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; }

 SinIntegral

 http://functions.wolfram.com/06.37.21.0064.01

 Input Form

 Integrate[z^3 ExpIntegralEi[b z] SinIntegral[a z], z] == (-(I/4)) ((1/b^4) (3 (ExpIntegralEi[((-I) a + b) z] - ExpIntegralEi[(I a + b) z])) - (1/(2 b^3)) ((6 E^(((-I) a + b) z))/((-I) a + b) - (6 E^((I a + b) z))/(I a + b) - (3 b Gamma[2, (-I) a z - b z])/(I a + b)^2 - (3 b Gamma[2, I a z - b z])/ (a + I b)^2 - (b^2 Gamma[3, (-I) a z - b z])/(I a + b)^3 + (b^2 Gamma[3, I a z - b z])/((-I) a + b)^3) - (1/(2 a^4)) ((6 a E^(((-I) a + b) z))/(a + I b) - (6 a E^((I a + b) z))/(a - I b) - 6 ExpIntegralEi[((-I) a + b) z] + 6 ExpIntegralEi[(I a + b) z] + (3 a^2 Gamma[2, (-I) a z - b z])/ (I a + b)^2 + (3 a^2 Gamma[2, I a z - b z])/(a + I b)^2 - (a^3 Gamma[3, (-I) a z - b z])/(a - I b)^3 + (a^3 Gamma[3, I a z - b z])/ (a + I b)^3 - ExpIntegralEi[b z] Gamma[4, (-I) a z] + ExpIntegralEi[b z] Gamma[4, I a z]) + I (z^4 ExpIntegralEi[b z] + Gamma[4, (-b) z]/b^4) SinIntegral[a z])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "3"], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List["b", " ", "z"]], "]"]], RowBox[List["SinIntegral", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["\[ImaginaryI]", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", SuperscriptBox["b", "4"]], RowBox[List["(", RowBox[List["3", " ", RowBox[List["(", RowBox[List[RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b"]], ")"]], " ", "z"]], "]"]], "-", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]], ")"]], " ", "z"]], "]"]]]], ")"]]]], ")"]]]], "-", RowBox[List[FractionBox["1", RowBox[List["2", " ", SuperscriptBox["b", "3"]]]], RowBox[List["(", RowBox[List[FractionBox[RowBox[List["6", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b"]], ")"]], " ", "z"]]]]], RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b"]]], "-", FractionBox[RowBox[List["6", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]], ")"]], " ", "z"]]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]]], "-", FractionBox[RowBox[List["3", " ", "b", " ", RowBox[List["Gamma", "[", RowBox[List["2", ",", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]], "-", RowBox[List["b", " ", "z"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]], ")"]], "2"]], "-", FractionBox[RowBox[List["3", " ", "b", " ", RowBox[List["Gamma", "[", RowBox[List["2", ",", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]], "-", RowBox[List["b", " ", "z"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], "2"]], "-", FractionBox[RowBox[List[SuperscriptBox["b", "2"], " ", RowBox[List["Gamma", "[", RowBox[List["3", ",", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]], "-", RowBox[List["b", " ", "z"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]], ")"]], "3"]], "+", FractionBox[RowBox[List[SuperscriptBox["b", "2"], " ", RowBox[List["Gamma", "[", RowBox[List["3", ",", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]], "-", RowBox[List["b", " ", "z"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b"]], ")"]], "3"]]]], ")"]]]], "-", RowBox[List[FractionBox["1", RowBox[List["2", " ", SuperscriptBox["a", "4"]]]], RowBox[List["(", RowBox[List[FractionBox[RowBox[List["6", " ", "a", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b"]], ")"]], " ", "z"]]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]], "-", FractionBox[RowBox[List["6", " ", "a", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]], ")"]], " ", "z"]]]]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]], "-", RowBox[List["6", " ", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b"]], ")"]], " ", "z"]], "]"]]]], "+", RowBox[List["6", " ", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]], ")"]], " ", "z"]], "]"]]]], "+", FractionBox[RowBox[List["3", " ", SuperscriptBox["a", "2"], " ", RowBox[List["Gamma", "[", RowBox[List["2", ",", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]], "-", RowBox[List["b", " ", "z"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]], ")"]], "2"]], "+", FractionBox[RowBox[List["3", " ", SuperscriptBox["a", "2"], " ", RowBox[List["Gamma", "[", RowBox[List["2", ",", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]], "-", RowBox[List["b", " ", "z"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], "2"]], "-", FractionBox[RowBox[List[SuperscriptBox["a", "3"], " ", RowBox[List["Gamma", "[", RowBox[List["3", ",", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]], "-", RowBox[List["b", " ", "z"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], "3"]], "+", FractionBox[RowBox[List[SuperscriptBox["a", "3"], " ", RowBox[List["Gamma", "[", RowBox[List["3", ",", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]], "-", RowBox[List["b", " ", "z"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], "3"]], "-", RowBox[List[RowBox[List["ExpIntegralEi", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["4", ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]]]], "]"]]]], "+", RowBox[List[RowBox[List["ExpIntegralEi", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["4", ",", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]]], "]"]]]]]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["z", "4"], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List["b", " ", "z"]], "]"]]]], "+", FractionBox[RowBox[List["Gamma", "[", RowBox[List["4", ",", RowBox[List[RowBox[List["-", "b"]], " ", "z"]]]], "]"]], SuperscriptBox["b", "4"]]]], ")"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["a", " ", "z"]], "]"]]]]]], ")"]]]]]]]]

 MathML Form

 z 3 Ei ( b z ) Si ( a z ) z - 4 ( - 1 2 b 3 ( Γ ( 3 , a z - b z ) b 2 ( b - a ) 3 - Γ ( 3 , - b z + a ( - ) z ) b 2 ( b + a ) 3 - 3 Γ ( 2 , - b z + a ( - ) z ) b ( b + a ) 2 - 3 Γ ( 2 , a z - b z ) b ( a + b ) 2 + 6 ( b - a ) z b - a - 6 ( b + a ) z b + a ) + 3 ( Ei ( ( b - a ) z ) - Ei ( ( b + a ) z ) ) b 4 - 1 2 a 4 ( Γ ( 3 , a z - b z ) a 3 ( a + b ) 3 - Γ ( 3 , - b z + a ( - ) z ) a 3 ( a - b ) 3 + 3 Γ ( 2 , - b z + a ( - ) z ) a 2 ( b + a ) 2 + 3 Γ ( 2 , a z - b z ) a 2 ( a + b ) 2 + 6 ( b - a ) z a a + b - 6 ( b + a ) z a a - b - 6 Ei ( ( b - a ) z ) + 6 Ei ( ( b + a ) z ) - Ei ( b z ) Γ ( 4 , - a z ) + Ei ( b z ) Γ ( 4 , a z ) ) + ( Ei ( b z ) z 4 + Γ ( 4 , - b z ) b 4 ) Si ( a z ) ) z z 3 ExpIntegralEi b z SinIntegral a z -1 4 -1 -1 1 2 b 3 -1 Gamma 3 a z -1 b z b 2 b -1 a 3 -1 -1 Gamma 3 -1 b z a -1 z b 2 b a 3 -1 -1 3 Gamma 2 -1 b z a -1 z b b a 2 -1 -1 3 Gamma 2 a z -1 b z b a b 2 -1 6 b -1 a z b -1 a -1 -1 6 b a z b a -1 3 ExpIntegralEi b -1 a z -1 ExpIntegralEi b a z b 4 -1 -1 1 2 a 4 -1 Gamma 3 a z -1 b z a 3 a b 3 -1 -1 Gamma 3 -1 b z a -1 z a 3 a -1 b 3 -1 3 Gamma 2 -1 b z a -1 z a 2 b a 2 -1 3 Gamma 2 a z -1 b z a 2 a b 2 -1 6 b -1 a z a a b -1 -1 6 b a z a a -1 b -1 -1 6 ExpIntegralEi b -1 a z 6 ExpIntegralEi b a z -1 ExpIntegralEi b z Gamma 4 -1 a z ExpIntegralEi b z Gamma 4 a z ExpIntegralEi b z z 4 Gamma 4 -1 b z b 4 -1 SinIntegral a z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "3"], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["-", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["3", " ", RowBox[List["(", RowBox[List[RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b"]], ")"]], " ", "z"]], "]"]], "-", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]], ")"]], " ", "z"]], "]"]]]], ")"]]]], SuperscriptBox["b", "4"]], "-", FractionBox[RowBox[List[FractionBox[RowBox[List["6", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b"]], ")"]], " ", "z"]]]]], RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b"]]], "-", FractionBox[RowBox[List["6", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]], ")"]], " ", "z"]]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]]], "-", FractionBox[RowBox[List["3", " ", "b", " ", RowBox[List["Gamma", "[", RowBox[List["2", ",", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]], "-", RowBox[List["b", " ", "z"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]], ")"]], "2"]], "-", FractionBox[RowBox[List["3", " ", "b", " ", RowBox[List["Gamma", "[", RowBox[List["2", ",", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]], "-", RowBox[List["b", " ", "z"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], "2"]], "-", FractionBox[RowBox[List[SuperscriptBox["b", "2"], " ", RowBox[List["Gamma", "[", RowBox[List["3", ",", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]], "-", RowBox[List["b", " ", "z"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]], ")"]], "3"]], "+", FractionBox[RowBox[List[SuperscriptBox["b", "2"], " ", RowBox[List["Gamma", "[", RowBox[List["3", ",", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]], "-", RowBox[List["b", " ", "z"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b"]], ")"]], "3"]]]], RowBox[List["2", " ", SuperscriptBox["b", "3"]]]], "-", FractionBox[RowBox[List[FractionBox[RowBox[List["6", " ", "a", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b"]], ")"]], " ", "z"]]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]], "-", FractionBox[RowBox[List["6", " ", "a", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]], ")"]], " ", "z"]]]]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]], "-", RowBox[List["6", " ", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b"]], ")"]], " ", "z"]], "]"]]]], "+", RowBox[List["6", " ", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]], ")"]], " ", "z"]], "]"]]]], "+", FractionBox[RowBox[List["3", " ", SuperscriptBox["a", "2"], " ", RowBox[List["Gamma", "[", RowBox[List["2", ",", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]], "-", RowBox[List["b", " ", "z"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]], ")"]], "2"]], "+", FractionBox[RowBox[List["3", " ", SuperscriptBox["a", "2"], " ", RowBox[List["Gamma", "[", RowBox[List["2", ",", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]], "-", RowBox[List["b", " ", "z"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], "2"]], "-", FractionBox[RowBox[List[SuperscriptBox["a", "3"], " ", RowBox[List["Gamma", "[", RowBox[List["3", ",", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]], "-", RowBox[List["b", " ", "z"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], "3"]], "+", FractionBox[RowBox[List[SuperscriptBox["a", "3"], " ", RowBox[List["Gamma", "[", RowBox[List["3", ",", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]], "-", RowBox[List["b", " ", "z"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], "3"]], "-", RowBox[List[RowBox[List["ExpIntegralEi", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["4", ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]]]], "]"]]]], "+", RowBox[List[RowBox[List["ExpIntegralEi", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["4", ",", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]]], "]"]]]]]], RowBox[List["2", " ", SuperscriptBox["a", "4"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["z", "4"], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List["b", " ", "z"]], "]"]]]], "+", FractionBox[RowBox[List["Gamma", "[", RowBox[List["4", ",", RowBox[List[RowBox[List["-", "b"]], " ", "z"]]]], "]"]], SuperscriptBox["b", "4"]]]], ")"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["a", " ", "z"]], "]"]]]]]], ")"]]]]]]]]

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

 2001-10-29