Why does the Cauchy integral theorem require the first term to vanish?

In summary, the conversation discusses the proof of the residue theorem and the confusion over the first term vanishing in the Cauchy integral theorem. The suggestion is made to correct the justification below equation (3) on the MathWorld page and appreciation is expressed for hosting the only elementary proof of the residue theorem available online.
  • #1
antonantal
243
21
I was looking at the proof of the residue theorem on MathWorld: http://mathworld.wolfram.com/ResidueTheorem.html
and got stucked on the 3rd relation.

I don't understand why the Cauchy integral theorem requires that the first term vanishes.
From the Cauchy integral theorem, the contour integral along any path not enclosing a pole is 0. But in this case, the contour [tex]\gamma[/tex] encloses [tex]z_{0}[/tex] which is a pole of [tex](z-z_{0})^{n}[/tex] for [tex]n \in \{-\infty,...,-2\}[/tex]
 
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  • #2
Perhaps they meant the Cauchy Integral Formula
 
  • #3
I agree with nicksauce. To expand his point go to this page and scroll down to equation 19 at the bottom:

http://mathworld.wolfram.com/CauchyIntegralFormula.html

In this case [itex]f(z) = 1[/itex] for all z, and so the sum of these terms vanishes because the derivatives of a constant are all zero.

Update: I sent a message to the mathworld team suggesting the following correction:

Below equation (3) the given justification: "The Cauchy integral theorem requires that the first and last terms vanish" should be replaced with something to the effect of "The first term is zero because of the Cauchy Integral Formula, while the last term is zero because of the Cauchy Integral Theorem." Thank you for hosting the only elementary proof of even this much of the residue theorem that I could find anywhere on the web.
 
Last edited:
  • #4
Thanks! Good idea to send that suggestion to the MathWorld team too.
 

What is the Cauchy integral theorem?

The Cauchy integral theorem, also known as Cauchy's integral formula, is a fundamental theorem in complex analysis that states that the integral of a complex-valued function along a closed path in the complex plane is equal to the sum of the function's values inside the path. It is named after the French mathematician Augustin-Louis Cauchy.

What is the significance of the Cauchy integral theorem?

The Cauchy integral theorem is significant because it allows for the evaluation of complex integrals using only knowledge of a function's values on a closed path. This makes it a powerful tool in complex analysis and has many applications in physics, engineering, and other fields.

Can the Cauchy integral theorem be extended to non-closed paths?

No, the Cauchy integral theorem only holds for closed paths. However, there is a similar theorem called the Cauchy integral formula for derivatives that applies to non-closed paths, but it is more complex and has more restrictions.

How is the Cauchy integral theorem related to Cauchy's residue theorem?

The Cauchy residue theorem is a consequence of the Cauchy integral theorem and states that the integral of a function over a closed path can be computed by summing the residues of the function's poles within the path. This makes it a useful tool for calculating complex integrals.

What are the applications of the Cauchy integral theorem?

The Cauchy integral theorem has numerous applications in complex analysis, including calculating complex integrals, solving differential equations, and finding the values of analytic functions. It is also used in physics to solve problems involving electric and magnetic fields and in engineering for circuit analysis.

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