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 Erfc

 http://functions.wolfram.com/06.27.21.0099.01

 Input Form

 Integrate[z^n E^(b z) Sinh[c z] Erfc[a z], z] == (1/2) Erfc[a z] ((-b + c)^(-1 - n) Gamma[1 + n, (-b + c) z] - (-b - c)^(-1 - n) Gamma[1 + n, (-(b + c)) z]) + (1/(2 Sqrt[Pi])) (a (-b - c)^(-1 - n) E^((b + c)^2/(4 a^2)) n! Sum[(1/m!) ((-(b + c))^m (-a^2)^((1/2) (-1 - m)) Sum[Binomial[m, k] (-((b + c)/(2 Sqrt[-a^2])))^(m - k) (Sqrt[-a^2] z + (b + c)/(2 Sqrt[-a^2]))^(1 + k) (-(Sqrt[-a^2] z + (b + c)/(2 Sqrt[-a^2]))^2)^((1/2) (-1 - k)) Gamma[(1 + k)/2, -(Sqrt[-a^2] z + (b + c)/(2 Sqrt[-a^2]))^2], {k, 0, m}]), {m, 0, n}]) - (1/(2 Sqrt[Pi])) (a (-b + c)^(-1 - n) E^((b - c)^2/(4 a^2)) n! Sum[(1/m!) ((-(b - c))^m (-a^2)^((1/2) (-1 - m)) Sum[Binomial[m, k] (-((b - c)/(2 Sqrt[-a^2])))^(m - k) (Sqrt[-a^2] z + (b - c)/(2 Sqrt[-a^2]))^(1 + k) (-(Sqrt[-a^2] z + (b - c)/(2 Sqrt[-a^2]))^2)^((1/2) (-1 - k)) Gamma[(1 + k)/2, -(Sqrt[-a^2] z + (b - c)/(2 Sqrt[-a^2]))^2], {k, 0, m}]), {m, 0, n}]) /; Element[n, Integers] && n >= 0

 Standard Form

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FractionBox[RowBox[List["b", "-", "c"]], RowBox[List["2", SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]]]]]]], ")"]], "2"]]]]], "]"]]]]]]]], ")"]]]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 z n b z sinh ( c z ) erfc ( a z ) z 1 2 erfc ( a z ) ( ( c - b ) - n - 1 Γ ( n + 1 , ( c - b ) z ) - ( - b - c ) - n - 1 Γ ( n + 1 , - ( b + c ) z ) ) - 1 2 π ( a ( - b + c ) - n - 1 ( b - c ) 2 4 a 2 n ! m = 0 n 1 m ! ( ( - ( b - c ) ) 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 - c 2 - a 2 ) m - k ( b - c 2 - a 2 + - a 2 z ) k + 1 ( - ( b - c 2 - a 2 + - a 2 z ) 2 ) 1 2 ( - k - 1 ) Γ ( k + 1 2 , - ( b - c 2 - a 2 + - a 2 z ) 2 ) ) ) + 1 2 π ( a ( - c - b ) - n - 1 ( b + c ) 2 4 a 2 n ! m = 0 n 1 m ! ( ( - ( b + c ) ) 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 + c 2 - a 2 ) m - k ( b + c 2 - a 2 + - a 2 z ) k + 1 ( - ( b + c 2 - a 2 + - a 2 z ) 2 ) 1 2 ( - k - 1 ) Γ ( k + 1 2 , - ( b + c 2 - a 2 + - a 2 z ) 2 ) ) ) /; n Condition z z n b z c z Erfc a z 1 2 Erfc a z c -1 b -1 n -1 Gamma n 1 c -1 b z -1 -1 b -1 c -1 n -1 Gamma n 1 -1 b c z -1 1 2 1 2 -1 a -1 b c -1 n -1 b -1 c 2 4 a 2 -1 n m 0 n 1 m -1 -1 b -1 c m -1 a 2 1 2 -1 m -1 k 0 m Binomial m k -1 b -1 c 2 -1 a 2 1 2 -1 m -1 k b -1 c 2 -1 a 2 1 2 -1 -1 a 2 1 2 z k 1 -1 b -1 c 2 -1 a 2 1 2 -1 -1 a 2 1 2 z 2 1 2 -1 k -1 Gamma k 1 2 -1 -1 b -1 c 2 -1 a 2 1 2 -1 -1 a 2 1 2 z 2 1 2 1 2 -1 a -1 c -1 b -1 n -1 b c 2 4 a 2 -1 n m 0 n 1 m -1 -1 b c m -1 a 2 1 2 -1 m -1 k 0 m Binomial m k -1 b c 2 -1 a 2 1 2 -1 m -1 k b c 2 -1 a 2 1 2 -1 -1 a 2 1 2 z k 1 -1 b c 2 -1 a 2 1 2 -1 -1 a 2 1 2 z 2 1 2 -1 k -1 Gamma k 1 2 -1 -1 b c 2 -1 a 2 1 2 -1 -1 a 2 1 2 z 2 n [/itex]

 Rule Form

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

 2001-10-29