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 one direct function and elementary functions > Involving rational functions > Involving (a z2+b)-n





http://functions.wolfram.com/01.03.21.0142.01









  


  










Input Form





Integrate[E^(c z)/(a z^2 + b)^2, z] == (2 a Sqrt[b] E^(c ((I Sqrt[b])/Sqrt[a] + z)) z - I (Sqrt[a] - I Sqrt[b] c) E^((2 I Sqrt[b] c)/Sqrt[a]) (b + a z^2) ExpIntegralEi[ c (-((I Sqrt[b])/Sqrt[a]) + z)] + I (Sqrt[a] + I Sqrt[b] c) (b + a z^2) ExpIntegralEi[c ((I Sqrt[b])/Sqrt[a] + z)])/E^((I Sqrt[b] c)/Sqrt[a])/ (4 a b^(3/2) (b + a z^2))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["a", " ", SuperscriptBox["z", "2"]]], "+", "b"]], ")"]], "2"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["b"], " ", "c"]], SqrtBox["a"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a", " ", SqrtBox["b"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["b"]]], SqrtBox["a"]], "+", "z"]], ")"]]]]], " ", "z"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[SqrtBox["a"], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["b"], " ", "c"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", SqrtBox["b"], " ", "c"]], SqrtBox["a"]]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["b"]]], SqrtBox["a"]]]], "+", "z"]], ")"]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[SqrtBox["a"], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox["b"], " ", "c"]]]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List["c", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["b"]]], SqrtBox["a"]], "+", "z"]], ")"]]]], "]"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["4", " ", "a", " ", SuperscriptBox["b", RowBox[List["3", "/", "2"]]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", SuperscriptBox["z", "2"]]]]], ")"]]]], ")"]]]]]]]]










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> <mfrac> <msup> <mi> &#8519; </mi> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> b </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mi> b </mi> </msqrt> <mo> &#8290; </mo> <mi> c </mi> </mrow> <msqrt> <mi> a </mi> </msqrt> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <msqrt> <mi> b </mi> </msqrt> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mi> b </mi> </msqrt> </mrow> <msqrt> <mi> a </mi> </msqrt> </mfrac> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msqrt> <mi> a </mi> </msqrt> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mi> b </mi> </msqrt> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> Ei </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mi> b </mi> </msqrt> </mrow> <msqrt> <mi> a </mi> </msqrt> </mfrac> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mi> a </mi> </msqrt> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mi> b </mi> </msqrt> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mi> b </mi> </msqrt> <mo> &#8290; </mo> <mi> c </mi> </mrow> <msqrt> <mi> a </mi> </msqrt> </mfrac> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> Ei </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mi> b </mi> </msqrt> </mrow> <msqrt> <mi> a </mi> </msqrt> </mfrac> </mrow> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </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 /> <exponentiale /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> b </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> <apply> <power /> <ci> b </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> b </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> c </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> z </ci> </apply> </apply> </apply> <ci> z </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> </apply> </apply> <imaginaryi /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> b </ci> </apply> <apply> <ci> ExpIntegralEi </ci> <apply> <times /> <ci> c </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> b </ci> </apply> <apply> <ci> ExpIntegralEi </ci> <apply> <times /> <ci> c </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["c_", " ", "z_"]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["a_", " ", SuperscriptBox["z_", "2"]]], "+", "b_"]], ")"]], "2"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["b"], " ", "c"]], SqrtBox["a"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a", " ", SqrtBox["b"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["b"]]], SqrtBox["a"]], "+", "z"]], ")"]]]]], " ", "z"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[SqrtBox["a"], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["b"], " ", "c"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", SqrtBox["b"], " ", "c"]], SqrtBox["a"]]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["b"]]], SqrtBox["a"]]]], "+", "z"]], ")"]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[SqrtBox["a"], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox["b"], " ", "c"]]]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List["c", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["b"]]], SqrtBox["a"]], "+", "z"]], ")"]]]], "]"]]]]]], ")"]]]], RowBox[List["4", " ", "a", " ", SuperscriptBox["b", RowBox[List["3", "/", "2"]]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", SuperscriptBox["z", "2"]]]]], ")"]]]]]]]]]










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





2002-12-18





© 1998- Wolfram Research, Inc.