Expanding function with spherical harmonics

In summary, expanding functions with spherical harmonics involves representing a function as a sum of multiple functions, each multiplied by a spherical harmonic, which is a special type of mathematical function used to describe the behavior of waves on a sphere. This approach allows for a more efficient and accurate representation of a function, particularly in three-dimensional space. Spherical harmonics are commonly used in fields such as physics, geophysics, and computer graphics to model and analyze complex functions.
  • #1
Integratethis
2
0

Homework Statement


The function cos(theta)*cos(phi) in spherical coordinates cannot be expanded to a series of spherical harmonics. Explain why.

Homework Equations


As far as I can recall, the spherical harmonics are a complete set over a sphere, meaning every function which is SI over a sphere can be expanded to such a series...including this one.

The Attempt at a Solution


Tried everything I can think of - got to the point in which I need to prove that the coefficient for each Y(l,m=+-1) = 0, but couldn't find a way around it.
 
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  • #2
You have to check if the function is a member of [itex] L^2 (S^2) [/itex]. Then apply the theory from http://en.wikipedia.org/wiki/Spherical_harmonics - real form and expansion of real functions in terms of real forms of spherical harmonics.

And pay attention to integrations, that's all.
 
  • #3
Integratethis said:
Tried everything I can think of - got to the point in which I need to prove that the coefficient for each Y(l,m=+-1) = 0, but couldn't find a way around it.
You can't prove that because it's not true. The coefficient of Y21, for instance, is not 0. I'm not sure what the problem is getting at because, as you said, the spherical harmonics are a complete set and the function is square-integrable on the sphere.
 

FAQ: Expanding function with spherical harmonics

1. What is the purpose of expanding a function with spherical harmonics?

Expanding a function with spherical harmonics allows for the representation of a function in terms of a sum of spherical harmonics, which are mathematical functions used to describe the physical properties of objects in a spherical coordinate system. This representation can provide insights into the behavior and properties of the function in question.

2. How are spherical harmonics used in scientific research?

Spherical harmonics are used in a variety of scientific fields, including physics, chemistry, and geophysics, to describe the behavior of physical systems with spherical symmetry. They are also used in computer graphics and image processing to analyze and manipulate spherical data.

3. Can any function be expanded with spherical harmonics?

Yes, any function that is defined on a sphere or has spherical symmetry can be expanded with spherical harmonics. This includes functions in three-dimensional space, such as electromagnetic fields, as well as functions on the surface of a sphere, such as temperature distributions on a planet.

4. How does the expansion of a function with spherical harmonics help in solving differential equations?

Expanding a function with spherical harmonics can help in solving differential equations by providing a new representation of the function that can simplify the equations and make them easier to solve. Additionally, the orthogonality properties of spherical harmonics can be used to separate the variables in differential equations, making them easier to solve analytically.

5. Are there any limitations to using spherical harmonics to expand functions?

One limitation is that spherical harmonics can only be used to expand functions on a sphere or with spherical symmetry. Additionally, the accuracy of the expansion may decrease as the function becomes more complex or if there are discontinuities in the function. In these cases, other techniques may be needed to accurately represent the function.

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