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http://functions.wolfram.com/06.34.20.0005.02
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D[ExpIntegralE[\[Nu], z], {\[Nu], n}] ==
Sum[z^(\[Nu] - 1) Binomial[n, k] Log[z]^k D[Gamma[1 - \[Nu]],
{\[Nu], n - k}], {k, 0, n}] +
n! Sum[((-1)^(n - k) z^k)/((\[Nu] - k - 1)^(n + 1) k!),
{k, 0, Infinity}] /; !(Element[\[Nu], Integers] && \[Nu] > 0) &&
Element[n, Integers] && n >= 0
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<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> ∂ </mo> <mi> n </mi> </msup> <mrow> <msub> <semantics> <mi> E </mi> <annotation encoding='Mathematica'> TagBox["E", ExpIntegralE] </annotation> </semantics> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> ν </mi> <mi> n </mi> </msup> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msup> <mi> z </mi> <mrow> <mi> ν </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> n </mi> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <msup> <mi> log </mi> <mi> k </mi> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </msup> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> ν </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> ν </mi> <mo> ∉ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> ν </ci> <degree> <ci> n </ci> </degree> </bvar> <apply> <ci> ExpIntegralE </ci> <ci> ν </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Binomial </ci> <ci> n </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <ln /> <ci> z </ci> </apply> <ci> k </ci> </apply> <apply> <partialdiff /> <bvar> <ci> ν </ci> <degree> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </degree> </bvar> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <factorial /> <ci> n </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <factorial /> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <notin /> <ci> ν </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["\[Nu]_", ",", "n_"]], "}"]]]]], RowBox[List["ExpIntegralE", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[SuperscriptBox["z", RowBox[List["\[Nu]", "-", "1"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], "k"], " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["\[Nu]", ",", RowBox[List["n", "-", "k"]]]], "}"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]]]]]]]], "+", RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "k"]]], " ", SuperscriptBox["z", "k"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", "k", "-", "1"]], ")"]], RowBox[List["n", "+", "1"]]], " ", RowBox[List["k", "!"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Nu]", ">", "0"]]]], ")"]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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