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http://functions.wolfram.com/01.03.23.0007.01
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Sum[z^k E^(k w), {k, 0, Infinity}] == 1/(1 - E^w z) /; Log[Abs[z]] + Re[w] < 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SuperscriptBox["z", "k"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["k", " ", "w"]]]]]]], "\[Equal]", FractionBox["1", RowBox[List["1", "-", RowBox[List[SuperscriptBox["\[ExponentialE]", "w"], " ", "z"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Log", "[", RowBox[List["Abs", "[", "z", "]"]], "]"]], "+", RowBox[List["Re", "[", "w", "]"]]]], "<", "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> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <msup> <mi> z </mi> <mi> k </mi> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> w </mi> </mrow> </msup> </mrow> </mrow> <mo> ⩵ </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msup> <mi> ⅇ </mi> <mi> w </mi> </msup> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> z </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> </mrow> <mo> < </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> k </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> k </ci> <ci> w </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <ci> w </ci> </apply> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <lt /> <apply> <plus /> <apply> <ln /> <apply> <abs /> <ci> z </ci> </apply> </apply> <apply> <real /> <ci> w </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "0"]], "\[Infinity]"], RowBox[List[SuperscriptBox["z_", "k_"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["k_", " ", "w_"]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", RowBox[List["1", "-", RowBox[List[SuperscriptBox["\[ExponentialE]", "w"], " ", "z"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Log", "[", RowBox[List["Abs", "[", "z", "]"]], "]"]], "+", RowBox[List["Re", "[", "w", "]"]]]], "<", "0"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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