- #1
daudaudaudau
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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.
Unraveling the Complex Derivative of Arctan(z)
- Thread starter daudaudaudau
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In summary, your CAS does not agree with you that the derivative of arctan(z) is just 1/(1+z^2). The derivative of the complex argument of z, Arg(z) is equal to 1/(z^2+1) because this is the derivative of arctan(z).
- #2
HallsofIvy
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The derivative of arctan(z) is 1/(1+ z^2)+ C no matter what number field z is in. What does your CAS say?
- #3
daudaudaudau
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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) ?
- #4
Hurkyl
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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.
- #5
daudaudaudau
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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 ?
- #6
Hurkyl
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Arg is differentiable as a function on the plane. It's just not differentiable as a complex function.daudaudaudau said:
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- #8
Hurkyl
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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]
andD[Arg[x*(I+1)+1],x]/.x->a
simplify to?- #9
- #10
Hurkyl
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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].
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].
- #11
daudaudaudau
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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. :)
to get the "right" result. :)
Related to Unraveling the Complex Derivative of Arctan(z)
1. What is a complex argument derivative?
A complex argument derivative is a mathematical concept that involves finding the rate of change of a function with respect to a complex variable. It is similar to finding the derivative of a real-valued function, but it takes into account the imaginary component of the complex variable.
2. How is a complex argument derivative calculated?
To calculate a complex argument derivative, we use the Cauchy-Riemann equations, which relate the real and imaginary parts of a complex function. These equations involve taking partial derivatives of the function with respect to both the real and imaginary components of the complex variable.
3. What is the importance of complex argument derivatives?
Complex argument derivatives have many applications in mathematics and physics, particularly in the fields of complex analysis and signal processing. They are also important in understanding the behavior of complex functions and their corresponding graphs.
4. Are there any limitations to calculating complex argument derivatives?
Yes, there are certain limitations to calculating complex argument derivatives. These include cases where the function is not differentiable at a certain point, or when the function is not analytic (meaning it cannot be expressed as a power series). In these cases, alternative methods must be used to find the complex argument derivative.
5. Can complex argument derivatives be visualized?
Yes, complex argument derivatives can be visualized using color mapping techniques. By assigning a specific color to each point in the complex plane based on its corresponding derivative value, we can create a visual representation of the function's complex argument derivative. This can help in gaining a better understanding of the behavior of the function.
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