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/06.37.21.0063.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2001-10-29