Limit question (from complex analysis)

In summary, the conversation is about a limit problem involving a continuous function and a fixed value. The goal is to show that the limit of a specific expression involving an integral and a 1/h term is equal to 0. The key to solving the problem is the continuity of the function. The idea is to use the ML Estimate to show that the limit goes to 0.
  • #1
synapsis
10
0

Homework Statement



This seems to be just a simple limit problem and I feel like I should know it but I'm just not seeing it.

I have a continuous function f, and a fixed w

I want to show that the limit (as h goes to 0) of the absolute value of:

(1/h)*integral[ f(z)-f(w) ]dz = 0 (the integral is over a contour)


Homework Equations



I believe the key to the problem is that f is continuous.

The Attempt at a Solution



For any a>0 there exists a b>0 such that z within b of w implies f(z) within a of f(w).

The problem is it seems to me like the 1/h term is going to infinity while the integral term is going to 0, which is indeterminate so I don't know how to get that the limit goes to 0.
 
Physics news on Phys.org
  • #2
Oh, I have an idea. The contour I'm integrating over is the line connecting w+h to w. So I believe I can use the ML Estimate to show the limit goes to 0...
 

1. What is a limit in complex analysis?

A limit in complex analysis refers to the behavior of a function as the input approaches a certain value in the complex plane. It is similar to the concept of a limit in real analysis, but in complex analysis, the input and output are both complex numbers.

2. How is a limit in complex analysis calculated?

To calculate a limit in complex analysis, we use the same techniques as in real analysis, such as evaluating the function at points closer and closer to the input value and seeing if there is a consistent output. We also use the concept of epsilon-delta proofs to rigorously prove the existence of a limit.

3. What is the Cauchy-Riemann condition in relation to limits in complex analysis?

The Cauchy-Riemann condition is a necessary condition for a complex function to be differentiable at a point. It states that the partial derivatives of the function with respect to the real and imaginary parts of the input must exist and satisfy certain relationships. This condition is important in calculating limits in complex analysis, as it ensures the function behaves in a consistent and predictable manner.

4. Can a limit in complex analysis not exist?

Yes, a limit in complex analysis can fail to exist. This can happen when the function behaves differently along different paths approaching the input value, or when the function oscillates infinitely close to the input value. In these cases, the limit is said to be undefined.

5. How are limits used in the study of complex analysis?

Limits are an essential tool in the study of complex analysis. They help us understand the behavior of complex functions and their derivatives, which in turn allows us to find critical points, maxima and minima, and other important characteristics of these functions. Limits also play a crucial role in proving theorems and developing new theories in complex analysis.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
934
  • Calculus and Beyond Homework Help
Replies
21
Views
404
  • Calculus and Beyond Homework Help
Replies
7
Views
389
  • Calculus and Beyond Homework Help
Replies
3
Views
811
  • Calculus and Beyond Homework Help
Replies
17
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
961
  • Calculus and Beyond Homework Help
Replies
20
Views
1K
  • Calculus and Beyond Homework Help
Replies
27
Views
606
  • Calculus and Beyond Homework Help
Replies
16
Views
1K
Back
Top