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Sin






Mathematica Notation

Traditional Notation









Elementary Functions > Sin[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving power function > Involving power > Involving zalpha-1and arguments a z





http://functions.wolfram.com/01.06.21.0051.01









  


  










Input Form





Integrate[z^(n + 1/2) Sin[a z], z] == (-(1/(2 Sqrt[a^2 z^2]))) (I (-1)^n (I a)^(-1 - n) Sqrt[z] ((Sqrt[I a z] Erfc[Sqrt[(-I) a z]] + (-1)^n Sqrt[(-I) a z] Erfc[Sqrt[I a z]]) Gamma[3/2 + n] + E^(I a z) Sqrt[I a z] (Sum[((-I) a z)^(1/2 + k)/Pochhammer[3/2 + n, k - n], {k, 0, n}] - Sum[((-I) a z)^(1/2 + k)/Pochhammer[3/2 + n, k - n], {k, 1 + n, -1}]) + ((-1)^n Sqrt[(-I) a z] (Sum[(I a z)^(1/2 + k)/Pochhammer[3/2 + n, k - n], {k, 0, n}] - Sum[(I a z)^(1/2 + k)/Pochhammer[3/2 + n, k - n], {k, 1 + n, -1}]))/ E^(I a z))) /; Element[n, Integers]










Standard Form





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MathML Form







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<apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </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_", "+", FractionBox["1", "2"]]]], " ", RowBox[List["Sin", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "a"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]], " ", RowBox[List["Erfc", "[", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]]], " ", RowBox[List["Erfc", "[", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]], "]"]]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "n"]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["k", "-", "n"]]]], "]"]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["1", "+", "n"]]]], RowBox[List["-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["k", "-", "n"]]]], "]"]]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["k", "-", "n"]]]], "]"]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["1", "+", "n"]]]], RowBox[List["-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["k", "-", "n"]]]], "]"]]]]]]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]]]










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





2002-12-18