma[1 + k, (-(b - c)) z^2], {k, 0, Infinity}] - (1/(2 Sqrt[Pi] (b + c))) Sum[(a^(2 k + 1)/((b + c)^k ((1 + 2 k) k!))) Gamma[1 + k, (-(b + 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["Sinh", "[", RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "]"]], " ", RowBox[List["Erfc", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[SqrtBox["\[Pi]"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[SuperscriptBox["c", "2"], "-", SuperscriptBox["b", "2"]]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["b", "-", "c"]]], " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List["b", "-", "c"]]], " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], " ", SqrtBox[RowBox[List["b", "+", "c"]]], " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List["b", "+", "c"]]], " ", "z"]], "]"]]]]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["2", SqrtBox["\[Pi]"], RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "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", "-", "c"]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]]]]]], "-", RowBox[List[FractionBox["1", RowBox[List["2", SqrtBox["\[Pi]"], RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "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", "+", "c"]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]]]]]]]]]]]]

 MathML Form

 b z 2 sinh ( c z 2 ) erfc ( a z ) z π 4 ( c 2 - b 2 ) ( b - c ( b + c ) erfi ( b - c z ) + ( c - b ) b + c erfi ( b + c z ) ) - 1 2 π ( b + c ) k = 0 ( b + c ) - k a 2 k + 1 Γ ( k + 1 , - ( b + c ) z 2 ) ( 2 k + 1 ) k ! + 1 2 π ( b - c ) k = 0 ( b - c ) - k a 2 k + 1 Γ ( k + 1 , - ( b - c ) z 2 ) ( 2 k + 1 ) k ! z b z 2 c z 2 Erfc a z 1 2 4 c 2 -1 b 2 -1 b -1 c 1 2 b c Erfi b -1 c 1 2 z c -1 b b c 1 2 Erfi b c 1 2 z -1 1 2 1 2 b c -1 k 0 b c -1 k a 2 k 1 Gamma k 1 -1 b c z 2 2 k 1 k -1 1 2 1 2 b -1 c -1 k 0 b -1 c -1 k a 2 k 1 Gamma k 1 -1 b -1 c z 2 2 k 1 k -1 [/itex]

 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["Sinh", "[", 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", "-", "c"]]], " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List["b", "-", "c"]]], " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], " ", SqrtBox[RowBox[List["b", "+", "c"]]], " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List["b", "+", "c"]]], " ", "z"]], "]"]]]]]], ")"]]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List[SuperscriptBox["c", "2"], "-", SuperscriptBox["b", "2"]]], ")"]]]]], "+", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "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", "-", "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", "-", "c"]], ")"]]]]], "-", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "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", "+", "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", "+", "c"]], ")"]]]]]]]]]]]

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

 2001-10-29