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/06.37.21.0024.01

 Input Form

 Integrate[z^n Sin[b z] SinIntegral[a z], z] == (-(I/4)) (I b)^(-2 - n) b n! ((-1)^n ExpIntegralEi[(-I) (a - b) z] - ExpIntegralEi[I (a - b) z] + ExpIntegralEi[(-I) (a + b) z] - ExpIntegralEi[I (a + b) z]/(-1)^n + (1/Gamma[2 + n]) (2 I (1 + n) ((-1)^n Gamma[1 + n, (-I) b z] + Gamma[1 + n, I b z]) SinIntegral[a z]) + E^(I (a - b) z) Sum[((b/(b - a))^m Sum[((-(I a) + I b)^k z^k)/k!, {k, 0, -1 + m}])/m, {m, 1, n}] - Sum[((b/(b + a))^m Sum[((I a + I b)^k z^k)/k!, {k, 0, -1 + m}])/m, {m, 1, n}]/E^(I (a + b) z) + (-1)^n E^(I (a + b) z) Sum[((b/(b + a))^m Sum[((-(I a) - I b)^k z^k)/k!, {k, 0, -1 + m}])/m, {m, 1, n}] - ((-1)^n Sum[((b/(b - a))^m Sum[((I a - I b)^k z^k)/k!, {k, 0, -1 + m}])/ m, {m, 1, n}])/E^(I (a - b) z)) /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], " ", RowBox[List["Sin", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["\[ImaginaryI]", "4"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b"]], ")"]], RowBox[List[RowBox[List["-", "2"]], "-", "n"]]], " ", "b", " ", RowBox[List["n", "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", "z"]], "]"]]]], "-", RowBox[List["ExpIntegralEi", "[", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", "z"]], "]"]], "+", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", "z"]], "]"]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["-", "n"]]], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", "z"]], "]"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["Gamma", "[", RowBox[List["2", "+", "n"]], "]"]]], RowBox[List["(", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", "z"]]]], "]"]]]], "+", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List["\[ImaginaryI]", " ", "b", " ", "z"]]]], "]"]]]], ")"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["a", " ", "z"]], "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", "z"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["b", RowBox[List["b", "-", "a"]]], ")"]], "m"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", "m"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "a"]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], "k"], " ", SuperscriptBox["z", "k"]]], RowBox[List["k", "!"]]]]]]], "m"]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", "z"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["b", RowBox[List["b", "+", "a"]]], ")"]], "m"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", "m"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], "k"], " ", SuperscriptBox["z", "k"]]], RowBox[List["k", "!"]]]]]]], "m"]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", "z"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["b", RowBox[List["b", "+", "a"]]], ")"]], "m"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", "m"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "a"]], ")"]]]], "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], "k"], " ", SuperscriptBox["z", "k"]]], RowBox[List["k", "!"]]]]]]], "m"]]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", "z"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["b", RowBox[List["b", "-", "a"]]], ")"]], "m"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", "m"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], "k"], " ", SuperscriptBox["z", "k"]]], RowBox[List["k", "!"]]]]]]], "m"]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 z n sin ( b z ) Si ( a z ) z - 4 ( b ) - n - 2 b n ! ( ( - 1 ) n Ei ( - ( a - b ) z ) - Ei ( ( a - b ) z ) + Ei ( - ( a + b ) z ) - ( - 1 ) - n Ei ( ( a + b ) z ) + 1 Γ ( n + 2 ) ( 2 ( n + 1 ) ( ( - 1 ) n Γ ( n + 1 , - b z ) + Γ ( n + 1 , b z ) ) Si ( a z ) ) - - ( a + b ) z m = 1 n ( b a + b ) m k = 0 m - 1 ( a + b ) k z k k ! m + ( a - b ) z m = 1 n ( b b - a ) m k = 0 m - 1 ( - ( a ) + b ) k z k k ! m - ( - 1 ) n - ( a - b ) z m = 1 n ( b b - a ) m k = 0 m - 1 ( a - b ) k z k k ! m + ( - 1 ) n ( a + b ) z m = 1 n ( b a + b ) m k = 0 m - 1 ( - ( a ) - b ) k z k k ! m ) /; n Condition z z n b z SinIntegral a z -1 4 -1 b -1 n -2 b n -1 n ExpIntegralEi -1 a -1 b z -1 ExpIntegralEi a -1 b z ExpIntegralEi -1 a b z -1 -1 -1 n ExpIntegralEi a b z 1 Gamma n 2 -1 2 n 1 -1 n Gamma n 1 -1 b z Gamma n 1 b z SinIntegral a z -1 -1 a b z m 1 n b a b -1 m k 0 m -1 a b k z k k -1 m -1 a -1 b z m 1 n b b -1 a -1 m k 0 m -1 -1 a b k z k k -1 m -1 -1 -1 n -1 a -1 b z m 1 n b b -1 a -1 m k 0 m -1 a -1 b k z k k -1 m -1 -1 n a b z m 1 n b a b -1 m k 0 m -1 -1 a -1 b k z k k -1 m -1 n [/itex]

 Rule Form

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

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

 2001-10-29