Cauchy Integral Formula and Electrodynamics

In summary, the conversation discusses the possibility of using Cauchy Integral Formulas from complex variables to solve for an E field from a charge density function. The formulas involve evaluating the n derivative of a function at a specific point and using a curl type formula. However, it is suggested that a divergence type formula may be more suitable for this problem. The conversation also mentions the similarity between the Cauchy residue theorem and Gauss's theorem, which may be useful in solving electrodynamics problems in two dimensions.
  • #1
sinyud
23
0
Is it possible to solve for an E field from a charge density function using the Cauchy Integral Formulas from complex variables?

Cauchy Integral Formula about a closed loop in the complex plane
(Integral[f[z]/ (z-z0)^(n+1)dz = 2 pi i /n! d^n f(z0)/dz ])

that is the n derivative of f with respect to z evaluated at z0
 
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  • #2
If I remember that was a curl type formula and you need a divergence type formula. It may not require math that complicated. It is just that doing the integral might be awkward. In the formula E=Integral (rho/r^2) dr.
 
  • #3
I was thinking that just like the Gauss's theorem (the surface integral version of the Div[E] = rho/ epsilon) picks out charges which are in effect mathematical singularities, so to the cauchy residue theorem picks out every 1/z of a function.

Can this similarity be used to solve electrodynamics problems in two dimensions?
 

Related to Cauchy Integral Formula and Electrodynamics

1. What is the Cauchy Integral Formula?

The Cauchy Integral Formula is a fundamental theorem in complex analysis that allows for the calculation of the value of a complex function at any point inside a closed contour, in terms of the values of the function on the contour itself. It is also known as the Cauchy-Goursat theorem.

2. How does the Cauchy Integral Formula relate to Electrodynamics?

In Electrodynamics, the Cauchy Integral Formula is a useful tool for calculating the electric potential at any point in space, given the distribution of electric charge. This is because the electric potential can be represented as a complex function, and the Cauchy Integral Formula can then be applied to calculate its value at any point.

3. What are the key assumptions in the Cauchy Integral Formula?

The Cauchy Integral Formula relies on three main assumptions: 1) the function being integrated is analytic (i.e. differentiable) in the region of interest, 2) the contour of integration is simple and closed, and 3) the point of interest is contained within the contour.

4. Can the Cauchy Integral Formula be applied to non-analytic functions?

No, the Cauchy Integral Formula only applies to analytic functions. If a function is not analytic, then the Cauchy Integral Formula cannot be used to calculate its value at a specific point inside a contour.

5. What are some practical applications of the Cauchy Integral Formula?

The Cauchy Integral Formula has numerous applications in mathematics and physics. In addition to its use in Electrodynamics, it is also used in solving problems in fluid dynamics, signal processing, and quantum mechanics. It is also an important tool in the study of complex analysis and has applications in other areas of mathematics, such as number theory and differential equations.

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