Contour Integrals in complex analysis questions

[nabeel17]

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.

 

[elegysix]

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.

 

[K^2]

Have you been able to follow proof of Cauchy's Integral Theorem? Because that pretty much answers the "why" question.

 

[nabeel17]

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?

 

[dauto]

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.