How do I differentiate complex numbers in complex analysis?

In summary, the conversation discusses the calculation of the limit of a complex function and how it can be simplified by considering z as r*exp{iArg(z)}. The speaker also mentions the importance of considering how the variable tends to zero in multivariable calculus.
  • #1
Wiemster
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I can't follow a vertain step in my book in complex analysis (Kwok):

[tex]\frac{d \bar{z}}{dz} = lim _{\Delta z \rightarrow 0} \frac{\Delta \bar{z}}{\Delta z} = lim _{\Delta z \rightarrow 0} e^{-2i Arg \Delta z}[/tex]

I can't see how you could arrive at this last expression, could anyone give me a hint as how to do this?
 
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  • #2
I guess you're thinking about this in the form z=x+iy. If you can't get that to work, think of z as r*exp{iArg(z)} instead, because then it is immediate.
 
  • #3
How is it then immediate? I can't unambiguously see what happens when I vary z as in real analysis, do you vary r and argz independently in some way?
 
  • #4
For any complex number w*/w (* for conjugate), is precisely exp{-2iArg(w)}. That is why I said it was immediate.

The point is that that limit is not well defined. It depends on how delta(z) tends to zero. Just as in real analysis you need limits to be independent of the way you let things tend to zero (or some point). This is multivariable calc with some extra structure.
 
Last edited:
  • #5
Aha, now I see... and because this is so for any complex number it also holds in the variation of this number. Great, thx!
 

1. What is complex differentiation?

Complex differentiation is a mathematical concept that deals with finding the derivative of a function whose input and output are complex numbers. It is an extension of the concept of differentiation in real numbers and is used extensively in fields such as complex analysis and engineering.

2. How is complex differentiation different from real differentiation?

While real differentiation deals with finding the slope of a curve at a point, complex differentiation involves finding the rate of change of a complex-valued function in a specific direction. This is because complex numbers have both real and imaginary components, so the concept of slope does not apply in the same way as in real numbers.

3. Why is complex differentiation important?

Complex differentiation is important because it allows us to study and understand complex-valued functions and their behavior in a more precise manner. It is also a fundamental tool in solving complex differential equations, which have numerous applications in physics, engineering, and other fields.

4. What are the basic rules of complex differentiation?

The basic rules of complex differentiation are similar to those in real differentiation, with a few key differences. The power rule, product rule, quotient rule, and chain rule all apply in complex differentiation. However, the derivative of a complex conjugate is the complex conjugate of the derivative, and the derivative of an exponential function is the function itself multiplied by the derivative of the exponent.

5. How is complex differentiation used in practical applications?

Complex differentiation has numerous practical applications in various fields such as physics, engineering, and economics. For example, it is used in signal processing to analyze and manipulate complex signals, in control theory to design and analyze complex control systems, and in fluid mechanics to model and simulate complex fluid flow. It is also used in optimization problems, where complex variables are used to represent constraints and objective functions.

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