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Exp




Mathematica Notation

Traditional Notation









Elementary Functions > Exp[z] > Integration > Indefinite integration > Involving functions of the direct function and a power function > Involving powers of the direct function and a power function > Involving powers of exp and power > Involving zalpha-1and arguments a z+b





http://functions.wolfram.com/01.03.21.0532.01









  


  








Input Form





Integrate[z^(\[Alpha] - 1) (E^(a z + b))^\[Nu], z] == ((-E^((-a) z \[Nu])) (E^(b + a z))^\[Nu] z^\[Alpha] Gamma[\[Alpha], (-a) z \[Nu]])/((-a) z \[Nu])^\[Alpha]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", 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["-", "a"]], " ", "z", " ", "\[Nu]"]]]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]]], ")"]], "\[Nu]"], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", "z", " ", "\[Nu]"]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["-", "a"]], " ", "z", " ", "\[Nu]"]]]], "]"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> &#8747; </mo> <mrow> <mrow> <msup> <mi> z </mi> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> </msup> <mo> ) </mo> </mrow> <mi> &#957; </mi> </msup> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msup> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> </msup> <mo> ) </mo> </mrow> <mi> &#957; </mi> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> &#945; </mi> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#945; </mi> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <exponentiale /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <ci> &#957; </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> a </ci> </apply> <ci> z </ci> <ci> &#957; </ci> </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> &#957; </ci> </apply> <apply> <power /> <ci> z </ci> <ci> &#945; </ci> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> z </ci> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> <apply> <ci> Gamma </ci> <ci> &#945; </ci> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> z </ci> <ci> &#957; </ci> </apply> </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["\[Alpha]_", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["a_", " ", "z_"]], "+", "b_"]]], ")"]], "\[Nu]_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a"]], " ", "z", " ", "\[Nu]"]]]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]]], ")"]], "\[Nu]"], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", "z", " ", "\[Nu]"]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["-", "a"]], " ", "z", " ", "\[Nu]"]]]], "]"]]]]]]]]


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





2002-12-18