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 SinIntegral

 http://functions.wolfram.com/06.37.21.0030.01

 Input Form

 Integrate[z^n Cos[a z] SinIntegral[a z], z] == (I^n/4) a^(-n - 1) (-2 I (Gamma[1 + n, (-I) a z] - (-1)^n Gamma[1 + n, I a z]) SinIntegral[a z] + n! ((-1)^n ExpIntegralEi[-2 I a z] + ExpIntegralEi[2 I a z] - (1 + (-1)^n) Log[z] - 2 Sum[(((-I) a z)^k/(2 k) + 2^(-1 - k) Gamma[k, -2 I a z])/k!, {k, 1, n}] - 2 (-1)^n Sum[((I a z)^k/(2 k) + 2^(-1 - k) Gamma[k, 2 I a z])/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["Cos", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["SinIntegral", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox["\[ImaginaryI]", "n"], "4"], SuperscriptBox["a", RowBox[List[RowBox[List["-", "n"]], "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", RowBox[List["(", " ", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]]]], "]"]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]]], "]"]]]]]], ")"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "a", " ", "z"]], "]"]]]], "+", " ", RowBox[List["ExpIntegralEi", "[", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "z"]], "]"]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List["2", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox[RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]], ")"]], "k"], RowBox[List["2", " ", "k"]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List["k", ",", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "a", " ", "z"]]]], "]"]]]]]], RowBox[List["k", "!"]]]]]]], "-", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox[RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]], ")"]], "k"], RowBox[List["2", " ", "k"]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List["k", ",", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "z"]]]], "]"]]]]]], RowBox[List["k", "!"]]]]]]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

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

 Rule Form

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

 2001-10-29