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 Erf

 http://functions.wolfram.com/06.25.21.0042.01

 Input Form

 Integrate[z^n Cos[b z] Erf[a z], z] == (I/(b^(2 n) (2 b Sqrt[Pi]))) Exp[-(b^2/(4 a^2))] ((-E^(b^2/(4 a^2))) Sqrt[Pi] Erf[a z] (I b)^n (Gamma[1 + n, (-I) b z] - (-1)^n Gamma[1 + n, I b z]) + a ((-I) b)^n n! Sum[((I b)^m/m!) (-a^2)^((1/2) (-1 - m)) Sum[Binomial[m, k] (-(((-I) b)/(2 Sqrt[-a^2])))^(m - k) (Sqrt[-a^2] z - (I b)/(2 Sqrt[-a^2]))^(1 + k) (-(Sqrt[-a^2] z - (I b)/(2 Sqrt[-a^2]))^2)^((1/2) (-1 - k)) Gamma[(1 + k)/2, -(Sqrt[-a^2] z - (I b)/(2 Sqrt[-a^2]))^2], {k, 0, m}], {m, 0, n}] - a (I b)^n n! Sum[(((-I) b)^m/m!) (-a^2)^((1/2) (-1 - m)) Sum[Binomial[m, k] (-((I b)/(2 Sqrt[-a^2])))^(m - k) (Sqrt[-a^2] z + (I b)/(2 Sqrt[-a^2]))^(1 + k) (-(Sqrt[-a^2] z + (I b)/(2 Sqrt[-a^2]))^2)^((1/2) (-1 - k)) Gamma[(1 + k)/2, -(Sqrt[-a^2] z + (I b)/(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 cos ( b z ) erf ( a z ) z b - 2 n 2 b π exp ( - b 2 4 a 2 ) ( a n ! ( - b ) n m = 0 n ( b ) m ( - a 2 ) 1 2 ( - m - 1 ) m ! 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 2 - a 2 ) m - k ( - a 2 z - b 2 - a 2 ) k + 1 ( - ( - a 2 z - b 2 - a 2 ) 2 ) 1 2 ( - k - 1 ) Γ ( k + 1 2 , - ( - a 2 z - b 2 - a 2 ) 2 ) - a n ! ( b ) n m = 0 n ( - b ) m ( - a 2 ) 1 2 ( - m - 1 ) m ! 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 2 - a 2 ) m - k ( - a 2 z + b 2 - a 2 ) k + 1 ( - ( - a 2 z + b 2 - a 2 ) 2 ) 1 2 ( - k - 1 ) Γ ( k + 1 2 , - ( - a 2 z + b 2 - a 2 ) 2 ) ) - π ( b ) n b 2 4 a 2 erf ( a z ) ( Γ ( n + 1 , - b z ) - ( - 1 ) n Γ ( n + 1 , b z ) ) /; n z n cos ( b z ) erf ( a z ) z b - 2 n 2 b π exp ( - b 2 4 a 2 ) ( a n ! ( - b ) n m = 0 n ( b ) m ( - a 2 ) 1 2 ( - m - 1 ) m ! 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 2 - a 2 ) m - k ( - a 2 z - b 2 - a 2 ) k + 1 ( - ( - a 2 z - b 2 - a 2 ) 2 ) 1 2 ( - k - 1 ) Γ ( k + 1 2 , - ( - a 2 z - b 2 - a 2 ) 2 ) - a n ! ( b ) n m = 0 n ( - b ) m ( - a 2 ) 1 2 ( - m - 1 ) m ! 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 2 - a 2 ) m - k ( - a 2 z + b 2 - a 2 ) k + 1 ( - ( - a 2 z + b 2 - a 2 ) 2 ) 1 2 ( - k - 1 ) Γ ( k + 1 2 , - ( - a 2 z + b 2 - a 2 ) 2 ) ) - π ( b ) n b 2 4 a 2 erf ( a z ) ( Γ ( n + 1 , - b z ) - ( - 1 ) n Γ ( n + 1 , b z ) ) /; n [/itex]

 Rule Form

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

 2001-10-29