Specializing Cauchy's Formula: Integral Limits 0-2\pi

  • Thread starter sinned4789
  • Start date
  • Tags
    Formula
In summary, Cauchy's Formula is a useful mathematical tool for calculating complex integrals by relating the values of a function on the boundary of a region to its values inside the region. Specializing it to integral limits of 0-2π allows for the calculation of integrals around a closed path, useful in many applications such as finding residues and solving differential equations. To use Cauchy's Formula, the contour of integration must be parameterized and the function must be expressed in terms of the parameter. However, there are limitations to using Cauchy's Formula, such as the need for the function to be analytic and difficulties in finding the correct contour for certain functions.
  • #1
sinned4789
7
0

Homework Statement


specialize cauchy's formula to the case when z is the center of the circle and show that

f(z) = (1/2[tex]\pi[/tex])[tex]\int[/tex] f(z + re^(it)) dt
integral limits are 0 to 2[tex]\pi[/tex]


Homework Equations


cauchy's formula
f(z) = (1/i2*pi) [tex]\int[/tex] f([tex]\zeta[/tex])/([tex]\xi[/tex] - z) d[tex]\zeta[/tex]

it is 2*pi not 2^pi just to be sure


The Attempt at a Solution



i have no idea how to do this
any input or hints would be appreciated
 
Physics news on Phys.org
  • #2
Just put zeta=z+r*e^(i*t). It's a simple substitution problem.
 

1. What is Cauchy's Formula?

Cauchy's Formula is a mathematical formula that relates the values of a function on the boundary of a region to its values inside the region. It is commonly used in complex analysis to calculate the value of a complex integral.

2. What is the significance of specializing Cauchy's Formula to integral limits 0-2π?

Specializing Cauchy's Formula to integral limits of 0-2π allows for the calculation of complex integrals around a closed path, known as a contour, that encircles the origin. This is useful in many applications, including calculating the residues and solving differential equations.

3. How do you calculate a complex integral using Cauchy's Formula?

To calculate a complex integral using Cauchy's Formula, you first need to parameterize the contour of integration and then express the function being integrated in terms of the parameter. Next, you apply Cauchy's Formula, which involves integrating the function over the contour and multiplying it by the derivative of the parameter with respect to the complex variable. Finally, you can evaluate the integral using the values of the function on the boundary of the region.

4. What are the main applications of specializing Cauchy's Formula?

Specializing Cauchy's Formula has many applications in mathematics, physics, and engineering. Some common uses include solving differential equations, finding the residues of complex functions, and calculating the Fourier transform of functions.

5. Are there any limitations to using Cauchy's Formula?

While Cauchy's Formula is a powerful tool for calculating complex integrals, it does have some limitations. It can only be applied to functions that are analytic within the region of integration, and it may not work for functions with singularities or branch points. Additionally, it can be challenging to find the correct contour of integration for certain functions.

Similar threads

  • Calculus and Beyond Homework Help
Replies
3
Views
810
  • Calculus and Beyond Homework Help
Replies
16
Views
898
  • Calculus and Beyond Homework Help
Replies
3
Views
487
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
911
  • Calculus and Beyond Homework Help
Replies
5
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
987
  • Calculus and Beyond Homework Help
Replies
32
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
514
  • Topology and Analysis
Replies
2
Views
574
Back
Top