Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Exp






Mathematica Notation

Traditional Notation









Elementary Functions > Exp[z] > Integration > Indefinite integration > Involving functions of the direct function > Involving products of powers of the direct function > Involving product of powers of two direct functions > Involving (eb zr)mu (ec zr+g)nu





http://functions.wolfram.com/01.03.21.0385.01









  


  










Input Form





Integrate[(E^(b z^r))^\[Mu] (E^(c z^r))^\[Nu], z] == (-(1/r)) (((E^(b z^r))^\[Mu] (E^(c z^r))^\[Nu] z Gamma[1/r, (-z^r) (b \[Mu] + c \[Nu])])/(E^(z^r (b \[Mu] + c \[Nu])) ((-z^r) (b \[Mu] + c \[Nu]))^r^(-1)))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", SuperscriptBox["z", "r"]]]], ")"]], "\[Mu]"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", SuperscriptBox["z", "r"]]]], ")"]], "\[Nu]"], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", "r"]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", SuperscriptBox["z", "r"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[Nu]"]]]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", SuperscriptBox["z", "r"]]]], ")"]], "\[Mu]"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", SuperscriptBox["z", "r"]]]], ")"]], "\[Nu]"], " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["z", "r"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[Nu]"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "/", "r"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "r"], ",", RowBox[List[RowBox[List["-", SuperscriptBox["z", "r"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[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> <mrow> <mo> ( </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> </msup> <mo> ) </mo> </mrow> <mi> &#956; </mi> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> c </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </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> <mo> - </mo> <mfrac> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> </msup> <mo> ) </mo> </mrow> <mi> &#956; </mi> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> c </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> </msup> <mo> ) </mo> </mrow> <mi> &#957; </mi> </msup> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> / </mo> <mi> r </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mi> r </mi> </mfrac> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> r </mi> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </apply> </apply> <ci> &#956; </ci> </apply> <apply> <power /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </apply> </apply> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> b </ci> <ci> &#956; </ci> </apply> <apply> <times /> <ci> c </ci> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </apply> </apply> <ci> &#956; </ci> </apply> <apply> <power /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </apply> </apply> <ci> &#957; </ci> </apply> <ci> z </ci> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> b </ci> <ci> &#956; </ci> </apply> <apply> <times /> <ci> c </ci> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> r </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> r </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> b </ci> <ci> &#956; </ci> </apply> <apply> <times /> <ci> c </ci> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <apply> <power /> <ci> r </ci> <cn type='integer'> -1 </cn> </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[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["b_", " ", SuperscriptBox["z_", "r_"]]]], ")"]], "\[Mu]_"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["c_", " ", SuperscriptBox["z_", "r_"]]]], ")"]], "\[Nu]_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", SuperscriptBox["z", "r"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[Nu]"]]]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", SuperscriptBox["z", "r"]]]], ")"]], "\[Mu]"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", SuperscriptBox["z", "r"]]]], ")"]], "\[Nu]"], " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["z", "r"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[Nu]"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "/", "r"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "r"], ",", RowBox[List[RowBox[List["-", SuperscriptBox["z", "r"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[Nu]"]]]], ")"]]]]]], "]"]]]], "r"]]]]]]]










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





2002-12-18





© 1998- Wolfram Research, Inc.