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 ExpIntegralE

 http://functions.wolfram.com/06.34.20.0006.02

 Input Form

 D[ExpIntegralE[\[Nu], z], {\[Nu], n}] == z^(\[Nu] - 1) Sum[(-1)^k Binomial[n, k] Log[z]^(n - k) (Derivative[k][Gamma][1 - \[Nu]] - z^(1 - \[Nu]) Sum[(-1)^(k - j) Binomial[k, j] (k - j)! Gamma[1 - \[Nu]]^(k - j + 1) Log[z]^j HypergeometricPFQRegularized[{Subscript[a, 1], Subscript[a, 2], \[Ellipsis], Subscript[a, k - j + 1]}, {1 + Subscript[a, 1], 1 + Subscript[a, 2], \[Ellipsis], 1 + Subscript[a, k - j + 1]}, -z], {j, 0, k}]), {k, 0, n}] /; Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, n + 1] == 1 - \[Nu] && Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["\[Nu]", ",", "n"]], "}"]]], RowBox[List["ExpIntegralE", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Nu]", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], RowBox[List["n", "-", "k"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", "k", ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["1", "-", "\[Nu]"]], "]"]], "-", RowBox[List[SuperscriptBox["z", RowBox[List["1", "-", "\[Nu]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "j"]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]], RowBox[List["k", "-", "j", "+", "1"]]], " ", SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], "j"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["k", "-", "j", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", RowBox[List["k", "-", "j", "+", "1"]]]]]]], "}"]], ",", RowBox[List["-", "z"]]]], "]"]]]]]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "\[Equal]", SubscriptBox["a", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["n", "+", "1"]]], "\[Equal]", RowBox[List["1", "-", "\[Nu]"]]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 n E TagBox["E", ExpIntegralE] ν ( z ) ν n z ν - 1 k = 0 n ( - 1 ) k ( n k ) 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]] log n - k ( z ) ( Γ ( k ) TagBox[RowBox[List["(", "k", ")"]], Derivative] ( 1 - ν ) - z 1 - ν j = 0 k ( - 1 ) k - j ( k j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( k - j ) ! Γ ( 1 - ν ) k - j + 1 log j ( z ) k - j + 1 F ~ k - j + 1 ( a 1 , a 2 , , a k - j + 1 TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", RowBox[List["k", "-", "j", "+", "1"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]] ; a 1 + 1 , a 2 + 1 , , a k - j + 1 + 1 TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "1"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "2"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", RowBox[List["k", "-", "j", "+", "1"]]], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]] ; - z TagBox[RowBox[List["-", "z"]], HypergeometricPFQRegularized, Rule[Editable, True]] ) ) /; a 1 a 2 a n + 1 1 - ν n FormBox RowBox RowBox FractionBox RowBox SuperscriptBox n RowBox SubscriptBox TagBox E ExpIntegralE ν ( z ) RowBox SuperscriptBox ν n RowBox SuperscriptBox z RowBox ν - 1 RowBox UnderoverscriptBox RowBox k = 0 n ErrorBox RowBox SuperscriptBox RowBox ( RowBox - 1 ) k TagBox RowBox ( GridBox TagBox n Rule Editable TagBox k Rule Editable ) InterpretTemplate Function Binomial Slot 1 Slot 2 Rule Editable RowBox SuperscriptBox log RowBox n - k ( z ) RowBox ( RowBox RowBox SuperscriptBox Γ TagBox RowBox ( k ) Derivative ( RowBox 1 - ν ) - RowBox SuperscriptBox z RowBox 1 - ν RowBox UnderoverscriptBox RowBox j = 0 k RowBox SuperscriptBox RowBox ( RowBox - 1 ) RowBox k - j TagBox RowBox ( GridBox TagBox k Rule Editable TagBox j Rule Editable ) InterpretTemplate Function Binomial Slot 1 Slot 2 Rule Editable RowBox RowBox ( RowBox k - j ) ! SuperscriptBox RowBox Γ ( RowBox 1 - ν ) RowBox k - j + 1 RowBox SuperscriptBox log j ( z ) RowBox RowBox SubscriptBox RowBox k - j + 1 SubscriptBox OverscriptBox F ~ RowBox k - j + 1 RowBox ( RowBox TagBox TagBox RowBox TagBox SubscriptBox a 1 HypergeometricPFQRegularized Rule Editable , TagBox SubscriptBox a 2 HypergeometricPFQRegularized Rule Editable , TagBox HypergeometricPFQRegularized Rule Editable , TagBox SubscriptBox a RowBox k - j + 1 HypergeometricPFQRegularized Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQRegularized Rule Editable ; TagBox TagBox RowBox TagBox RowBox SubscriptBox a 1 + 1 HypergeometricPFQRegularized Rule Editable , TagBox RowBox SubscriptBox a 2 + 1 HypergeometricPFQRegularized Rule Editable , TagBox HypergeometricPFQRegularized Rule Editable , TagBox RowBox SubscriptBox a RowBox k - j + 1 + 1 HypergeometricPFQRegularized Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQRegularized Rule Editable ; TagBox RowBox - z HypergeometricPFQRegularized Rule Editable ) ) /; RowBox RowBox SubscriptBox a 1 SubscriptBox a 2 SubscriptBox a RowBox n + 1 RowBox 1 - ν RowBox n TraditionalForm [/itex]

 Rule Form

 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[SuperscriptBox["z", RowBox[List["\[Nu]", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], RowBox[List["n", "-", "k"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", "k", ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["1", "-", "\[Nu]"]], "]"]], "-", RowBox[List[SuperscriptBox["z", RowBox[List["1", "-", "\[Nu]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "j"]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]], RowBox[List["k", "-", "j", "+", "1"]]], " ", SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], "j"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["k", "-", "j", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", RowBox[List["k", "-", "j", "+", "1"]]]]]]], "}"]], ",", RowBox[List["-", "z"]]]], "]"]]]]]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "\[Equal]", SubscriptBox["a", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["n", "+", "1"]]], "\[Equal]", RowBox[List["1", "-", "\[Nu]"]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2001-10-29

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