Complex Analysis: Examples & Questions Solved with Poisson's & Cauchy's Formulas

In summary, The speakers discuss the uses of Poisson's formula and its relationship to Cauchy's formula, noting its application in solving the two dimensional Laplace equation. They also discuss the utility of the inversion with respect to a circle in simplifying problems.
  • #1
esisk
44
0
Hope this does not sound vague!

1) I a looking at the Poisson's formula for the disk. Can somebody give me an example how one uses this, or a question where we use it to solve the problem. What is it exactly saying that Cauchy's formula is not saying? Thank you

2) Can somebody give me an example wherby I use the inversion with respect to a circle (unit circle or otherwise) and the problem becomes easier. I guess I am asking: how do I make use of this notion.
Thank you
 
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  • #2
Cauchy's formula gives you values inside the disk if you know complex values on the boundary. Poisson's formula allows you to reconstruct the function from just the real component. As such, it has some use outside complex analysis, because it allows you to solve the two dimensional Laplace equation.
 

1. What is Complex Analysis?

Complex Analysis is a branch of mathematics that deals with the study of complex numbers and their functions. It is used to understand and analyze the behavior of functions that involve complex numbers.

2. What are some examples of problems solved using Poisson's and Cauchy's Formulas?

Poisson's and Cauchy's Formulas are commonly used to solve problems involving harmonic functions, such as finding the solution to Laplace's equation in two dimensions. They are also used to solve problems in potential theory, fluid dynamics, and electromagnetism.

3. How do Poisson's and Cauchy's Formulas differ?

Poisson's Formula is used to find the solution to Laplace's equation in two dimensions, while Cauchy's Formula is used to find the solution to Laplace's equation in three dimensions. Additionally, Cauchy's Formula involves the use of the Cauchy-Riemann equations, which relate the real and imaginary parts of a complex function.

4. Can Poisson's and Cauchy's Formulas be used to solve real-world problems?

Yes, Poisson's and Cauchy's Formulas have many applications in physics, engineering, and other fields. For example, they can be used to model the flow of fluids, the distribution of electrical charges, and the behavior of heat in different materials.

5. What is the significance of Cauchy's Integral Theorem in Complex Analysis?

Cauchy's Integral Theorem is a fundamental result in Complex Analysis that states that the value of a complex integral is completely determined by the values of the function being integrated and its derivatives at a finite number of points. This theorem is essential in many areas of mathematics and physics, including the study of complex functions and their properties.

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