Contour Integrals in complex analysis questions

In summary: What I recall from doing those was that when you do the contour integral, you're basically finding something analogous to flux. This is one of those hazy areas for me, but I hope it helps.
  • #1
nabeel17
57
1
I am confused as to what we are obtaining when taking these contour integrals.

I know that the close loop contour integral of a holomorphic function is 0. Is this analogous to the closed loop of integral of a conservative force which also gives 0?

Also when I am integrating around a function and there is a singularity in my contour, it gives me a value according to Cauchy integral theorem and 0 if no singularity is inside. Why is this? In this case does the function still have to be holomorphic and is there a relation to dirac delta function, since it looks somewhat similar. What is the difference whether there is a singularity inside or not and exactly WHAT am i getting when I calculate the integral (Area under something? or what..).

A lot of questions, but I'd like to know what I'm doing since the math itself is not too hard but I have no idea the physical meaning.
 
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  • #2
What I recall from doing those was that when you do the contour integral, you're basically finding something analogous to flux.
This is one of those hazy areas for me, but I hope it helps. I think we talked about those for a week in one of my calculus classes and I haven't seen them since.
 
  • #3
Have you been able to follow proof of Cauchy's Integral Theorem? Because that pretty much answers the "why" question.
 
  • #4
K^2 said:
Have you been able to follow proof of Cauchy's Integral Theorem? Because that pretty much answers the "why" question.

No, I am trying to follow it in my textbook but it is not clear to me. Do you have a good link or textbook you can refer?
 
  • #5
nabeel17 said:
I am confused as to what we are obtaining when taking these contour integrals.

I know that the close loop contour integral of a holomorphic function is 0. Is this analogous to the closed loop of integral of a conservative force which also gives 0?

Also when I am integrating around a function and there is a singularity in my contour, it gives me a value according to Cauchy integral theorem and 0 if no singularity is inside. Why is this? In this case does the function still have to be holomorphic
Yes the function still must be holomorphic everywhere except at the pole
and is there a relation to dirac delta function, since it looks somewhat similar. What is the difference whether there is a singularity inside or not and exactly WHAT am i getting when I calculate the integral (Area under something? or what..).
Thinking of an integral as an area under something is a clutch. A clutch might help you walk but you might have to get rid of it if you want to run.
A lot of questions, but I'd like to know what I'm doing since the math itself is not too hard but I have no idea the physical meaning.
 

1. What is a contour integral in complex analysis?

A contour integral in complex analysis is a type of line integral that is calculated along a path in the complex plane. It involves integrating a function of a complex variable over a curve or contour in the complex plane. It is used to evaluate complex functions and is an important tool in complex analysis.

2. How do you calculate a contour integral?

To calculate a contour integral, you first need to parameterize the contour. This involves expressing the curve as a function of a complex variable, usually denoted by z. Then, you integrate the function of z along the contour with respect to z. This can be done using the fundamental theorem of calculus or by using Cauchy's integral formula.

3. What is the significance of contour integrals in complex analysis?

Contour integrals have many applications in complex analysis. They are used to calculate residues, which are important in finding the poles of complex functions. They also help in evaluating complex integrals and in solving differential equations. Moreover, contour integrals are essential in understanding the behavior of complex functions and their singularities.

4. What are the types of contours used in contour integrals?

There are two main types of contours used in contour integrals: closed contours and open contours. Closed contours form a complete loop and are used to enclose singularities of a complex function. Open contours do not form a loop and are used to integrate over a specific part of the complex plane, such as a line or curve.

5. How are contour integrals related to Cauchy's integral theorem and Cauchy's integral formula?

Contour integrals are closely related to Cauchy's integral theorem and Cauchy's integral formula. These theorems state that for a function that is analytic on a simply connected region, the contour integral along any closed contour is equal to zero. Cauchy's integral formula is used to evaluate contour integrals by expressing the function in terms of its derivatives at a point inside the contour. This makes it a powerful tool in solving complex integrals.

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