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http://functions.wolfram.com/01.04.16.0111.01
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Sum[Subscript[a, k] Log[Subscript[z, k]], {k, 1, n}] ==
Log[Product[Subscript[z, k]^Subscript[a, k], {k, 1, n}]] -
2 I Pi Floor[(Pi - Sum[Im[Subscript[a, k] Log[Subscript[z, k]]],
{k, 1, n}])/(2 Pi)]
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Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[SubscriptBox["a", "k"], " ", RowBox[List["Log", "[", SubscriptBox["z", "k"], "]"]]]]]], "\[Equal]", RowBox[List[RowBox[List["Log", "[", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "n"], SubsuperscriptBox["z", "k", SubscriptBox["a", "k"]]]], "]"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List["Im", "[", RowBox[List[SubscriptBox["a", "k"], " ", RowBox[List["Log", "[", SubscriptBox["z", "k"], "]"]]]], "]"]]]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]]]]]
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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> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> <mo>  </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <msubsup> <mi> z </mi> <mi> k </mi> <msub> <mi> a </mi> <mi> k </mi> </msub> </msubsup> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <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> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> k </ci> </apply> <apply> <ln /> <apply> <ci> Subscript </ci> <ci> z </ci> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <ln /> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <ci> k </ci> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <pi /> <apply> <floor /> <apply> <times /> <apply> <plus /> <pi /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <imaginary /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> k </ci> </apply> <apply> <ln /> <apply> <ci> Subscript </ci> <ci> z </ci> <ci> k </ci> </apply> </apply> </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> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "1"]], "n_"], RowBox[List[SubscriptBox["a_", "k_"], " ", RowBox[List["Log", "[", SubscriptBox["z_", "k_"], "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Log", "[", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "n"], SubsuperscriptBox["zz", "k", SubscriptBox["aa", "k"]]]], "]"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List["Im", "[", RowBox[List[SubscriptBox["aa", "k"], " ", RowBox[List["Log", "[", SubscriptBox["zz", "k"], "]"]]]], "]"]]]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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