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; }

 Erf

 http://functions.wolfram.com/06.25.21.0064.01

 Input Form

 Integrate[z^n E^(b z) Cos[c z] Erf[a z], z] == (-(1/2)) (-b - I c)^(-1 - n) Erf[a z] Gamma[1 + n, (-b - I c) z] - (1/2) (-b + I c)^(-1 - n) Erf[a z] Gamma[1 + n, (-b + I c) z] - (1/(2 Sqrt[Pi])) (a (-b + I c)^(-1 - n) E^((b - I c)^2/(4 a^2)) n! Sum[(1/m!) ((-(b - I c))^m (-a^2)^((1/2) (-1 - m)) Sum[Binomial[m, k] (-((b - I c)/(2 Sqrt[-a^2])))^(m - k) (Sqrt[-a^2] z + (b - I c)/(2 Sqrt[-a^2]))^(1 + k) (-(Sqrt[-a^2] z + (b - I c)/(2 Sqrt[-a^2]))^2)^((1/2) (-1 - k)) Gamma[(1 + k)/2, -(Sqrt[-a^2] z + (b - I c)/(2 Sqrt[-a^2]))^2], {k, 0, m}]), {m, 0, n}]) - (1/(2 Sqrt[Pi])) (a (-b - I c)^(-1 - n) E^((b + I c)^2/(4 a^2)) n! Sum[(1/m!) ((-(b + I c))^m (-a^2)^((1/2) (-1 - m)) Sum[Binomial[m, k] (-((b + I c)/(2 Sqrt[-a^2])))^(m - k) (Sqrt[-a^2] z + (b + I c)/(2 Sqrt[-a^2]))^(1 + k) (-(Sqrt[-a^2] z + (b + I c)/(2 Sqrt[-a^2]))^2)^((1/2) (-1 - k)) Gamma[(1 + k)/2, -(Sqrt[-a^2] z + (b + I c)/(2 Sqrt[-a^2]))^2], {k, 0, m}]), {m, 0, n}]) /; Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 z n b z cos ( c z ) erf ( a z ) z - 1 2 erf ( a z ) Γ ( n + 1 , ( c - b ) z ) ( c - b ) - n - 1 - 1 2 ( - b - c ) - n - 1 erf ( a z ) Γ ( n + 1 , ( - b - c ) z ) - 1 2 π ( a ( - b - c ) - n - 1 ( b + c ) 2 4 a 2 n ! m = 0 n 1 m ! ( ( - ( b + c ) ) m ( - a 2 ) 1 2 ( - m - 1 ) k = 0 m ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - b + c 2 - a 2 ) m - k ( b + c 2 - a 2 + - a 2 z ) k + 1 ( - ( b + c 2 - a 2 + - a 2 z ) 2 ) 1 2 ( - k - 1 ) Γ ( k + 1 2 , - ( b + c 2 - a 2 + - a 2 z ) 2 ) ) ) - 1 2 π ( a ( c - b ) - n - 1 ( b - c ) 2 4 a 2 n ! m = 0 n 1 m ! ( ( - ( b - c ) ) m ( - a 2 ) 1 2 ( - m - 1 ) k = 0 m ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - b - c 2 - a 2 ) m - k ( b - c 2 - a 2 + - a 2 z ) k + 1 ( - ( b - c 2 - a 2 + - a 2 z ) 2 ) 1 2 ( - k - 1 ) Γ ( k + 1 2 , - ( b - c 2 - a 2 + - a 2 z ) 2 ) ) ) /; n Condition z z n b z c z Erf a z -1 1 2 Erf a z Gamma n 1 c -1 b z c -1 b -1 n -1 -1 1 2 -1 b -1 c -1 n -1 Erf a z Gamma n 1 -1 b -1 c z -1 1 2 1 2 -1 a -1 b -1 c -1 n -1 b c 2 4 a 2 -1 n m 0 n 1 m -1 -1 b c m -1 a 2 1 2 -1 m -1 k 0 m Binomial m k -1 b c 2 -1 a 2 1 2 -1 m -1 k b c 2 -1 a 2 1 2 -1 -1 a 2 1 2 z k 1 -1 b c 2 -1 a 2 1 2 -1 -1 a 2 1 2 z 2 1 2 -1 k -1 Gamma k 1 2 -1 -1 b c 2 -1 a 2 1 2 -1 -1 a 2 1 2 z 2 -1 1 2 1 2 -1 a c -1 b -1 n -1 b -1 c 2 4 a 2 -1 n m 0 n 1 m -1 -1 b -1 c m -1 a 2 1 2 -1 m -1 k 0 m Binomial m k -1 b -1 c 2 -1 a 2 1 2 -1 m -1 k b -1 c 2 -1 a 2 1 2 -1 -1 a 2 1 2 z k 1 -1 b -1 c 2 -1 a 2 1 2 -1 -1 a 2 1 2 z 2 1 2 -1 k -1 Gamma k 1 2 -1 -1 b -1 c 2 -1 a 2 1 2 -1 -1 a 2 1 2 z 2 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "n_"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b_", " ", "z_"]]], " ", RowBox[List["Cos", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", RowBox[List["Erf", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["Erf", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", "z"]]]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", 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 Date Added to functions.wolfram.com (modification date)

 2001-10-29