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ExpIntegralE






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > ExpIntegralE[nu,z] > Series representations > Generalized power series > Expansions at generic point nu==nu0 > For the function itself





http://functions.wolfram.com/06.34.06.0018.01









  


  










Input Form





ExpIntegralE[\[Nu], z] \[Proportional] ExpIntegralE[Subscript[\[Nu], 0], z] - (1/z) Gamma[1 - Subscript[\[Nu], 0]] (z Gamma[1 - Subscript[\[Nu], 0]] HypergeometricPFQRegularized[ {1 - Subscript[\[Nu], 0], 1 - Subscript[\[Nu], 0]}, {2 - Subscript[\[Nu], 0], 2 - Subscript[\[Nu], 0]}, -z] - z^Subscript[\[Nu], 0] (EulerGamma - HarmonicNumber[ -Subscript[\[Nu], 0]] + Log[z])) (\[Nu] - Subscript[\[Nu], 0]) + (1/(2 z)) Gamma[1 - Subscript[\[Nu], 0]] (-2 z Gamma[1 - Subscript[\[Nu], 0]]^2 HypergeometricPFQRegularized[ {1 - Subscript[\[Nu], 0], 1 - Subscript[\[Nu], 0], 1 - Subscript[\[Nu], 0]}, {2 - Subscript[\[Nu], 0], 2 - Subscript[\[Nu], 0], 2 - Subscript[\[Nu], 0]}, -z] + z^Subscript[\[Nu], 0] ((EulerGamma - HarmonicNumber[-Subscript[\[Nu], 0]] + Log[z])^2 + PolyGamma[1, 1 - Subscript[\[Nu], 0]])) (\[Nu] - Subscript[\[Nu], 0])^ 2 + O[(\[Nu] - Subscript[\[Nu], 0])^3]










Standard Form





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MathML Form







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</ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> &#957; </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> &#957; </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> &#957; </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ExpIntegralE", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["ExpIntegralE", "[", RowBox[List[SubscriptBox["\[Nu]\[Nu]", "0"], ",", "z"]], "]"]], "-", FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], ",", RowBox[List["1", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], ",", RowBox[List["2", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]]]], "}"]], ",", RowBox[List["-", "z"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["z", SubscriptBox["\[Nu]\[Nu]", "0"]], " ", RowBox[List["(", RowBox[List["EulerGamma", "-", RowBox[List["HarmonicNumber", "[", RowBox[List["-", SubscriptBox["\[Nu]\[Nu]", "0"]]], "]"]], "+", RowBox[List["Log", "[", "z", "]"]]]], ")"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], ")"]]]], "z"], "+", FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "z", " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], "]"]], "2"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], ",", RowBox[List["1", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], ",", RowBox[List["1", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], ",", RowBox[List["2", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], ",", RowBox[List["2", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]]]], "}"]], ",", RowBox[List["-", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["z", SubscriptBox["\[Nu]\[Nu]", "0"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["EulerGamma", "-", RowBox[List["HarmonicNumber", "[", RowBox[List["-", SubscriptBox["\[Nu]\[Nu]", "0"]]], "]"]], "+", RowBox[List["Log", "[", "z", "]"]]]], ")"]], "2"], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]]]], "]"]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], ")"]], "2"]]], RowBox[List["2", " ", "z"]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], "]"]], "3"]]]]]]]










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





2007-05-02