Logarithm: Complex characteristics (formula 01.05.19.0006)

Log

$Mathematica Notation$

Elementary Functions

Log[a,z]

Complex characteristics

Signum value

http://functions.wolfram.com/01.05.19.0006.01

Input Form

Sign[Log[a, z]] == (ArcTan[Re[a], Im[a]] ArcTan[Re[z], Im[z]] + Log[Abs[a]] Log[Abs[z]] - I ((-ArcTan[Re[z], Im[z]]) Log[Abs[a]] + ArcTan[Re[a], Im[a]] Log[Abs[z]]))/ (Sqrt[ArcTan[Re[a], Im[a]]^2 + Log[Abs[a]]^2] Sqrt[ArcTan[Re[z], Im[z]]^2 + Log[Abs[z]]^2])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["Sign", "[", RowBox[List["Log", "[", RowBox[List["a", ",", "z"]], "]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["Re", "[", "a", "]"]], ",", RowBox[List["Im", "[", "a", "]"]]]], "]"]], " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ",", RowBox[List["Im", "[", "z", "]"]]]], "]"]]]], "+", RowBox[List[RowBox[List["Log", "[", RowBox[List["Abs", "[", "a", "]"]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["Abs", "[", "z", "]"]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ",", RowBox[List["Im", "[", "z", "]"]]]], "]"]]]], " ", RowBox[List["Log", "[", RowBox[List["Abs", "[", "a", "]"]], "]"]]]], "+", RowBox[List[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["Re", "[", "a", "]"]], ",", RowBox[List["Im", "[", "a", "]"]]]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["Abs", "[", "z", "]"]], "]"]]]]]], ")"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["Re", "[", "a", "]"]], ",", RowBox[List["Im", "[", "a", "]"]]]], "]"]], "2"], "+", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["Abs", "[", "a", "]"]], "]"]], "2"]]]], SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ",", RowBox[List["Im", "[", "z", "]"]]]], "]"]], "2"], "+", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["Abs", "[", "z", "]"]], "]"]], "2"]]]]]], ")"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> sgn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> log </mi> <mi> a </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> a </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <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> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <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> </mrow> <mo> - </mo> <mrow> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> a </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <msup> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> a </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <msup> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <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> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Sign </ci> <apply> <log /> <logbase> <ci> a </ci> </logbase> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <arctan /> <apply> <real /> <ci> a </ci> </apply> <apply> <imaginary /> <ci> a </ci> </apply> </apply> <apply> <arctan /> <apply> <real /> <ci> z </ci> </apply> <apply> <imaginary /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <ln /> <apply> <abs /> <ci> a </ci> </apply> </apply> <apply> <ln /> <apply> <abs /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <plus /> <apply> <times /> <apply> <arctan /> <apply> <real /> <ci> a </ci> </apply> <apply> <imaginary /> <ci> a </ci> </apply> </apply> <apply> <ln /> <apply> <abs /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <arctan /> <apply> <real /> <ci> z </ci> </apply> <apply> <imaginary /> <ci> z </ci> </apply> </apply> <apply> <ln /> <apply> <abs /> <ci> a </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <arctan /> <apply> <real /> <ci> a </ci> </apply> <apply> <imaginary /> <ci> a </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ln /> <apply> <abs /> <ci> a </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <arctan /> <apply> <real /> <ci> z </ci> </apply> <apply> <imaginary /> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ln /> <apply> <abs /> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sign", "[", RowBox[List["Log", "[", RowBox[List["a_", ",", "z_"]], "]"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["Re", "[", "a", "]"]], ",", RowBox[List["Im", "[", "a", "]"]]]], "]"]], " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ",", RowBox[List["Im", "[", "z", "]"]]]], "]"]]]], "+", RowBox[List[RowBox[List["Log", "[", RowBox[List["Abs", "[", "a", "]"]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["Abs", "[", "z", "]"]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ",", RowBox[List["Im", "[", "z", "]"]]]], "]"]]]], " ", RowBox[List["Log", "[", RowBox[List["Abs", "[", "a", "]"]], "]"]]]], "+", RowBox[List[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["Re", "[", "a", "]"]], ",", RowBox[List["Im", "[", "a", "]"]]]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["Abs", "[", "z", "]"]], "]"]]]]]], ")"]]]]]], RowBox[List[SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["Re", "[", "a", "]"]], ",", RowBox[List["Im", "[", "a", "]"]]]], "]"]], "2"], "+", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["Abs", "[", "a", "]"]], "]"]], "2"]]]], " ", SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ",", RowBox[List["Im", "[", "z", "]"]]]], "]"]], "2"], "+", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["Abs", "[", "z", "]"]], "]"]], "2"]]]]]]]]]]]

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

2007-05-02

Log[z]