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variants of this functions
Log




Mathematica Notation

Traditional Notation









Elementary Functions > Log[z] > Representations through equivalent functions > With related functions > Involving other related functions





http://functions.wolfram.com/01.04.27.0060.01









  


  








Input Form





Log[ProductLog[-1, z]] == Log[z] - ProductLog[-1, z] - 2 I Pi /; !IntervalMemberQ[Interval[{-Infinity, -(1/E)}], z] && !IntervalMemberQ[Interval[{0, Infinity}], z]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Log", "[", RowBox[List["ProductLog", "[", RowBox[List[RowBox[List["-", "1"]], ",", "z"]], "]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["Log", "[", "z", "]"]], "-", RowBox[List["ProductLog", "[", RowBox[List[RowBox[List["-", "1"]], ",", "z"]], "]"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]"]]]]]], "/;", " ", RowBox[List[FormBox[FormBox[RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", FractionBox["1", "\[ExponentialE]"]]]]], "}"]], "]"]], ",", "z"]], "]"]], "]"]], TraditionalForm], TraditionalForm], "\[And]", FormBox[FormBox[RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["0", ",", "\[Infinity]"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]], TraditionalForm], TraditionalForm]]]]]]]








MathML Form







<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> &#8289; </mo> <mo> ( </mo> <mrow> <mstyle fontweight='normal' fontstyle='normal'> <msub> <mi> W </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> </mstyle> <mo> ( </mo> <mi fontweight='normal' fontstyle='italic'> z </mi> <mo fontweight='normal' fontstyle='normal'> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mstyle fontweight='normal' fontstyle='normal'> <msub> <mi> W </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> </mstyle> <mo> ( </mo> <mi fontweight='normal' fontstyle='italic'> z </mi> <mo fontweight='normal' fontstyle='normal'> ) </mo> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mrow> </mrow> <mo> /; </mo> <mtext> &#8203;&#8203; </mtext> <mrow> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> &#8519; </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8743; </mo> <mtext> &#8203; </mtext> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mstyle fontweight='normal' fontstyle='normal'> <msub> <mi> W </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> </mstyle> <mo> ( </mo> <mi fontweight='normal' fontstyle='italic'> z </mi> <mo fontweight='normal' fontstyle='normal'> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mstyle fontweight='normal' fontstyle='normal'> <msub> <mi> W </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> </mstyle> <mo> ( </mo> <mi fontweight='normal' fontstyle='italic'> z </mi> <mo fontweight='normal' fontstyle='normal'> ) </mo> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mrow> </mrow> <mo> /; </mo> <mtext> &#8203;&#8203; </mtext> <mrow> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> &#8519; </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8743; </mo> <mtext> &#8203; </mtext> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Log", "[", RowBox[List["ProductLog", "[", RowBox[List[RowBox[List["-", "1"]], ",", "z_"]], "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Log", "[", "z", "]"]], "-", RowBox[List["ProductLog", "[", RowBox[List[RowBox[List["-", "1"]], ",", "z"]], "]"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]"]]]], "/;", RowBox[List[RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", FractionBox["1", "\[ExponentialE]"]]]]], "}"]], "]"]], ",", "z"]], "]"]]]], "&&", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["0", ",", "\[Infinity]"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]]]


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





2001-10-29