How and when to use Cauchy's integral formula

In summary, the conversation discusses the use of Cauchy integral formula to evaluate integrals and its applications in physics and signal theory. The formula is used when the integral matches the pattern of a Cauchy integral, and sometimes the contour needs to be closed to evaluate the integral on the closing path. The values for f(z), z, and z0 can be checked for accuracy. It is recommended to type out work instead of attaching handwritten photos for better assistance.
  • #1
MissP.25_5
331
0
Hello.
How do I know when to use Cauchy integral formula. Why do we use the formula in this question? As you can see in my attempt, I got stuck.
Are my values for f(z), z, z​0 here correct?
 

Attachments

  • IMG_6430.jpg
    IMG_6430.jpg
    30.6 KB · Views: 432
Last edited:
Physics news on Phys.org
  • #2
You got the right answer, since f(zo)=1.
You use the Cauchy integral when you have to evaluate an integral that matches the pattern of a Cauchy integral!
Sometimes, the contour needs to be closed to get a match and the integral on the closing path can be evaluated by some other means.
The Cauchy integral is very useful in physics and in signal theory.
 
  • #3
maajdl said:
You got the right answer, since f(zo)=1.
You use the Cauchy integral when you have to evaluate an integral that matches the pattern of a Cauchy integral!
Sometimes, the contour needs to be closed to get a match and the integral on the closing path can be evaluated by some other means.
The Cauchy integral is very useful in physics and in signal theory.

No, I just solved it and f(z) is not 1, but 1/(z-1).
 
  • #4
You proved that your integral is 2 Pi I * the Cauchy integral of f(z)=1 around z=I .
Therefore, your integral is 2 Pi I f(I) = 2 Pi I .
 
  • #5
MissP.25_5 said:
Hello.
How do I know when to use Cauchy integral formula. Why do we use the formula in this question? As you can see in my attempt, I got stuck.
Are my values for f(z), z, z​0 here correct?

I never look at photo attachments of handwritten work; in fact, if you read the "PF Guidelines" post by vela, you will see that you are not supposed to use them except for very special circumstances---for several good reasons. You should take the trouble to type out your work if you want the helpers to take the trouble to offer free assistance.
 
  • #6
Ray Vickson said:
I never look at photo attachments of handwritten work; in fact, if you read the "PF Guidelines" post by vela, you will see that you are not supposed to use them except for very special circumstances---for several good reasons. You should take the trouble to type out your work if you want the helpers to take the trouble to offer free assistance.

Exactly! Thread locked. Please create a thread where you write out your work.
 

FAQ: How and when to use Cauchy's integral formula

1. What is Cauchy's integral formula?

Cauchy's integral formula is a mathematical theorem that relates the values of an analytic function inside a closed contour to the values of the function on the contour itself. It is named after the French mathematician Augustin-Louis Cauchy.

2. When should Cauchy's integral formula be used?

Cauchy's integral formula is used when solving complex integrals over closed paths in the complex plane. It is also used to evaluate derivatives of complex functions and to solve boundary value problems.

3. How do you use Cauchy's integral formula?

To use Cauchy's integral formula, you first need to identify the function and the contour over which you want to integrate. Then, you need to ensure that the function is analytic (i.e. it is differentiable at every point inside the contour). Finally, you can apply the formula to calculate the value of the integral.

4. What are the key components of Cauchy's integral formula?

The key components of Cauchy's integral formula are the function, the contour, and the Cauchy integral formula itself, which includes a complex integral and the function's values on the contour.

5. Can Cauchy's integral formula be used for any type of function?

No, Cauchy's integral formula can only be used for analytic functions, meaning they are differentiable at every point inside the contour. Non-analytic functions, such as those with singularities or discontinuities, cannot be evaluated using this formula.

Similar threads

Complex Integration Along Given Path
Replies
3
Views
1K
Can I Use Substitution to Prove the Residue Theorem Integral?
Replies
5
Views
516
Understanding the Cauchy Integration Formula for Analytic Functions
Replies
4
Views
1K
Derive an upper bound for |f(i)|
Replies
16
Views
1K
Using Cauchy's integral formula to evaluate integrals
Replies
1
Views
2K
Two ways of integration giving different results
Replies
9
Views
1K
Prerequisites to Cauchy Integral Formula
Replies
4
Views
1K
Complex integration (Using Cauchy Integral formula)
Replies
3
Views
2K
Contour Integral of |z| = 2 using Cauchy's Formula
Replies
1
Views
1K
How Do You Solve a Complex Integral Using Cauchy-Goursat's Theorem?
Replies
32
Views
2K

Hot Threads

Recent Insights

Back
Top