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 Erfi

 http://functions.wolfram.com/06.28.21.0076.01

 Input Form

 Integrate[z^n Sinh[b z] Erfi[a z], z] == ((-1)^(n - 1)/(b^(2 n) (2 b Sqrt[Pi]))) Exp[-(b^2/(4 a^2))] ((-E^(b^2/(4 a^2))) Sqrt[Pi] Erfi[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 sinh ( b z ) erfi ( a z ) z ( - 1 ) n - 1 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 erfi ( 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 Erfi a z -1 n -1 b -2 n 2 b 1 2 -1 -1 b 2 4 a 2 -1 -1 a n b n m 0 n 1 m -1 -1 b m a 2 1 2 -1 m -1 k 0 m Binomial m k -1 b 2 a 2 1 2 -1 m -1 k a 2 1 2 z b 2 a 2 1 2 -1 k 1 -1 a 2 1 2 z b 2 a 2 1 2 -1 2 1 2 -1 k -1 Gamma k 1 2 -1 -1 a 2 1 2 z b 2 a 2 1 2 -1 2 -1 -1 b n b 2 4 a 2 -1 1 2 Erfi a z Gamma n 1 b z -1 n Gamma n 1 -1 b z -1 a -1 b n n m 0 n 1 m -1 b m a 2 1 2 -1 m -1 k 0 m Binomial m k b 2 a 2 1 2 -1 m -1 k a 2 1 2 z -1 b 2 a 2 1 2 -1 k 1 -1 a 2 1 2 z -1 b 2 a 2 1 2 -1 2 1 2 -1 k -1 Gamma k 1 2 -1 -1 a 2 1 2 z -1 b 2 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

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