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 > Definite integration > Multiple integrals





http://functions.wolfram.com/01.03.21.0800.01









  


  










Input Form





Integrate[(1 - s + s t)^n/E^(z s t), {t, 0, 1}, {s, 0, 1}] == (E^z Gamma[n + 1, z] (EulerGamma + Gamma[0, z] + Log[z]))/z^(n + 1) - (n!/z^(n + 1)) Sum[(z^k HarmonicNumber[k])/k!, {k, 1, n}] /; Element[z, Reals] && z > 0 && Element[\[Lambda], Integers] && z >= 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[List[SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "s", "+", RowBox[List["s", " ", "t"]]]], ")"]], "n"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "z"]], " ", "s", " ", "t"]]], RowBox[List["\[DifferentialD]", "s"]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", "z"], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["n", "+", "1"]], ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List["EulerGamma", "+", RowBox[List["Gamma", "[", RowBox[List["0", ",", "z"]], "]"]], "+", RowBox[List["Log", "[", "z", "]"]]]], ")"]]]], SuperscriptBox["z", RowBox[List["n", "+", "1"]]]], "-", RowBox[List[FractionBox[RowBox[List["n", "!"]], SuperscriptBox["z", RowBox[List["n", "+", "1"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox[RowBox[List[SuperscriptBox["z", "k"], " ", RowBox[List["HarmonicNumber", "[", "k", "]"]]]], RowBox[List["k", "!"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["z", "\[Element]", "Reals"]], "\[And]", RowBox[List["z", ">", "0"]], "\[And]", RowBox[List["\[Lambda]", "\[Element]", "Integers"]], "\[And]", RowBox[List["z", "\[GreaterEqual]", "0"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mn> 1 </mn> </msubsup> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mn> 1 </mn> </msubsup> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> s </mi> <mo> + </mo> <mrow> <mi> s </mi> <mo> &#8290; </mo> <mi> t </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> &#8290; </mo> <mi> s </mi> <mo> &#8290; </mo> <mi> t </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> s </mi> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msup> <mi> &#8519; </mi> <mi> z </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[EulerGamma]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> <msup> <mi> z </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mfrac> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <msup> <mi> z </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mfrac> <mrow> <msup> <mi> z </mi> <mi> k </mi> </msup> <mo> &#8290; </mo> <msub> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mi> k </mi> </msub> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> z </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#955; </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> s </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <cn type='integer'> 1 </cn> </uplimit> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <cn type='integer'> 1 </cn> </uplimit> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> <apply> <times /> <ci> s </ci> <ci> t </ci> </apply> </apply> <ci> n </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <ci> s </ci> <ci> t </ci> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <ci> z </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> <apply> <plus /> <apply> <ci> Gamma </ci> <cn type='integer'> 0 </cn> <ci> z </ci> </apply> <apply> <ln /> <ci> z </ci> </apply> <eulergamma /> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <factorial /> <ci> n </ci> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> k </ci> </apply> <apply> <ci> HarmonicNumber </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> z </ci> <reals /> </apply> <apply> <gt /> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <in /> <ci> &#955; </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[List[SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "s_", "+", RowBox[List["s_", " ", "t_"]]]], ")"]], "n_"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "z_"]], " ", "s_", " ", "t_"]]]]], RowBox[List["\[DifferentialD]", "s_"]], RowBox[List["\[DifferentialD]", "t_"]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", "z"], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["n", "+", "1"]], ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List["EulerGamma", "+", RowBox[List["Gamma", "[", RowBox[List["0", ",", "z"]], "]"]], "+", RowBox[List["Log", "[", "z", "]"]]]], ")"]]]], SuperscriptBox["z", RowBox[List["n", "+", "1"]]]], "-", FractionBox[RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox[RowBox[List[SuperscriptBox["z", "k"], " ", RowBox[List["HarmonicNumber", "[", "k", "]"]]]], RowBox[List["k", "!"]]]]]]], SuperscriptBox["z", RowBox[List["n", "+", "1"]]]]]], "/;", RowBox[List[RowBox[List["z", "\[Element]", "Reals"]], "&&", RowBox[List["z", ">", "0"]], "&&", RowBox[List["\[Lambda]", "\[Element]", "Integers"]], "&&", RowBox[List["z", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2002-12-18





© 1998- Wolfram Research, Inc.