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 Erf

 http://functions.wolfram.com/06.25.21.0086.01

 Input Form

 Integrate[z^n Cosh[b z] Erf[a z], z] == ((-1)^n/(b^(2 n) (2 b Sqrt[Pi]))) Exp[b^2/(4 a^2)] ((-E^(-(b^2/(4 a^2)))) Sqrt[Pi] Erf[a z] (-b)^n (Gamma[1 + n, b z] - (-1)^n Gamma[1 + n, (-b) z]) + a b^n n! Sum[((-b)^m/m!) (-a^2)^((1/2) (-1 - m)) Sum[Binomial[m, k] (-(b/(2 Sqrt[-a^2])))^(m - k) (Sqrt[-a^2] z + b/(2 Sqrt[-a^2]))^(1 + k) (-(Sqrt[-a^2] z + b/(2 Sqrt[-a^2]))^2)^((1/2) (-1 - k)) Gamma[(1 + k)/2, -(Sqrt[-a^2] z + b/(2 Sqrt[-a^2]))^2], {k, 0, m}], {m, 0, n}] - a (-b)^n n! Sum[(b^m/m!) (-a^2)^((1/2) (-1 - m)) Sum[Binomial[m, k] (b/(2 Sqrt[-a^2]))^(m - k) (Sqrt[-a^2] z - b/(2 Sqrt[-a^2]))^(1 + k) (-(Sqrt[-a^2] z - b/(2 Sqrt[-a^2]))^2)^((1/2) (-1 - k)) Gamma[(1 + k)/2, -(Sqrt[-a^2] z - 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 cosh ( b z ) erf ( a z ) z ( - 1 ) n 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 cosh ( b z ) erf ( a z ) z ( - 1 ) n 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

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