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variants of this functions
ArcTan






Mathematica Notation

Traditional Notation









Elementary Functions > ArcTan[x,y] > Complex characteristics > Conjugate value





http://functions.wolfram.com/01.15.19.0006.01









  


  










Input Form





Conjugate[ArcTan[x, y]] == ArcTan[(Cos[(1/2) ArcTan[-Im[x]^2 - Im[y]^2 + Re[x]^2 + Re[y]^2, 2 Im[x] Re[x] + 2 Im[y] Re[y]]] (-Im[y] + Re[x]))/ ((2 Im[x] Re[x] + 2 Im[y] Re[y])^2 + (-Im[x]^2 - Im[y]^2 + Re[x]^2 + Re[y]^2)^2)^(1/4) + ((Im[x] + Re[y]) Sin[(1/2) ArcTan[-Im[x]^2 - Im[y]^2 + Re[x]^2 + Re[y]^2, 2 Im[x] Re[x] + 2 Im[y] Re[y]]])/ ((2 Im[x] Re[x] + 2 Im[y] Re[y])^2 + (-Im[x]^2 - Im[y]^2 + Re[x]^2 + Re[y]^2)^2)^(1/4), (Cos[(1/2) ArcTan[-Im[x]^2 - Im[y]^2 + Re[x]^2 + Re[y]^2, 2 Im[x] Re[x] + 2 Im[y] Re[y]]] (Im[x] + Re[y]))/ ((2 Im[x] Re[x] + 2 Im[y] Re[y])^2 + (-Im[x]^2 - Im[y]^2 + Re[x]^2 + Re[y]^2)^2)^(1/4) - ((-Im[y] + Re[x]) Sin[(1/2) ArcTan[-Im[x]^2 - Im[y]^2 + Re[x]^2 + Re[y]^2, 2 Im[x] Re[x] + 2 Im[y] Re[y]]])/ ((2 Im[x] Re[x] + 2 Im[y] Re[y])^2 + (-Im[x]^2 - Im[y]^2 + Re[x]^2 + Re[y]^2)^2)^(1/4)] + I Log[Sqrt[(-Im[y] + Re[x])^2 + (Im[x] + Re[y])^2]/ ((2 Im[x] Re[x] + 2 Im[y] Re[y])^2 + (-Im[x]^2 - Im[y]^2 + Re[x]^2 + Re[y]^2)^2)^(1/4)]










Standard Form





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MathML Form







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</mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mroot> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <msup> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <msup> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mn> 4 </mn> </mroot> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <msup> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <msup> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> , </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mroot> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <msup> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <msup> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mn> 4 </mn> </mroot> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <msqrt> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </msqrt> <mroot> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <msup> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <msup> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mn> 4 </mn> </mroot> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> OverBar </ci> <apply> <arctan /> <ci> x </ci> <ci> y </ci> </apply> </apply> <apply> <plus /> <apply> <arctan /> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <imaginary /> <ci> x </ci> </apply> <apply> <real /> <ci> y </ci> </apply> </apply> <apply> <sin /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <arctan /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <imaginary /> <ci> x </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <imaginary /> <ci> y </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <real /> <ci> x </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <real /> <ci> y </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <imaginary /> <ci> x </ci> </apply> <apply> <real /> <ci> x </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <imaginary /> <ci> y </ci> </apply> <apply> <real /> <ci> y </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <imaginary /> <ci> x </ci> </apply> <apply> <real /> <ci> x 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<power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <imaginary /> <ci> x </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <imaginary /> <ci> y </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <real /> <ci> x </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <real /> <ci> y </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <ln /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <real /> <ci> x </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <imaginary /> <ci> y </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <imaginary /> <ci> x </ci> </apply> <apply> <real /> <ci> y </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <imaginary /> <ci> x </ci> </apply> <apply> <real /> <ci> x </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <imaginary /> <ci> y </ci> </apply> <apply> <real /> <ci> y </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <imaginary /> <ci> x </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <imaginary /> <ci> y </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <real /> <ci> x </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <real /> <ci> y </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29





© 1998- Wolfram Research, Inc.