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 Erf

 http://functions.wolfram.com/06.25.21.0032.01

 Input Form

 Integrate[z^n Sin[b z] Erf[a z], z] == (1/(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 sin ( 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 1 m ! ( b ) 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 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 ) + - ( b ) n exp ( b 2 4 a 2 ) π erf ( a z ) ( Γ ( n + 1 , - b z ) + ( - 1 ) n Γ ( n + 1 , b z ) ) - a ( b ) n n ! m = 0 n 1 m ! ( - b ) 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 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 ) ) /; n Condition z z n b z Erf a z b -2 n 2 b 1 2 -1 -1 b 2 4 a 2 -1 -1 a n -1 b n m 0 n 1 m -1 b m -1 a 2 1 2 -1 m -1 k 0 m Binomial m k -1 -1 b 2 -1 a 2 1 2 -1 m -1 k -1 a 2 1 2 z -1 b 2 -1 a 2 1 2 -1 k 1 -1 -1 a 2 1 2 z -1 b 2 -1 a 2 1 2 -1 2 1 2 -1 k -1 Gamma k 1 2 -1 -1 -1 a 2 1 2 z -1 b 2 -1 a 2 1 2 -1 2 -1 b n b 2 4 a 2 -1 1 2 Erf a z Gamma n 1 -1 b z -1 n Gamma n 1 b z -1 a b n n m 0 n 1 m -1 -1 b m -1 a 2 1 2 -1 m -1 k 0 m Binomial m k -1 b 2 -1 a 2 1 2 -1 m -1 k -1 a 2 1 2 z b 2 -1 a 2 1 2 -1 k 1 -1 -1 a 2 1 2 z b 2 -1 a 2 1 2 -1 2 1 2 -1 k -1 Gamma k 1 2 -1 -1 -1 a 2 1 2 z b 2 -1 a 2 1 2 -1 2 n [/itex]

 Rule Form

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

 2001-10-29