html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Erfi

 http://functions.wolfram.com/06.28.21.0096.01

 Input Form

 Integrate[E^(b z^2) Sinh[c z^2] Erfi[a z], z] == (-(1/(2 Sqrt[Pi] (b - c)))) Sum[(a^(2 k + 1)/((c - b)^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["Erfi", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[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["c", "-", "b"]], ")"]], 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[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 ) erfi ( a z ) 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 ( c - b ) - k a 2 k + 1 Γ ( k + 1 , - ( b - c ) z 2 ) ( 2 k + 1 ) k ! z b z 2 c z 2 Erfi a z 1 2 1 2 b c -1 k 0 -1 b -1 c -1 k a 2 k 1 Gamma k 1 -1 b c z 2 2 k 1 k -1 -1 1 2 1 2 b -1 c -1 k 0 c -1 b -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["Erfi", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["c", "-", "b"]], ")"]], 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[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