Legendre Polynomials - expansion of an isotropic function on a sphere

In summary, the speaker is seeking help with a problem involving an integral and is unsure of what to do. They mention attaching equations as a separate file and believe something is missing in the integration, which they believe will result in an identity of Cl=Cl. They clarify that their goal is to find C_{l} and ask for assistance in finding their mistake.
  • #1
petmal
10
0
Hello.
I don't know what to do with one integral. I am sure it is something very simple, but I just don't see it...

For some reason I am not able to post the equations, so I am attaching them as a separatre file.

Many thanks for help.
 

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  • #2
Something is missing. It looks like the integration will give you an identity Cl=Cl.
 
  • #3
mathman said:
Something is missing. It looks like the integration will give you an identity Cl=Cl.

Well, I want to get [tex]C_{l}[/tex], right? That is why I apply [tex]P_{l}[/tex] on the correlation function (1) and integrate - it removes [tex]P_{l}[/tex] and the sum leaving [tex]C_{l}[/tex] * some factor in [tex]l[/tex]. Where do you see a mistake?

Thanks.
 

FAQ: Legendre Polynomials - expansion of an isotropic function on a sphere

What are Legendre Polynomials?

Legendre polynomials are a set of orthogonal polynomials that are used to expand an isotropic function on a sphere. They are named after French mathematician Adrien-Marie Legendre and are widely used in physics and engineering.

How are Legendre Polynomials used to expand an isotropic function on a sphere?

Legendre polynomials are used as a basis for expanding an isotropic function on a sphere. This means that any isotropic function on a sphere can be expressed as a linear combination of Legendre polynomials. This expansion is similar to the Fourier series expansion of a function in terms of sine and cosine functions.

Why are Legendre Polynomials important in physics and engineering?

Legendre polynomials are important because they provide a systematic way of expanding functions on a sphere, which is a common shape in many physical and engineering problems. They are also used in solving differential equations, especially those that involve spherical geometries.

How are Legendre Polynomials related to the Laplace equation?

The Laplace equation is a partial differential equation that describes the behavior of certain physical systems. Legendre polynomials are solutions to this equation when the physical system is defined on a sphere. This makes them useful in solving problems involving spherical symmetry.

Are there any real-world applications of Legendre Polynomials?

Yes, Legendre polynomials have many practical applications in various fields. They are used in geodesy for modeling the Earth's gravitational field, in meteorology for studying atmospheric pressure and temperature, and in geophysics for analyzing the Earth's magnetic field. They are also used in image processing, signal analysis, and quantum mechanics.

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