Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











ExpIntegralE






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > ExpIntegralE[nu,z] > Differentiation > Fractional integro-differentiation > With respect to nu





http://functions.wolfram.com/06.34.20.0009.01









  


  










Input Form





D[ExpIntegralE[\[Nu], z], {\[Nu], \[Alpha]}] == (1/(\[Nu]^\[Alpha] z)) Sum[Derivative[k][Gamma][1] (-\[Nu])^k Hypergeometric1F1Regularized[1 + k, 1 + k - \[Alpha], \[Nu] Log[z]], {k, 0, Infinity}] - Sum[((-z)^k/(k + 1)!) (Sqrt[\[Nu]]/Sqrt[1 + k])^\[Alpha] BesselI[-\[Alpha], (2 Sqrt[\[Nu]])/Sqrt[1 + k]], {k, 0, Infinity}]/ \[Nu]^\[Alpha] /; !(Element[\[Nu], Integers] && \[Nu] > 0)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["\[Nu]", ",", "\[Alpha]"]], "}"]]], RowBox[List["ExpIntegralE", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[SuperscriptBox["\[Nu]", RowBox[List["-", "\[Alpha]"]]], "z"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", "k", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "1", "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "k"], RowBox[List["Hypergeometric1F1Regularized", "[", RowBox[List[RowBox[List["1", "+", "k"]], ",", RowBox[List["1", "+", "k", "-", "\[Alpha]"]], ",", RowBox[List["\[Nu]", " ", RowBox[List["Log", "[", "z", "]"]]]]]], "]"]]]]]]]], "-", " ", RowBox[List[SuperscriptBox["\[Nu]", RowBox[List["-", "\[Alpha]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "k"], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "!"]]], SuperscriptBox[RowBox[List["(", FractionBox[SqrtBox["\[Nu]"], SqrtBox[RowBox[List["1", "+", "k"]]]], ")"]], "\[Alpha]"], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], ",", FractionBox[RowBox[List["2", " ", SqrtBox["\[Nu]"]]], SqrtBox[RowBox[List["1", "+", "k"]]]]]], "]"]]]]]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List[RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]], "\[And]", RowBox[List["\[Nu]", ">", "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> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> &#945; </mi> </msup> <mrow> <msub> <semantics> <mi> E </mi> <annotation encoding='Mathematica'> TagBox[&quot;E&quot;, ExpIntegralE] </annotation> </semantics> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> &#957; </mi> <mi> &#945; </mi> </msup> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <msup> <mi> &#957; </mi> <mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> </msup> <mi> z </mi> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mrow> <msup> <mi> &#915; </mi> <semantics> <mrow> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, &quot;k&quot;, &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 1 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;1&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[TagBox[RowBox[List[&quot;k&quot;, &quot;+&quot;, &quot;1&quot;]], Hypergeometric1F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1Regularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[&quot;k&quot;, &quot;-&quot;, &quot;\[Alpha]&quot;, &quot;+&quot;, &quot;1&quot;]], Hypergeometric1F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1Regularized, Rule[Editable, False]], &quot;;&quot;, TagBox[RowBox[List[&quot;\[Nu]&quot;, &quot; &quot;, RowBox[List[&quot;log&quot;, &quot;(&quot;, &quot;z&quot;, &quot;)&quot;]]]], Hypergeometric1F1Regularized, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1Regularized] </annotation> </semantics> </mrow> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mi> &#957; </mi> <mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mtext> </mtext> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <msqrt> <mi> &#957; </mi> </msqrt> <msqrt> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mi> &#945; </mi> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> I </mi> <mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> </msub> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mi> &#957; </mi> </msqrt> </mrow> <msqrt> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> &#957; </mi> <mo> &#8713; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> &#957; </ci> <degree> <ci> &#945; </ci> </degree> </bvar> <apply> <ci> ExpIntegralE </ci> <ci> &#957; </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> D </ci> <apply> <ci> Gamma </ci> <cn type='integer'> 1 </cn> </apply> <list> <cn type='integer'> 1 </cn> <ci> k </ci> </list> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <ci> k </ci> </apply> <apply> <ci> Hypergeometric1F1Regularized </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <ci> &#957; </ci> <apply> <ln /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> &#957; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#945; </ci> </apply> <apply> <ci> BesselI </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <notin /> <ci> &#957; </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["\[Nu]_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["ExpIntegralE", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[Nu]", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", "k", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "1", "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "k"], " ", RowBox[List["Hypergeometric1F1Regularized", "[", RowBox[List[RowBox[List["1", "+", "k"]], ",", RowBox[List["1", "+", "k", "-", "\[Alpha]"]], ",", RowBox[List["\[Nu]", " ", RowBox[List["Log", "[", "z", "]"]]]]]], "]"]]]]]]]], "z"], "-", RowBox[List[SuperscriptBox["\[Nu]", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["(", FractionBox[SqrtBox["\[Nu]"], SqrtBox[RowBox[List["1", "+", "k"]]]], ")"]], "\[Alpha]"], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], ",", FractionBox[RowBox[List["2", " ", SqrtBox["\[Nu]"]]], SqrtBox[RowBox[List["1", "+", "k"]]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "!"]]]]]]]]], "/;", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Nu]", ">", "0"]]]], ")"]]]]]]]]]]










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





2001-10-29