Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Complex argument derivative

  1. Oct 13, 2008 #1
    If z is a complex number, isn't the derivative of arctan(z) just 1/(1+z^2) ? That's what I would think, but my CAS does not agree with me.
     
  2. jcsd
  3. Oct 13, 2008 #2

    HallsofIvy

    User Avatar
    Science Advisor

    The derivative of arctan(z) is 1/(1+ z^2)+ C no matter what number field z is in. What does your CAS say?
     
  4. Oct 13, 2008 #3
    I meant to write: Isn't the derivative of the complex argument of z, Arg(z) equal to 1/(z^2+1) beacuse this is the derivative of arctan(z) ?
     
  5. Oct 13, 2008 #4

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    I don't see your line of reasoning.


    Anyways, the facts of the situation are that Arg(z) isn't complex-differentiable. (Try computing it directly) You have a huge clue that something's wrong: Arg(z) is a strictly real-valued function, yet your alledged derivative can take complex values.
     
  6. Oct 13, 2008 #5
    Yes, that made no sense, sorry. What about if A and B are a complex constants and x is a real number. Then I suppose the derivative of Arg(A+C*x) exists ?
     
  7. Oct 13, 2008 #6

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Arg is differentiable as a function on the plane. It's just not differentiable as a complex function.
     
  8. Oct 13, 2008 #7
    What about this attached screenshot then? Why does the derivative even have an imaginary part?
     

    Attached Files:

  9. Oct 13, 2008 #8

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    My best guess is that you gave a, or maybe x, a numerical value earlier in your session. What does it think
    D[Arg[x*(I+1)+1],x]​
    and
    D[Arg[x*(I+1)+1],x]/.x->a​
    simplify to?
     
  10. Oct 13, 2008 #9
    I started a new session, and this is what it looks like. D[Arg[...]] makes no sense, but D[ArcTan[...]] looks about right. Isn't Arg using ArcTan ?
     

    Attached Files:

  11. Oct 13, 2008 #10

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Aha, I see what's going on.

    Mathematica's documentation says that it only 'evaluates' Arg when it has a numerical result. It's internal thinking about the function changed in the following way:


    When you first asked for the derivative, it made a purely formal calculation via the chain rule, probably thinking of it as a 'formal' complex derivative. (Note that mathematica will do the same thing with any formal symbol. Try asking for D[f[x], x] when you haven't defined f)

    But when you replaced x with an actual number, you kicked in the programming for Arg, so it happily returned (1+i) times whatever it thinks the derivative of Arg should be. (I can't explain the extra factor of -1/2, though))


    If you want to insist on working with Arg directly, you might be able to get better results in one of the following ways:
    . Use assumptions to tell Mathematica that x is a real number.
    . Try replacing x in the expression with Abs[x], or maybe Re[x].
     
  12. Oct 14, 2008 #11
    Yes, that seems to be the case. Apparently one has to write D[ComplexExpand[Arg[x*(1 + I) + 1], TargetFunctions -> {Re, Im}], x]

    to get the "right" result. :)
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook




Loading...
Similar Threads for Complex argument derivative Date
Argument value in complex numbers in polar form. Sep 10, 2009
Zeta function for complex argument Jul 23, 2009
Derivating polynomial with complex argument Mar 28, 2007
Complex analysis - argument principle May 2, 2006