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http://functions.wolfram.com/01.04.06.0021.01
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Log[f[z]] == 2 I Pi Floor[(Pi - Arg[f[Subscript[z, 0]]] -
Arg[f[z]/f[Subscript[z, 0]]])/(2 Pi)] + Log[f[Subscript[z, 0]]] +
Sum[((-1)^k/(k + 1)) (Derivative[u][f][Subscript[z, 0]]/
(f[Subscript[z, 0]] u!))^(k + 1) Subscript[p, k + 1, s - u k]
(z - Subscript[z, 0])^(s + u), {s, 0, Infinity}, {k, 0, s/u}] /;
f[Subscript[z, 0]] != 0 && (Derivative[k][f][Subscript[z, 0]] == 0 /;
1 <= k <= u - 1) && Derivative[u][f][Subscript[z, 0]] != 0 &&
Subscript[p, j, 0] == 1 && Subscript[p, j, k] ==
(u!/(Derivative[u][f][Subscript[z, 0]] k))
Sum[((j m - k + m)/((m + u)! (m + 1)!)) Derivative[m + u][f][
Subscript[z, 0]] Subscript[p, j, k - m], {m, 1, k}] &&
Element[k, Integers] && k > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Log", "[", RowBox[List["f", "[", "z", "]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Arg", "[", RowBox[List["f", "[", SubscriptBox["z", "0"], "]"]], "]"]], "-", RowBox[List["Arg", "[", FractionBox[RowBox[List["f", "[", "z", "]"]], RowBox[List["f", "[", SubscriptBox["z", "0"], "]"]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List["Log", "[", RowBox[List["f", "[", SubscriptBox["z", "0"], "]"]], "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["s", "/", "u"]]], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List["k", "+", "1"]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List[SuperscriptBox["f", TagBox[RowBox[List["(", "u", ")"]], Derivative], Rule[MultilineFunction, None]], "[", SubscriptBox["z", "0"], "]"]], RowBox[List[RowBox[List["f", "[", SubscriptBox["z", "0"], "]"]], RowBox[List["u", "!"]]]]], ")"]], RowBox[List["k", "+", "1"]]], SubscriptBox["p", RowBox[List[RowBox[List["k", "+", "1"]], ",", RowBox[List["s", "-", RowBox[List["u", " ", "k"]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], RowBox[List["s", "+", "u"]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["f", "[", SubscriptBox["z", "0"], "]"]], "\[NotEqual]", "0"]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["f", TagBox[RowBox[List["(", "k", ")"]], Derivative], Rule[MultilineFunction, None]], "[", SubscriptBox["z", "0"], "]"]], "\[Equal]", "0"]], "/;", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", RowBox[List["u", "-", "1"]]]]]], ")"]], "\[And]", RowBox[List[RowBox[List[SuperscriptBox["f", TagBox[RowBox[List["(", "u", ")"]], Derivative], Rule[MultilineFunction, None]], "[", SubscriptBox["z", "0"], "]"]], "\[NotEqual]", "0"]], "\[And]", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "0"]]], "\[Equal]", "1"]], "\[And]", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "k"]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["u", "!"]], RowBox[List[RowBox[List[SuperscriptBox["f", TagBox[RowBox[List["(", "u", ")"]], Derivative], Rule[MultilineFunction, None]], "[", SubscriptBox["z", "0"], "]"]], "k"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "k"], RowBox[List[FractionBox[RowBox[List[RowBox[List["j", " ", "m"]], "-", "k", "+", "m"]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "+", "u"]], ")"]], "!"]], RowBox[List[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "!"]]]]], RowBox[List[SuperscriptBox["f", TagBox[RowBox[List["(", RowBox[List["m", "+", "u"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", SubscriptBox["z", "0"], "]"]], " ", SubscriptBox["p", RowBox[List["j", ",", RowBox[List["k", "-", "m"]]]]]]]]]]]]], "\[And]", RowBox[List["k", "\[Element]", "Integers"]], "\[And]", RowBox[List["k", ">", "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> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> f </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> π </mi> <mo> - </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> f </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> f </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mrow> <mi> f </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> f </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> s </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mfrac> <mi> s </mi> <mi> u </mi> </mfrac> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mtext> </mtext> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <msup> <mi> f </mi> <semantics> <mrow> <mo> ( </mo> <mi> u </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", "u", ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mrow> <mrow> <mi> f </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> u </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msub> <mi> p </mi> <mrow> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> s </mi> <mo> - </mo> <mrow> <mi> u </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> </mrow> </mrow> </msub> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> s </mi> <mo> + </mo> <mi> u </mi> </mrow> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> f </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> ≠ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msup> <mi> f </mi> <semantics> <mrow> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", "k", ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mn> 1 </mn> <mo> ≤ </mo> <mi> k </mi> <mo> ≤ </mo> <mrow> <mi> u </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <mrow> <msup> <mi> f </mi> <semantics> <mrow> <mo> ( </mo> <mi> u </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", "u", ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> ≠ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> p </mi> <mrow> <mi> j </mi> <mo> , </mo> <mn> 0 </mn> </mrow> </msub> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> p </mi> <mrow> <mi> j </mi> <mo> , </mo> <mi> k </mi> </mrow> </msub> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mrow> <mi> u </mi> <mo> ! </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mrow> <msup> <mi> f </mi> <semantics> <mrow> <mo> ( </mo> <mi> u </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", "u", ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> k </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> m </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <mfrac> <mrow> <mrow> <mi> j </mi> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> + </mo> <mi> m </mi> <mo> - </mo> <mi> k </mi> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> u </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> f </mi> <semantics> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> u </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", RowBox[List["m", "+", "u"]], ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <mi> p </mi> <mrow> <mi> j </mi> <mo> , </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> m </mi> </mrow> </mrow> </msub> </mrow> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> k </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ln /> <apply> <ci> f </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <pi /> <apply> <floor /> <apply> <times /> <apply> <plus /> <pi /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arg /> <apply> <times /> <ci> f </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arg /> <apply> <times /> <apply> <ci> f </ci> <ci> z </ci> </apply> <apply> <power /> <apply> <ci> f </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ln /> <apply> <ci> f </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <times /> <ci> s </ci> <apply> <power /> <ci> u </ci> <cn type='integer'> -1 </cn> </apply> </apply> </uplimit> <apply> <sum /> <bvar> <ci> s </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> D </ci> <apply> <ci> f </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <list> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <ci> u </ci> </list> </apply> <apply> <power /> <apply> <times /> <apply> <ci> f </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <factorial /> <ci> u </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> p </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> s </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> u </ci> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> s </ci> <ci> u </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <neq /> <apply> <ci> f </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> D </ci> <apply> <ci> f </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <list> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <ci> k </ci> </list> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <leq /> <cn type='integer'> 1 </cn> <ci> k </ci> <apply> <plus /> <ci> u </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <neq /> <apply> <ci> D </ci> <apply> <ci> f </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <list> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <ci> u </ci> </list> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> <ci> k </ci> </apply> <apply> <times /> <apply> <times /> <apply> <factorial /> <ci> u </ci> </apply> <apply> <power /> <apply> <times /> <apply> <ci> D </ci> <apply> <ci> f </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <list> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <ci> u </ci> </list> </apply> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> m </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> j </ci> <ci> m </ci> </apply> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <ci> u </ci> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> D </ci> <apply> <ci> f </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <list> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <plus /> <ci> m </ci> <ci> u </ci> </apply> </list> </apply> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> k </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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){kind=small}
