Proof of Cauchy integral formula

In summary, the conversation discusses proving the Cauchy integral formula for analytic functions. The deformation principle, which states that the path between limits of integration does not matter for analytic functions, is used in the proof. The formula for smooth functions is also mentioned, with the difference being that the second integral is zero for analytic functions. The term "max C(r, z_0)" is defined as the maximum modulus of a function on a circle of radius r centered at z_0.
  • #1
stgermaine
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Homework Statement


For an quiz for a diff eq class, I need to prove the Cauchy integral formula.
The assignment says prove the formula for analytic functions. Is the proof significantly different when the function is not analytic?

Homework Equations


Basically, a proof I found online says that I take the integral of a larger contour C that encloses the contour C_r which is the contour that wraps around the cirlce of radius r around z_0. By the deformation principle, if r is sufficiently small, then the integral of f(z) around C and C_r are equal. I've tried searching for 'deformation principle' but none was of much help.


3. Questions
http://www.math.colostate.edu/~achter/419f06/help/cauchy.pdf

What is the deformation principle? I don't know much about contour integrals, so if someone can relate it to just regular integrals, that'd be great.

http://www2.latech.edu/~schroder/slides/comp_var/14_Cauchy_Integral_Formula.pdf [Broken] Pg 43

What is max C(r, z_0) supposed to mean? I think the this proof uses the same principles as in the first one, but doesn't explicitly use the delta-epsilon limit.

Thank you
 
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  • #2
Cauchy's formula for smooth functions is

[tex]f(a) = \frac{1}{2\pi i}\int_{\partial D} \frac{f(z) dz}{z-a} + \frac{1}{2\pi i}\iint_D \frac{\partial f(z)}{\partial \bar{z}} \frac{dz\wedge d\bar{z}}{z-a}[/tex]

The proof is not that much different than for analytic functions other than for analytic functions the second integral is zero.

The deformation principle is that path between the limits of integration does not matter for analytic functions. For real integration this is also true (at least if the function is non pathological and the path does over emphasize a region).
For example if integrating from 1 to 4 is written I(1,4) we have
I(1,4)=I(1,2)+I(2,4)=I(1,3)+I(3,2)+I(2,4) and so on
For complex integrals there is more freedom because for each segment the path can be curved and twisted in endless ways.

max C(r, z_0) is the maximum modulus
the largest absolute value the function can obtain on the circle
 

1. What is the Cauchy integral formula?

The Cauchy integral formula is a fundamental theorem in complex analysis that relates the values of a function inside a closed curve to the values of the function on the boundary of the curve. It is named after French mathematician Augustin-Louis Cauchy and is a powerful tool for calculating complex integrals.

2. What is the significance of the Cauchy integral formula?

The Cauchy integral formula is significant because it allows us to evaluate complex integrals without having to explicitly calculate them. This makes it a useful tool in many branches of mathematics and physics, including electrical engineering and fluid dynamics.

3. Can the Cauchy integral formula be extended to higher dimensions?

Yes, the Cauchy integral formula can be extended to higher dimensions. In one dimension, it relates the values of a function on a curve to its values on the boundary of the curve. In higher dimensions, it relates the values of a function inside a closed surface to the values of the function on the boundary of the surface.

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

The key components of the Cauchy integral formula are the function being integrated, the closed curve or surface, and the values of the function on the boundary of the curve or surface. The formula also involves complex numbers, specifically the complex derivative of the function being integrated.

5. What are some applications of the Cauchy integral formula?

The Cauchy integral formula has many applications in mathematics and physics. It is used to solve problems in complex analysis, such as finding the residues of a function. It is also used in engineering, particularly in electrical engineering for calculating the behavior of electric circuits. Additionally, the Cauchy integral formula is used in fluid dynamics for calculating the flow of fluids around obstacles.

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