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/06.37.21.0061.01

 Input Form

 Integrate[z^n ExpIntegralEi[b z] SinIntegral[a z], z] == (1/(1 + n)) (z^(1 + n) ExpIntegralEi[b z] + (-b)^(-1 - n) Gamma[1 + n, (-b) z]) SinIntegral[a z] + (I/(1 + n)) ((1/2) (I a)^(-1 - n) ((-ExpIntegralEi[((-I) a + b) z]) n! + ExpIntegralEi[b z] Gamma[1 + n, I a z] + (-1)^n ((-ExpIntegralEi[(I a + b) z]) n! + ExpIntegralEi[b z] Gamma[1 + n, (-I) a z] + n! Sum[(a^k Gamma[k, (-(b + I a)) z])/ ((a - I b)^k k!), {k, 1, n}]) + n! Sum[(a^k Gamma[k, (I a - b) z])/((a + I b)^k k!), {k, 1, n}]) + (1/2) (-b)^(-1 - n) n! (-ExpIntegralEi[(-I) a z + b z] + ExpIntegralEi[I a z + b z] + Sum[(1/k!) (b^k (Gamma[k, (I a - b) z]/(b - I a)^k - Gamma[k, (-(I a + b)) z]/(b + I a)^k)), {k, 1, n}])) /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["1", "+", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List["1", "+", "n"]]], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List["b", " ", "z"]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "b"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List[RowBox[List["-", "b"]], " ", "z"]]]], "]"]]]]]], ")"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List[FractionBox["\[ImaginaryI]", RowBox[List["1", "+", "n"]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "a"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["(", 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["n", "!"]]]], "+", RowBox[List[RowBox[List["ExpIntegralEi", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]], ")"]], " ", "z"]], "]"]]]], " ", RowBox[List["n", "!"]]]], "+", RowBox[List[RowBox[List["ExpIntegralEi", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]]]], "]"]]]], "+", RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox[RowBox[List[SuperscriptBox["a", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], RowBox[List["-", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List["k", ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "a"]]]], ")"]]]], " ", "z"]]]], "]"]]]], RowBox[List["k", "!"]]]]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox[RowBox[List[SuperscriptBox["a", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], RowBox[List["-", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List["k", ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "-", "b"]], ")"]], " ", "z"]]]], "]"]]]], RowBox[List["k", "!"]]]]]]]]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "b"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["n", "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]], "+", RowBox[List["b", " ", "z"]]]], "]"]]]], "+", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]], "+", RowBox[List["b", " ", "z"]]]], "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[FractionBox["1", RowBox[List["k", "!"]]], RowBox[List["(", RowBox[List[SuperscriptBox["b", "k"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "a"]]]], ")"]], RowBox[List["-", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List["k", ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "-", "b"]], ")"]], " ", "z"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "a"]]]], ")"]], RowBox[List["-", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List["k", ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]], ")"]]]], " ", "z"]]]], "]"]]]]]], ")"]]]], ")"]]]]]]]], ")"]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 z n Ei ( b z ) Si ( a z ) z ( Γ ( n + 1 , - b z ) ( - b ) - n - 1 + z n + 1 Ei ( b z ) ) Si ( a z ) n + 1 + n + 1 ( ( a ) - n - 1 2 ( - Ei ( ( b - a ) z ) n ! + n ! k = 1 n a k ( a + b ) - k Γ ( k , ( a - b ) z ) k ! + Ei ( b z ) Γ ( n + 1 , a z ) + ( - 1 ) n ( - Ei ( ( b + a ) z ) n ! + n ! k = 1 n a k ( a - b ) - k Γ ( k , - ( b + a ) z ) k ! + Ei ( b z ) Γ ( n + 1 , - a z ) ) ) + 1 2 ( - b ) - n - 1 n ! ( - Ei ( b z - a z ) + Ei ( b z + a z ) + k = 1 n 1 k ! ( b k ( ( b - a ) - k Γ ( k , ( a - b ) z ) - ( b + a ) - k Γ ( k , - ( b + a ) z ) ) ) ) ) /; n Condition z z n ExpIntegralEi b z SinIntegral a z Gamma n 1 -1 b z -1 b -1 n -1 z n 1 ExpIntegralEi b z SinIntegral a z n 1 -1 n 1 -1 a -1 n -1 2 -1 -1 ExpIntegralEi b -1 a z n n k 1 n a k a b -1 k Gamma k a -1 b z k -1 ExpIntegralEi b z Gamma n 1 a z -1 n -1 ExpIntegralEi b a z n n k 1 n a k a -1 b -1 k Gamma k -1 b a z k -1 ExpIntegralEi b z Gamma n 1 -1 a z 1 2 -1 b -1 n -1 n -1 ExpIntegralEi b z -1 a z ExpIntegralEi b z a z k 1 n 1 k -1 b k b -1 a -1 k Gamma k a -1 b z -1 b a -1 k Gamma k -1 b a z n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "n_"], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List["1", "+", "n"]]], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List["b", " ", "z"]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "b"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List[RowBox[List["-", "b"]], " ", "z"]]]], "]"]]]]]], ")"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["a", " ", "z"]], "]"]]]], RowBox[List["1", "+", "n"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "a"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["(", 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["n", "!"]]]], "+", RowBox[List[RowBox[List["ExpIntegralEi", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]], ")"]], " ", "z"]], "]"]]]], " ", RowBox[List["n", "!"]]]], "+", RowBox[List[RowBox[List["ExpIntegralEi", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]]]], "]"]]]], "+", RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox[RowBox[List[SuperscriptBox["a", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], RowBox[List["-", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List["k", ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "a"]]]], ")"]]]], " ", "z"]]]], "]"]]]], RowBox[List["k", "!"]]]]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox[RowBox[List[SuperscriptBox["a", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], RowBox[List["-", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List["k", ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "-", "b"]], ")"]], " ", "z"]]]], "]"]]]], RowBox[List["k", "!"]]]]]]]]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "b"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["n", "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]], "+", RowBox[List["b", " ", "z"]]]], "]"]]]], "+", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]], "+", RowBox[List["b", " ", "z"]]]], "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox[RowBox[List[SuperscriptBox["b", "k"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "a"]]]], ")"]], RowBox[List["-", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List["k", ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "-", "b"]], ")"]], " ", "z"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "a"]]]], ")"]], RowBox[List["-", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List["k", ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]], ")"]]]], " ", "z"]]]], "]"]]]]]], ")"]]]], RowBox[List["k", "!"]]]]]]], ")"]]]]]], ")"]]]], RowBox[List["1", "+", "n"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2001-10-29