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 Erfc

 http://functions.wolfram.com/06.27.21.0075.01

 Input Form

 Integrate[z^n Sinh[b z] Erfc[a z], z] == (b^(-n - 1)/2) Erfc[a z] ((-1)^n Gamma[1 + n, (-b) z] + Gamma[1 + n, b z]) + (((-b)^(-n - 1) a n!)/(2 Sqrt[Pi])) Exp[b^2/(4 a^2)] (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}] + (-1)^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 ) erfc ( a z ) z 1 2 b - n - 1 erfc ( a z ) ( ( - 1 ) n Γ ( n + 1 , - b z ) + Γ ( n + 1 , b z ) ) + ( - b ) - n - 1 a n ! 2 π exp ( b 2 4 a 2 ) ( 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 ) + ( - 1 ) 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 Erfc a z 1 2 b -1 n -1 Erfc a z -1 n Gamma n 1 -1 b z Gamma n 1 b z -1 b -1 n -1 a n 2 1 2 -1 b 2 4 a 2 -1 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 -1 n m 0 n 1 m -1 b m -1 a 2 1 2 -1 m -1 k 0 m Binomial m k 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 n [/itex]

 Rule Form

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

 2001-10-29