html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Erfi

 http://functions.wolfram.com/06.28.21.0100.01

 Input Form

 Integrate[z^n E^(b z) Sinh[c z] Erfi[a z], z] == (1/2) (-b + c)^(-1 - n) Erfi[a z] Gamma[1 + n, (-b + c) z] - (1/2) (-b - c)^(-1 - n) Erfi[a z] Gamma[1 + n, (-b - c) z] - (1/(2 Sqrt[Pi])) ((a (-b - c)^(-1 - n) 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}])/E^((b + c)^2/(4 a^2))) + (1/(2 Sqrt[Pi])) ((a (-b + c)^(-1 - n) 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}])/E^((b - c)^2/(4 a^2))) /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", "z"]]], " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], RowBox[List["Erfi", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", "2"], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["Erfi", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], " ", "z"]]]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["Erfi", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", "c"]], ")"]], " ", "z"]]]], "]"]]]], "-", RowBox[List[FractionBox["1", RowBox[List["2", " ", SqrtBox["\[Pi]"]]]], RowBox[List["(", RowBox[List["a", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"], RowBox[List["4", " ", SuperscriptBox["a", "2"]]]]]]], " ", RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "n"], RowBox[List[FractionBox["1", RowBox[List["m", "!"]]], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]]]], ")"]], "m"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["a", "2"], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "m"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["b", "+", "c"]], RowBox[List["2", " ", SqrtBox[SuperscriptBox["a", "2"]]]]]]], ")"]], RowBox[List["m", "-", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[SuperscriptBox["a", "2"]], " ", "z"]], "+", FractionBox[RowBox[List["b", "+", "c"]], RowBox[List["2", " ", SqrtBox[SuperscriptBox["a", "2"]]]]]]], ")"]], RowBox[List["1", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[SuperscriptBox["a", "2"]], " ", "z"]], "+", FractionBox[RowBox[List["b", "+", "c"]], RowBox[List["2", " ", SqrtBox[SuperscriptBox["a", "2"]]]]]]], ")"]], "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "k"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "k"]], "2"], ",", RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[SuperscriptBox["a", "2"]], " ", "z"]], "+", FractionBox[RowBox[List["b", "+", "c"]], RowBox[List["2", SqrtBox[SuperscriptBox["a", "2"]]]]]]], ")"]], "2"]]]]], "]"]]]]]]]], ")"]]]]]]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["2", " ", SqrtBox["\[Pi]"]]]], RowBox[List["(", RowBox[List["a", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"], RowBox[List["4", " ", SuperscriptBox["a", "2"]]]]]]], " ", RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "n"], RowBox[List[FractionBox["1", RowBox[List["m", "!"]]], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]]]], ")"]], "m"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["a", "2"], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "m"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["b", "-", "c"]], RowBox[List["2", " ", SqrtBox[SuperscriptBox["a", "2"]]]]]]], ")"]], RowBox[List["m", "-", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[SuperscriptBox["a", "2"]], " ", "z"]], "+", FractionBox[RowBox[List["b", "-", "c"]], RowBox[List["2", " ", SqrtBox[SuperscriptBox["a", "2"]]]]]]], ")"]], RowBox[List["1", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[SuperscriptBox["a", "2"]], " ", "z"]], "+", FractionBox[RowBox[List["b", "-", "c"]], RowBox[List["2", " ", SqrtBox[SuperscriptBox["a", "2"]]]]]]], ")"]], "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "k"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "k"]], "2"], ",", RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[SuperscriptBox["a", "2"]], " ", "z"]], "+", FractionBox[RowBox[List["b", "-", "c"]], RowBox[List["2", SqrtBox[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 ) erfi ( a z ) z - 1 2 ( - c - b ) - n - 1 erfi ( a z ) Γ ( n + 1 , ( - c - b ) z ) + 1 2 ( - b + c ) - n - 1 erfi ( a z ) 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 Erfi a z -1 1 2 -1 c -1 b -1 n -1 Erfi a z Gamma n 1 -1 c -1 b z 1 2 -1 b c -1 n -1 Erfi a z Gamma n 1 -1 b c z 1 2 1 2 -1 a -1 b c -1 n -1 -1 b -1 c 2 4 a 2 -1 n m 0 n 1 m -1 -1 b -1 c m a 2 1 2 -1 m -1 k 0 m Binomial m k -1 b -1 c 2 a 2 1 2 -1 m -1 k b -1 c 2 a 2 1 2 -1 a 2 1 2 z k 1 -1 b -1 c 2 a 2 1 2 -1 a 2 1 2 z 2 1 2 -1 k -1 Gamma k 1 2 -1 -1 b -1 c 2 a 2 1 2 -1 a 2 1 2 z 2 -1 1 2 1 2 -1 a -1 c -1 b -1 n -1 -1 b c 2 4 a 2 -1 n m 0 n 1 m -1 -1 b c m a 2 1 2 -1 m -1 k 0 m Binomial m k -1 b c 2 a 2 1 2 -1 m -1 k b c 2 a 2 1 2 -1 a 2 1 2 z k 1 -1 b c 2 a 2 1 2 -1 a 2 1 2 z 2 1 2 -1 k -1 Gamma k 1 2 -1 -1 b c 2 a 2 1 2 -1 a 2 1 2 z 2 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "n_"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b_", " ", "z_"]]], " ", RowBox[List["Sinh", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", RowBox[List["Erfi", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["Erfi", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], " ", "z"]]]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["Erfi", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", "c"]], ")"]], " ", "z"]]]], "]"]]]], "-", FractionBox[RowBox[List["a", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"], RowBox[List["4", " ", SuperscriptBox["a", "2"]]]]]]], " ", RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]]]], ")"]], "m"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["a", "2"], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "m"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["b", "+", "c"]], RowBox[List["2", " ", SqrtBox[SuperscriptBox["a", "2"]]]]]]], ")"]], RowBox[List["m", "-", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[SuperscriptBox["a", "2"]], " ", "z"]], "+", FractionBox[RowBox[List["b", "+", "c"]], RowBox[List["2", " ", SqrtBox[SuperscriptBox["a", "2"]]]]]]], ")"]], RowBox[List["1", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[SuperscriptBox["a", "2"]], " ", "z"]], "+", FractionBox[RowBox[List["b", "+", "c"]], RowBox[List["2", " ", SqrtBox[SuperscriptBox["a", "2"]]]]]]], ")"]], "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "k"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "k"]], "2"], ",", RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[SuperscriptBox["a", "2"]], " ", "z"]], "+", FractionBox[RowBox[List["b", "+", "c"]], RowBox[List["2", " ", SqrtBox[SuperscriptBox["a", "2"]]]]]]], ")"]], "2"]]]]], "]"]]]]]]]], RowBox[List["m", "!"]]]]]]], RowBox[List["2", " ", SqrtBox["\[Pi]"]]]], "+", FractionBox[RowBox[List["a", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"], RowBox[List["4", " ", SuperscriptBox["a", "2"]]]]]]], " ", RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]]]], ")"]], "m"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["a", "2"], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "m"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["b", "-", "c"]], RowBox[List["2", " ", SqrtBox[SuperscriptBox["a", "2"]]]]]]], ")"]], RowBox[List["m", "-", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[SuperscriptBox["a", "2"]], " ", "z"]], "+", FractionBox[RowBox[List["b", "-", "c"]], RowBox[List["2", " ", SqrtBox[SuperscriptBox["a", "2"]]]]]]], ")"]], RowBox[List["1", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[SuperscriptBox["a", "2"]], " ", "z"]], "+", FractionBox[RowBox[List["b", "-", "c"]], RowBox[List["2", " ", SqrtBox[SuperscriptBox["a", "2"]]]]]]], ")"]], "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "k"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "k"]], "2"], ",", RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[SuperscriptBox["a", "2"]], " ", "z"]], "+", FractionBox[RowBox[List["b", "-", "c"]], RowBox[List["2", " ", SqrtBox[SuperscriptBox["a", "2"]]]]]]], ")"]], "2"]]]]], "]"]]]]]]]], RowBox[List["m", "!"]]]]]]], RowBox[List["2", " ", SqrtBox["\[Pi]"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2001-10-29