Need help with Cauchy Theorum/Formula

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In summary, The conversation is about someone asking for help with applying the formula for Cauchy's Theorum on the complex plane. They are struggling and looking for a good website with examples on the topic. The applications involve using the theorem with estimates for the size of an integral, which can provide insight into the behavior of complex functions as z gets larger. The person is advised to consult any book or webpage on complex variables for more information.
  • #1
chota
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Ok so I am asking for some help, please help.

I don't know how to apply the formula given for Cauchy's Theorum, on the complex plane

if anyone can show me a good website with examples on this topic, please tell me.. I am struggling and this is like a cry for help, please help.

thank you
 
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  • #2
the applications involve combining the theorem with the usual estimates for the size of an integral, leading to conclusions about the behavior of complex functions of z as z gets larger. any book on complex variable discusses these things as does no doubt any webpage on it.
 

Related to Need help with Cauchy Theorum/Formula

1. What is Cauchy's Theorem?

Cauchy's Theorem, also known as the Cauchy Integral Theorem, is a fundamental result in complex analysis that states that if a function is analytic within a closed contour, then the contour integral around that contour is equal to the sum of all the values of the function inside the contour.

2. What is the Cauchy Integral Formula?

The Cauchy Integral Formula is a special case of Cauchy's Theorem that states that if a function is analytic within a closed contour, then the value of the function at any point inside the contour can be calculated by integrating the function over the contour and dividing by 2πi.

3. How is Cauchy's Theorem useful?

Cauchy's Theorem has many applications in mathematics and physics. It provides a powerful tool for calculating complex integrals and solving differential equations. It is also used in the study of conformal mappings and the theory of residues.

4. What are the conditions for Cauchy's Theorem to hold?

There are several conditions that must be met for Cauchy's Theorem to hold. The first is that the function must be analytic within the contour. The contour must also be a simple closed curve, meaning it does not intersect itself. Finally, the contour must be positively oriented, meaning it is traversed counterclockwise.

5. Are there any generalizations of Cauchy's Theorem?

Yes, there are several generalizations of Cauchy's Theorem, including Cauchy's Residue Theorem and Cauchy's Integral Formula for Derivatives. These theorems extend the applicability of Cauchy's Theorem to more complex functions and contours.

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