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 ExpIntegralE

 http://functions.wolfram.com/06.34.06.0019.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ExpIntegralE", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List[SubscriptBox["\[Nu]\[Nu]", "0"], "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "s"], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "s"]], "]"]], " ", SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], RowBox[List["k", "-", "s"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", "s", ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["1", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], "]"]], "-", RowBox[List[SuperscriptBox["z", RowBox[List["1", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "s"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["s", "-", "j"]]], " ", RowBox[List["Binomial", "[", RowBox[List["s", ",", "j"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["s", "-", "j"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], "]"]], RowBox[List["s", "-", "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["s", "-", "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["s", "-", "j", "+", "1"]]]]]]], "}"]], ",", RowBox[List["-", "z"]]]], "]"]]]]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], ")"]], "k"]]]]], RowBox[List["k", "!"]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "\[Equal]", SubscriptBox["a", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["k", "+", "1"]]], "\[Equal]", RowBox[List["1", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]]]], "&&", RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02