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ExpIntegralE






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/06.34.20.0002.01









  


  










Input Form





D[ExpIntegralE[\[Nu], z], {\[Nu], 2}] == z^(-1 + \[Nu]) Gamma[1 - \[Nu]] (Pi^2 Cot[Pi \[Nu]]^2 + Pi^2 Csc[Pi \[Nu]]^2 - 2 Pi Cot[Pi \[Nu]] Log[z] + Log[z]^2 + 2 (Pi Cot[Pi \[Nu]] - Log[z]) PolyGamma[\[Nu]] + PolyGamma[\[Nu]]^2 - PolyGamma[1, \[Nu]]) + (1/(\[Nu] - 1)^3) (2 HypergeometricPFQ[{1 - \[Nu], 1 - \[Nu], 1 - \[Nu]}, {2 - \[Nu], 2 - \[Nu], 2 - \[Nu]}, -z])










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["\[Nu]_", ",", "2"]], "}"]]]]], RowBox[List["ExpIntegralE", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox[RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], "2"]]], "+", RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], "2"]]], "-", RowBox[List["2", " ", "\[Pi]", " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "+", SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], "2"], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], "-", RowBox[List["Log", "[", "z", "]"]]]], ")"]], " ", RowBox[List["PolyGamma", "[", "\[Nu]", "]"]]]], "+", SuperscriptBox[RowBox[List["PolyGamma", "[", "\[Nu]", "]"]], "2"], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", "\[Nu]"]], "]"]]]], ")"]]]], "+", FractionBox[RowBox[List["2", " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["1", "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "-", "\[Nu]"]], ",", RowBox[List["2", "-", "\[Nu]"]], ",", RowBox[List["2", "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", "z"]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", "1"]], ")"]], "3"]]]]]]]]










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





2001-10-29