Expansion of 1/|x-x'| into Legendre Polynomials

In summary, the expansion of the function \( \frac{1}{|x-x'|} \) into Legendre polynomials involves expressing this singular function as a series of orthogonal polynomials defined on a certain interval. The process utilizes the properties of Legendre polynomials, which are solutions to Legendre's differential equation and are particularly useful in problems involving spherical symmetry. This expansion is beneficial in various fields, including mathematical physics and potential theory, as it allows for the simplification of complex interactions in a mathematically rigorous manner. The coefficients of the expansion are determined through integration, leveraging the orthogonality of Legendre polynomials to isolate the contributions from each polynomial term.
  • #1
deuteron
60
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TL;DR Summary
.
we know that we can expand the following function in Legendre polynomials in the following way
1712516696799.png

in the script given yo us by my professor, ##\frac 1 {|\vec x -\vec x'|}## is expanded using geometric series in the following way:


1712516763072.png


However, I don't understand how ##\frac 1 {|\vec x -\vec x'|}## is equal to both the above, and the below:

1712517015521.png
 
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  • #2
Do you understand the meaning of ##\Theta(x-x')##?
It's not equal to both. It's equal to one or the other depending on which of ##x## and ##x'## is larger. The bottom expression summarizes in one line the two "für" cases above it.
 
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FAQ: Expansion of 1/|x-x'| into Legendre Polynomials

What is the significance of expanding 1/|x-x'| into Legendre polynomials?

The expansion of the function 1/|x-x'| into Legendre polynomials is significant in various fields such as physics and engineering, particularly in potential theory and electrostatics. This expansion allows for the representation of the potential due to a point charge in terms of a series of orthogonal functions, facilitating calculations in multi-dimensional spaces.

How is the expansion of 1/|x-x'| derived?

The expansion is derived using the properties of Legendre polynomials and the theory of spherical harmonics. It involves expressing the function in terms of a series of Legendre polynomials, which are orthogonal on the interval [-1, 1]. The coefficients of the expansion can be calculated using integral formulas that involve the inner product of the function with the Legendre polynomials.

What are the applications of this expansion in real-world problems?

This expansion has applications in gravitational and electrostatic potential calculations, where it is used to simplify the evaluation of potentials due to distributed charges or masses. It is also utilized in numerical methods for solving partial differential equations, particularly in problems involving spherical symmetry.

Are there any limitations to using Legendre polynomial expansions?

One limitation is that the convergence of the series can be slow for certain values of x and x', especially when they are close together, leading to numerical instability. Additionally, the applicability of the expansion is primarily confined to problems with spherical symmetry, which may not be suitable for all geometrical configurations.

Can the expansion be generalized to other functions or dimensions?

Yes, the expansion can be generalized to other functions using different families of orthogonal polynomials, such as Chebyshev or Hermite polynomials, depending on the problem's symmetry and boundary conditions. In higher dimensions, spherical harmonics can be used to expand functions defined on the surface of a sphere, extending the concept beyond one-dimensional cases.

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