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Cos






Mathematica Notation

Traditional Notation









Elementary Functions > Cos[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving exponential function and a power function > Involving exp and power > Involving zalpha-1 eb z cos(c z)





http://functions.wolfram.com/01.07.21.0297.01









  


  










Input Form





Integrate[(E^(b z) Cos[c z])/z^n, z] == -(((-(-1)^n) E^(2 I c z) ((-b + I c)^n (b + I c) ExpIntegralEi[ (b - I c) z] + (-b - I c)^n (b - I c) ExpIntegralEi[(b + I c) z]) + (-b - I c)^n (b - I c) E^((b + 3 I c) z) (-1 + n)! Sum[((-b - I c)^(k - n) z^(k - n))/Pochhammer[1 - n, k], {k, 1, -1 + n}] + (-b + I c)^n (b + I c) E^((b + I c) z) (-1 + n)! Sum[((-b + I c)^(k - n) z^(k - n))/Pochhammer[1 - n, k], {k, 1, -1 + n}])/E^(2 I c z))/(2 (b^2 + c^2) (-1 + n)!) /; Element[n, Integers] && n > 0










Standard Form





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MathML Form







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<times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["-", "n_"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b_", " ", "z_"]]], " ", RowBox[List["Cos", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "c", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "n"], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", "z"]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "n"], " ", RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", "z"]], "]"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "n"], " ", RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["3", " ", "\[ImaginaryI]", " ", "c"]]]], ")"]], " ", "z"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]], "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], RowBox[List["k", "-", "n"]]], " ", SuperscriptBox["z", RowBox[List["k", "-", "n"]]]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "n"]], ",", "k"]], "]"]]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "n"], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", "z"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]], "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], RowBox[List["k", "-", "n"]]], " ", SuperscriptBox["z", RowBox[List["k", "-", "n"]]]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "n"]], ",", "k"]], "]"]]]]]]]]], ")"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "+", SuperscriptBox["c", "2"]]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]], "!"]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]










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





2002-12-18