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Why does Arg'[1. + I] return -0.5?



Announcing the arrival of Valued Associate #679: Cesar Manara
Unicorn Meta Zoo #1: Why another podcast?NDSolve with Piecewise gives the incorrect answer randomlyWhy is NHoldFirst not propagated to symbolic derivatives?Numerical errors/inaccuracies in ProductLogWhy does this integral have a complex component?Why is (-1.)^2. a complex numberWhy does FindMinimum return 'The function value Null is not a real number'?Funny behavior when integratingWhy am I getting RootSearch::numb:Magnitude is returning a complex number if I precede it with Complex Expand?How to take derivative of the argument of an interpolating function










5












$begingroup$


From the document we know that




Arg[z] gives the gives the argument of the complex number z.




Then how about Arg'[z]? This seems to be meaningless, but Mathematica returns something if z is a non-exact number, for example



Arg'[1. + I]
(* -0.5 *)


So my question is:



  1. How is the numeric value of Arg'[z] defined?


  2. Why does Arg' behave like this? What's the potential usage of this behavior?










share|improve this question











$endgroup$











  • $begingroup$
    This can shed some light: Trace[ Arg'[1. + I], TraceInternal -> True ]
    $endgroup$
    – Kuba
    4 hours ago











  • $begingroup$
    It is impossible to understand what the output from Trace[ Arg'[1. + I], TraceInternal -> True ] mean. May be numerics gone mad or something :) so just change 1.0 to 1 in the example given and then Arg will no longer do what you show.
    $endgroup$
    – Nasser
    4 hours ago











  • $begingroup$
    @Kuba Oh blinding light…
    $endgroup$
    – xzczd
    4 hours ago










  • $begingroup$
    @Nasser and xzczd it looks like it calculates derivative value from set of points f or Arg, but I wasn't paying too much attention.
    $endgroup$
    – Kuba
    4 hours ago
















5












$begingroup$


From the document we know that




Arg[z] gives the gives the argument of the complex number z.




Then how about Arg'[z]? This seems to be meaningless, but Mathematica returns something if z is a non-exact number, for example



Arg'[1. + I]
(* -0.5 *)


So my question is:



  1. How is the numeric value of Arg'[z] defined?


  2. Why does Arg' behave like this? What's the potential usage of this behavior?










share|improve this question











$endgroup$











  • $begingroup$
    This can shed some light: Trace[ Arg'[1. + I], TraceInternal -> True ]
    $endgroup$
    – Kuba
    4 hours ago











  • $begingroup$
    It is impossible to understand what the output from Trace[ Arg'[1. + I], TraceInternal -> True ] mean. May be numerics gone mad or something :) so just change 1.0 to 1 in the example given and then Arg will no longer do what you show.
    $endgroup$
    – Nasser
    4 hours ago











  • $begingroup$
    @Kuba Oh blinding light…
    $endgroup$
    – xzczd
    4 hours ago










  • $begingroup$
    @Nasser and xzczd it looks like it calculates derivative value from set of points f or Arg, but I wasn't paying too much attention.
    $endgroup$
    – Kuba
    4 hours ago














5












5








5





$begingroup$


From the document we know that




Arg[z] gives the gives the argument of the complex number z.




Then how about Arg'[z]? This seems to be meaningless, but Mathematica returns something if z is a non-exact number, for example



Arg'[1. + I]
(* -0.5 *)


So my question is:



  1. How is the numeric value of Arg'[z] defined?


  2. Why does Arg' behave like this? What's the potential usage of this behavior?










share|improve this question











$endgroup$




From the document we know that




Arg[z] gives the gives the argument of the complex number z.




Then how about Arg'[z]? This seems to be meaningless, but Mathematica returns something if z is a non-exact number, for example



Arg'[1. + I]
(* -0.5 *)


So my question is:



  1. How is the numeric value of Arg'[z] defined?


  2. Why does Arg' behave like this? What's the potential usage of this behavior?







calculus-and-analysis numerics complex implementation-details






share|improve this question















share|improve this question













share|improve this question




share|improve this question








edited 4 hours ago







xzczd

















asked 4 hours ago









xzczdxzczd

27.9k576258




27.9k576258











  • $begingroup$
    This can shed some light: Trace[ Arg'[1. + I], TraceInternal -> True ]
    $endgroup$
    – Kuba
    4 hours ago











  • $begingroup$
    It is impossible to understand what the output from Trace[ Arg'[1. + I], TraceInternal -> True ] mean. May be numerics gone mad or something :) so just change 1.0 to 1 in the example given and then Arg will no longer do what you show.
    $endgroup$
    – Nasser
    4 hours ago











  • $begingroup$
    @Kuba Oh blinding light…
    $endgroup$
    – xzczd
    4 hours ago










  • $begingroup$
    @Nasser and xzczd it looks like it calculates derivative value from set of points f or Arg, but I wasn't paying too much attention.
    $endgroup$
    – Kuba
    4 hours ago

















  • $begingroup$
    This can shed some light: Trace[ Arg'[1. + I], TraceInternal -> True ]
    $endgroup$
    – Kuba
    4 hours ago











  • $begingroup$
    It is impossible to understand what the output from Trace[ Arg'[1. + I], TraceInternal -> True ] mean. May be numerics gone mad or something :) so just change 1.0 to 1 in the example given and then Arg will no longer do what you show.
    $endgroup$
    – Nasser
    4 hours ago











  • $begingroup$
    @Kuba Oh blinding light…
    $endgroup$
    – xzczd
    4 hours ago










  • $begingroup$
    @Nasser and xzczd it looks like it calculates derivative value from set of points f or Arg, but I wasn't paying too much attention.
    $endgroup$
    – Kuba
    4 hours ago
















$begingroup$
This can shed some light: Trace[ Arg'[1. + I], TraceInternal -> True ]
$endgroup$
– Kuba
4 hours ago





$begingroup$
This can shed some light: Trace[ Arg'[1. + I], TraceInternal -> True ]
$endgroup$
– Kuba
4 hours ago













$begingroup$
It is impossible to understand what the output from Trace[ Arg'[1. + I], TraceInternal -> True ] mean. May be numerics gone mad or something :) so just change 1.0 to 1 in the example given and then Arg will no longer do what you show.
$endgroup$
– Nasser
4 hours ago





$begingroup$
It is impossible to understand what the output from Trace[ Arg'[1. + I], TraceInternal -> True ] mean. May be numerics gone mad or something :) so just change 1.0 to 1 in the example given and then Arg will no longer do what you show.
$endgroup$
– Nasser
4 hours ago













$begingroup$
@Kuba Oh blinding light…
$endgroup$
– xzczd
4 hours ago




$begingroup$
@Kuba Oh blinding light…
$endgroup$
– xzczd
4 hours ago












$begingroup$
@Nasser and xzczd it looks like it calculates derivative value from set of points f or Arg, but I wasn't paying too much attention.
$endgroup$
– Kuba
4 hours ago





$begingroup$
@Nasser and xzczd it looks like it calculates derivative value from set of points f or Arg, but I wasn't paying too much attention.
$endgroup$
– Kuba
4 hours ago











3 Answers
3






active

oldest

votes


















6












$begingroup$

The internal Trace[] Kuba advises shows calculations consistent with the numeric approximation of the partial derivative with respect to the real part:



D[ComplexExpand[Arg[x + I y], TargetFunctions -> Re, Im], x] /. 
x -> 1 /. y -> 1
(* -(1/2) *)


This is what Mathematica does with the derivative of a numeric function with approximate input.



Other examples:



ClearAll[f, g];
f[x_?NumericQ] := Re[x]^2;
g[x_?NumericQ] := Im[x]^2;

f'[1. + I]
g'[1. + I]
(*
1.999999999999995`
-2.7506672371246275`*^-15
*)


It seems like the wrong way to evalutate Derivative.






share|improve this answer









$endgroup$












  • $begingroup$
    Funny, Abs'[2. + I] returns the input.
    $endgroup$
    – xzczd
    54 mins ago


















3












$begingroup$

The definition of the argument is $arg(z)=textIm(ln(z))$. Its partial derivative with respect to $z$ would then be



$$
fracpartial arg(z)partial z=
fracpartialpartial zfracln(z)-ln(z^*)2i
= -fraci2z.
$$



What you see looks like twice the real part of this expression:



With[z = 2. + 5 I,
Arg'[z], 2Re[-I/(2z)]]

(* -0.172414, -0.172414 *)


I don't know in which sense this is the "correct" answer. It could be that what is actually calculated is not the partial derivative with respect to $z$, but rather the sum of partial derivatives



$$
fracpartial arg(z)partial z
+fracpartial arg(z)partial z^*
= -fraci2z+fraci2z^*
= -fractextIm(z)
$$



With[z = 2. + 5 I,
Arg'[z], 2Re[-I/(2z)], -Im[z]/Abs[z]^2]

(* -0.172414, -0.172414, -0.172414 *)





share|improve this answer











$endgroup$












  • $begingroup$
    Er… how is derivative of $ln(z^*)$ defined here?
    $endgroup$
    – xzczd
    3 hours ago










  • $begingroup$
    Any idea why then With[z = 1.0 + I, Arg'[z]] not same as With[z = 1 + I, Arg'[z]]? Should not these give same result?
    $endgroup$
    – Nasser
    3 hours ago











  • $begingroup$
    @xzczd $z$ and $z^*$ are independent variables in complex analysis, so $partial z^*/partial z=0$ etc.
    $endgroup$
    – Roman
    3 hours ago










  • $begingroup$
    @Nasser they do give the same result when you apply N: Arg'[1 + I] // N also gives -0.5.
    $endgroup$
    – Roman
    3 hours ago


















0












$begingroup$

Funny Answer,
Just ignore the method it uses to calculate and use your own method of calculation based on the result..



The reason for using this is to extended divided by two.



Arg'[1./2]





share|improve this answer








New contributor




Zillinium is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






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    3 Answers
    3






    active

    oldest

    votes








    3 Answers
    3






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    6












    $begingroup$

    The internal Trace[] Kuba advises shows calculations consistent with the numeric approximation of the partial derivative with respect to the real part:



    D[ComplexExpand[Arg[x + I y], TargetFunctions -> Re, Im], x] /. 
    x -> 1 /. y -> 1
    (* -(1/2) *)


    This is what Mathematica does with the derivative of a numeric function with approximate input.



    Other examples:



    ClearAll[f, g];
    f[x_?NumericQ] := Re[x]^2;
    g[x_?NumericQ] := Im[x]^2;

    f'[1. + I]
    g'[1. + I]
    (*
    1.999999999999995`
    -2.7506672371246275`*^-15
    *)


    It seems like the wrong way to evalutate Derivative.






    share|improve this answer









    $endgroup$












    • $begingroup$
      Funny, Abs'[2. + I] returns the input.
      $endgroup$
      – xzczd
      54 mins ago















    6












    $begingroup$

    The internal Trace[] Kuba advises shows calculations consistent with the numeric approximation of the partial derivative with respect to the real part:



    D[ComplexExpand[Arg[x + I y], TargetFunctions -> Re, Im], x] /. 
    x -> 1 /. y -> 1
    (* -(1/2) *)


    This is what Mathematica does with the derivative of a numeric function with approximate input.



    Other examples:



    ClearAll[f, g];
    f[x_?NumericQ] := Re[x]^2;
    g[x_?NumericQ] := Im[x]^2;

    f'[1. + I]
    g'[1. + I]
    (*
    1.999999999999995`
    -2.7506672371246275`*^-15
    *)


    It seems like the wrong way to evalutate Derivative.






    share|improve this answer









    $endgroup$












    • $begingroup$
      Funny, Abs'[2. + I] returns the input.
      $endgroup$
      – xzczd
      54 mins ago













    6












    6








    6





    $begingroup$

    The internal Trace[] Kuba advises shows calculations consistent with the numeric approximation of the partial derivative with respect to the real part:



    D[ComplexExpand[Arg[x + I y], TargetFunctions -> Re, Im], x] /. 
    x -> 1 /. y -> 1
    (* -(1/2) *)


    This is what Mathematica does with the derivative of a numeric function with approximate input.



    Other examples:



    ClearAll[f, g];
    f[x_?NumericQ] := Re[x]^2;
    g[x_?NumericQ] := Im[x]^2;

    f'[1. + I]
    g'[1. + I]
    (*
    1.999999999999995`
    -2.7506672371246275`*^-15
    *)


    It seems like the wrong way to evalutate Derivative.






    share|improve this answer









    $endgroup$



    The internal Trace[] Kuba advises shows calculations consistent with the numeric approximation of the partial derivative with respect to the real part:



    D[ComplexExpand[Arg[x + I y], TargetFunctions -> Re, Im], x] /. 
    x -> 1 /. y -> 1
    (* -(1/2) *)


    This is what Mathematica does with the derivative of a numeric function with approximate input.



    Other examples:



    ClearAll[f, g];
    f[x_?NumericQ] := Re[x]^2;
    g[x_?NumericQ] := Im[x]^2;

    f'[1. + I]
    g'[1. + I]
    (*
    1.999999999999995`
    -2.7506672371246275`*^-15
    *)


    It seems like the wrong way to evalutate Derivative.







    share|improve this answer












    share|improve this answer



    share|improve this answer










    answered 1 hour ago









    Michael E2Michael E2

    151k12203483




    151k12203483











    • $begingroup$
      Funny, Abs'[2. + I] returns the input.
      $endgroup$
      – xzczd
      54 mins ago
















    • $begingroup$
      Funny, Abs'[2. + I] returns the input.
      $endgroup$
      – xzczd
      54 mins ago















    $begingroup$
    Funny, Abs'[2. + I] returns the input.
    $endgroup$
    – xzczd
    54 mins ago




    $begingroup$
    Funny, Abs'[2. + I] returns the input.
    $endgroup$
    – xzczd
    54 mins ago











    3












    $begingroup$

    The definition of the argument is $arg(z)=textIm(ln(z))$. Its partial derivative with respect to $z$ would then be



    $$
    fracpartial arg(z)partial z=
    fracpartialpartial zfracln(z)-ln(z^*)2i
    = -fraci2z.
    $$



    What you see looks like twice the real part of this expression:



    With[z = 2. + 5 I,
    Arg'[z], 2Re[-I/(2z)]]

    (* -0.172414, -0.172414 *)


    I don't know in which sense this is the "correct" answer. It could be that what is actually calculated is not the partial derivative with respect to $z$, but rather the sum of partial derivatives



    $$
    fracpartial arg(z)partial z
    +fracpartial arg(z)partial z^*
    = -fraci2z+fraci2z^*
    = -fractextIm(z)
    $$



    With[z = 2. + 5 I,
    Arg'[z], 2Re[-I/(2z)], -Im[z]/Abs[z]^2]

    (* -0.172414, -0.172414, -0.172414 *)





    share|improve this answer











    $endgroup$












    • $begingroup$
      Er… how is derivative of $ln(z^*)$ defined here?
      $endgroup$
      – xzczd
      3 hours ago










    • $begingroup$
      Any idea why then With[z = 1.0 + I, Arg'[z]] not same as With[z = 1 + I, Arg'[z]]? Should not these give same result?
      $endgroup$
      – Nasser
      3 hours ago











    • $begingroup$
      @xzczd $z$ and $z^*$ are independent variables in complex analysis, so $partial z^*/partial z=0$ etc.
      $endgroup$
      – Roman
      3 hours ago










    • $begingroup$
      @Nasser they do give the same result when you apply N: Arg'[1 + I] // N also gives -0.5.
      $endgroup$
      – Roman
      3 hours ago















    3












    $begingroup$

    The definition of the argument is $arg(z)=textIm(ln(z))$. Its partial derivative with respect to $z$ would then be



    $$
    fracpartial arg(z)partial z=
    fracpartialpartial zfracln(z)-ln(z^*)2i
    = -fraci2z.
    $$



    What you see looks like twice the real part of this expression:



    With[z = 2. + 5 I,
    Arg'[z], 2Re[-I/(2z)]]

    (* -0.172414, -0.172414 *)


    I don't know in which sense this is the "correct" answer. It could be that what is actually calculated is not the partial derivative with respect to $z$, but rather the sum of partial derivatives



    $$
    fracpartial arg(z)partial z
    +fracpartial arg(z)partial z^*
    = -fraci2z+fraci2z^*
    = -fractextIm(z)
    $$



    With[z = 2. + 5 I,
    Arg'[z], 2Re[-I/(2z)], -Im[z]/Abs[z]^2]

    (* -0.172414, -0.172414, -0.172414 *)





    share|improve this answer











    $endgroup$












    • $begingroup$
      Er… how is derivative of $ln(z^*)$ defined here?
      $endgroup$
      – xzczd
      3 hours ago










    • $begingroup$
      Any idea why then With[z = 1.0 + I, Arg'[z]] not same as With[z = 1 + I, Arg'[z]]? Should not these give same result?
      $endgroup$
      – Nasser
      3 hours ago











    • $begingroup$
      @xzczd $z$ and $z^*$ are independent variables in complex analysis, so $partial z^*/partial z=0$ etc.
      $endgroup$
      – Roman
      3 hours ago










    • $begingroup$
      @Nasser they do give the same result when you apply N: Arg'[1 + I] // N also gives -0.5.
      $endgroup$
      – Roman
      3 hours ago













    3












    3








    3





    $begingroup$

    The definition of the argument is $arg(z)=textIm(ln(z))$. Its partial derivative with respect to $z$ would then be



    $$
    fracpartial arg(z)partial z=
    fracpartialpartial zfracln(z)-ln(z^*)2i
    = -fraci2z.
    $$



    What you see looks like twice the real part of this expression:



    With[z = 2. + 5 I,
    Arg'[z], 2Re[-I/(2z)]]

    (* -0.172414, -0.172414 *)


    I don't know in which sense this is the "correct" answer. It could be that what is actually calculated is not the partial derivative with respect to $z$, but rather the sum of partial derivatives



    $$
    fracpartial arg(z)partial z
    +fracpartial arg(z)partial z^*
    = -fraci2z+fraci2z^*
    = -fractextIm(z)
    $$



    With[z = 2. + 5 I,
    Arg'[z], 2Re[-I/(2z)], -Im[z]/Abs[z]^2]

    (* -0.172414, -0.172414, -0.172414 *)





    share|improve this answer











    $endgroup$



    The definition of the argument is $arg(z)=textIm(ln(z))$. Its partial derivative with respect to $z$ would then be



    $$
    fracpartial arg(z)partial z=
    fracpartialpartial zfracln(z)-ln(z^*)2i
    = -fraci2z.
    $$



    What you see looks like twice the real part of this expression:



    With[z = 2. + 5 I,
    Arg'[z], 2Re[-I/(2z)]]

    (* -0.172414, -0.172414 *)


    I don't know in which sense this is the "correct" answer. It could be that what is actually calculated is not the partial derivative with respect to $z$, but rather the sum of partial derivatives



    $$
    fracpartial arg(z)partial z
    +fracpartial arg(z)partial z^*
    = -fraci2z+fraci2z^*
    = -fractextIm(z)
    $$



    With[z = 2. + 5 I,
    Arg'[z], 2Re[-I/(2z)], -Im[z]/Abs[z]^2]

    (* -0.172414, -0.172414, -0.172414 *)






    share|improve this answer














    share|improve this answer



    share|improve this answer








    edited 3 hours ago

























    answered 3 hours ago









    RomanRoman

    6,06511132




    6,06511132











    • $begingroup$
      Er… how is derivative of $ln(z^*)$ defined here?
      $endgroup$
      – xzczd
      3 hours ago










    • $begingroup$
      Any idea why then With[z = 1.0 + I, Arg'[z]] not same as With[z = 1 + I, Arg'[z]]? Should not these give same result?
      $endgroup$
      – Nasser
      3 hours ago











    • $begingroup$
      @xzczd $z$ and $z^*$ are independent variables in complex analysis, so $partial z^*/partial z=0$ etc.
      $endgroup$
      – Roman
      3 hours ago










    • $begingroup$
      @Nasser they do give the same result when you apply N: Arg'[1 + I] // N also gives -0.5.
      $endgroup$
      – Roman
      3 hours ago
















    • $begingroup$
      Er… how is derivative of $ln(z^*)$ defined here?
      $endgroup$
      – xzczd
      3 hours ago










    • $begingroup$
      Any idea why then With[z = 1.0 + I, Arg'[z]] not same as With[z = 1 + I, Arg'[z]]? Should not these give same result?
      $endgroup$
      – Nasser
      3 hours ago











    • $begingroup$
      @xzczd $z$ and $z^*$ are independent variables in complex analysis, so $partial z^*/partial z=0$ etc.
      $endgroup$
      – Roman
      3 hours ago










    • $begingroup$
      @Nasser they do give the same result when you apply N: Arg'[1 + I] // N also gives -0.5.
      $endgroup$
      – Roman
      3 hours ago















    $begingroup$
    Er… how is derivative of $ln(z^*)$ defined here?
    $endgroup$
    – xzczd
    3 hours ago




    $begingroup$
    Er… how is derivative of $ln(z^*)$ defined here?
    $endgroup$
    – xzczd
    3 hours ago












    $begingroup$
    Any idea why then With[z = 1.0 + I, Arg'[z]] not same as With[z = 1 + I, Arg'[z]]? Should not these give same result?
    $endgroup$
    – Nasser
    3 hours ago





    $begingroup$
    Any idea why then With[z = 1.0 + I, Arg'[z]] not same as With[z = 1 + I, Arg'[z]]? Should not these give same result?
    $endgroup$
    – Nasser
    3 hours ago













    $begingroup$
    @xzczd $z$ and $z^*$ are independent variables in complex analysis, so $partial z^*/partial z=0$ etc.
    $endgroup$
    – Roman
    3 hours ago




    $begingroup$
    @xzczd $z$ and $z^*$ are independent variables in complex analysis, so $partial z^*/partial z=0$ etc.
    $endgroup$
    – Roman
    3 hours ago












    $begingroup$
    @Nasser they do give the same result when you apply N: Arg'[1 + I] // N also gives -0.5.
    $endgroup$
    – Roman
    3 hours ago




    $begingroup$
    @Nasser they do give the same result when you apply N: Arg'[1 + I] // N also gives -0.5.
    $endgroup$
    – Roman
    3 hours ago











    0












    $begingroup$

    Funny Answer,
    Just ignore the method it uses to calculate and use your own method of calculation based on the result..



    The reason for using this is to extended divided by two.



    Arg'[1./2]





    share|improve this answer








    New contributor




    Zillinium is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.






    $endgroup$

















      0












      $begingroup$

      Funny Answer,
      Just ignore the method it uses to calculate and use your own method of calculation based on the result..



      The reason for using this is to extended divided by two.



      Arg'[1./2]





      share|improve this answer








      New contributor




      Zillinium is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.






      $endgroup$















        0












        0








        0





        $begingroup$

        Funny Answer,
        Just ignore the method it uses to calculate and use your own method of calculation based on the result..



        The reason for using this is to extended divided by two.



        Arg'[1./2]





        share|improve this answer








        New contributor




        Zillinium is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
        Check out our Code of Conduct.






        $endgroup$



        Funny Answer,
        Just ignore the method it uses to calculate and use your own method of calculation based on the result..



        The reason for using this is to extended divided by two.



        Arg'[1./2]






        share|improve this answer








        New contributor




        Zillinium is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
        Check out our Code of Conduct.









        share|improve this answer



        share|improve this answer






        New contributor




        Zillinium is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
        Check out our Code of Conduct.









        answered 19 mins ago









        ZilliniumZillinium

        1011




        1011




        New contributor




        Zillinium is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
        Check out our Code of Conduct.





        New contributor





        Zillinium is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
        Check out our Code of Conduct.






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        Check out our Code of Conduct.



























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