Complex Limit Definition: Trying to Remember

In summary: Then the denominator is \Delta x + \Delta y then?Sorry, you posted while I was editing my post- the denominator should be the distance from the point (\Delta x, \Delta y) to (0, 0):
  • #1
fauboca
158
0
Trying to remember how to use the definition of a complex limit.

[tex]\lim_{\Delta z\to 0}\frac{f(z+h)-f(z)}{\Delta z}[/tex]

[tex]f(z) = |z| = \sqrt{x^2+y^2}[/tex]

[tex]\Delta z = \Delta x + i\Delta y[/tex]

[tex]\lim_{\Delta x\to 0}\frac{\sqrt{(x+\Delta x)^2+(y+\Delta y)^2}- \sqrt{x^2+y^2}}{\Delta x}[/tex]

Is that correct? Or do I just have the delta x with the x? Or is there a x + delta x and y + delta y?

Thanks.
 
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  • #2
Well, that is almost the correct formula. The only things missing are that you need to take the limit as both [itex]\Delta x[/itex], [itex]\Delta y[/itex] go to 0 and the denominator must be [itex]\sqrt{(\Delta x)^2+ (/Delta y)^2}[/itex], the distance from [itex](\Delta x, \Delta y)[/itex] to (0, 0), and you have to take the limit without assuming a relation between [itex]\Delta x[/itex] and [itex]\Delta y[/itex]. Strictly spealing, we take the limit as the point [itex](\Delta x, \Delta y)[/itex] goes to (0, 0) along any possible path in two dimensions.
 
  • #3
HallsofIvy said:
Well, that is almost the correct formula. The only thing missing is that you need to take the limit as both [itex]\Delta x[/itex], [itex]\Delta y[/itex] go to 0. And you have to take the limit without assuming a relation between [itex]\Delta x[/itex] and [itex]\Delta y[/itex]. Strictly spealing, we take the limit as the point [itex](\Delta x, \Delta y)[/itex] goes to (0, 0) along any possible path in two dimensions.

So the denominator is [tex]\Delta x + \Delta y[/tex] then?
 
  • #4
Sorry, you posted while I was editing my post- the denominator should be the distance from the point [itex](\Delta x, \Delta y)[/itex] to (0, 0):
[tex]\sqrt{(\Delta x)^2+ (\Delta y)^2}[/tex]

That is difficult to calculate. What is often done is to asssume that the function is differentable so that the limits along, say, parallel to the real axis or parallel to the imaginary axis are sufficient.
 

1. What is a complex limit definition?

A complex limit definition is a mathematical concept used to describe the behavior of a function as it approaches a particular point on a complex plane. It is similar to the concept of a limit in real analysis, but it takes into account the imaginary part of a complex number.

2. How is a complex limit defined?

A complex limit is defined as the value that a function approaches as the input approaches a specified point on the complex plane. This can be written as "lim f(z) = L" where L is the limit and z is the complex variable.

3. What is the importance of understanding complex limit definitions?

Understanding complex limit definitions is crucial in many areas of mathematics, physics, and engineering. It allows us to analyze the behavior of functions in complex domains and make predictions about their behavior near certain points. It is also essential for solving complex differential equations and studying the behavior of complex systems.

4. How do you find the limit of a complex function?

To find the limit of a complex function, you can use various techniques such as algebraic manipulation, L'Hôpital's rule, or the squeeze theorem. It is important to carefully consider the real and imaginary parts of the function and look for any patterns or simplifications that can help determine the limit.

5. Are there any common misconceptions about complex limit definitions?

One common misconception about complex limit definitions is that they are only relevant in theoretical mathematics and have no practical applications. However, as mentioned before, they are crucial in various fields and have real-world implications. Another misconception is that they are only used in advanced mathematics, but in reality, they are introduced in many undergraduate mathematics courses.

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