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http://functions.wolfram.com/01.03.21.0717.01
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Integrate[z^n (E^(c z))^\[Mu] (E^(a z + b))^\[Nu], z] ==
(-E^((-z) (c \[Mu] + a \[Nu]))) (E^(b + a z))^\[Nu] (E^(c z))^\[Mu]
z^(1 + n) ((-z) (c \[Mu] + a \[Nu]))^(-1 - n)
(((-1)^n ExpIntegralEi[z (c \[Mu] + a \[Nu])])/(-1 - n)! +
E^(z (c \[Mu] + a \[Nu])) Sum[((-z) (c \[Mu] + a \[Nu]))^j/
Pochhammer[1 + n, j - n], {j, 0, n}] - E^(z (c \[Mu] + a \[Nu]))
Sum[((-z) (c \[Mu] + a \[Nu]))^j/Pochhammer[1 + n, j - n],
{j, 1 + n, -1}]) /; Element[n, Integers]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], ")"]], "\[Mu]"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["a", " ", "z"]], "+", "b"]]], ")"]], "\[Nu]"], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "z"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", "\[Mu]"]], "+", RowBox[List["a", " ", "\[Nu]"]]]], ")"]]]]]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]]], ")"]], "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], ")"]], "\[Mu]"], " ", SuperscriptBox["z", RowBox[List["1", "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "z"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", "\[Mu]"]], "+", RowBox[List["a", " ", "\[Nu]"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", "\[Mu]"]], "+", RowBox[List["a", " ", "\[Nu]"]]]], ")"]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "n"]], ")"]], "!"]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", "\[Mu]"]], "+", RowBox[List["a", " ", "\[Nu]"]]]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "z"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", "\[Mu]"]], "+", RowBox[List["a", " ", "\[Nu]"]]]], ")"]]]], ")"]], "j"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List["j", "-", "n"]]]], "]"]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", "\[Mu]"]], "+", RowBox[List["a", " ", "\[Nu]"]]]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", RowBox[List["1", "+", "n"]]]], RowBox[List["-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "z"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", "\[Mu]"]], "+", RowBox[List["a", " ", "\[Nu]"]]]], ")"]]]], ")"]], "j"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List["j", "-", "n"]]]], "]"]]]]]]]]], ")"]]]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mo> ∫ </mo> <mrow> <msup> <mi> z </mi> <mi> n </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ) </mo> </mrow> <mi> μ </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mi> b </mi> </mrow> </msup> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> </msup> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ) </mo> </mrow> <mi> μ </mi> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> Ei </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> + </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <mi> n </mi> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], RowBox[List["j", "-", "n"]]], Pochhammer] </annotation> </semantics> </mfrac> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <mi> n </mi> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], RowBox[List["j", "-", "n"]]], Pochhammer] </annotation> </semantics> </mfrac> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> <apply> <power /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <ci> μ </ci> </apply> <apply> <power /> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <ci> b </ci> </apply> </apply> <ci> ν </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <ci> c </ci> <ci> μ </ci> </apply> <apply> <times /> <ci> a </ci> <ci> ν </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <exponentiale /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <ci> ν </ci> </apply> <apply> <power /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <ci> μ </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <ci> c </ci> <ci> μ </ci> </apply> <apply> <times /> <ci> a </ci> <ci> ν </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <ci> ExpIntegralEi </ci> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> c </ci> <ci> μ </ci> </apply> <apply> <times /> <ci> a </ci> <ci> ν </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> c </ci> <ci> μ </ci> </apply> <apply> <times /> <ci> a </ci> <ci> ν </ci> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <ci> c </ci> <ci> μ </ci> </apply> <apply> <times /> <ci> a </ci> <ci> ν </ci> </apply> </apply> </apply> <ci> j </ci> </apply> <apply> <power /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> c </ci> <ci> μ </ci> </apply> <apply> <times /> <ci> a </ci> <ci> ν </ci> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </lowlimit> <uplimit> <cn type='integer'> -1 </cn> </uplimit> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <ci> c </ci> <ci> μ </ci> </apply> <apply> <times /> <ci> a </ci> <ci> ν </ci> </apply> </apply> </apply> <ci> j </ci> </apply> <apply> <power /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> j </ci> <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> <in /> <ci> n </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
