# Spherical Harmonics Homework: Calculate Coeficients & Express Function

• pgerlein
In summary, the conversation is about a person seeking help with calculating the expression of a function using spherical harmonics and expressing it in terms of the harmonics. They are also discussing the use of LaTeX for easier equation writing.
pgerlein

## Homework Statement

Hi! I need help with this. I have to calculate the expression of a function using spherical harmonics. The relevant equations are given below. An example of a function can be f(theta,phi) = sin(theta)... Can you help me to calculate de coeficients and to express the function? Thanks.

## Homework Equations

View attachment formulas.doc

## The Attempt at a Solution

Hola pgerlein. Welcome to PF

Why don't you show us what you've tried for your example ($f(\theta,\phi)=sin(\theta)$) and we'll know how to help you better.

To make equation writing easier, this forum supports LaTeX. You can find an introduction to using it by clicking here

I can´t understand the use of this base (spherical harmonics). If I solve the ecuation that gives the coefficients value, how can I express the function in terms of the spherical harmonics?

How would you express a function in terms of a Fourier series, for example? Essentially it's the same idea when you write any function in terms of an infinite series of basis functions.

## 1. What are spherical harmonics?

Spherical harmonics are mathematical functions that describe the behavior of waves on a spherical surface. They are used in many areas of science, including quantum mechanics, electromagnetics, and geophysics.

## 2. What is the purpose of calculating coefficients for spherical harmonics?

Calculating coefficients for spherical harmonics allows us to express a given function as a sum of simpler, spherical harmonic functions. This can help us better understand the behavior of the function and make predictions about its properties.

## 3. How do you calculate coefficients for spherical harmonics?

The coefficients for spherical harmonics can be calculated using various methods, such as the Gram-Schmidt process or solving a system of equations. The specific method used may depend on the function being analyzed and the desired level of accuracy.

## 4. Can you provide an example of calculating coefficients for spherical harmonics?

Sure, let's say we have a function f(x,y,z) on a unit sphere. We can calculate the coefficients clm for the spherical harmonics Ylm(θ,φ) that best approximate f(x,y,z) by solving the system of equations: ∫∫f(x,y,z)Ylm(θ,φ)sinθdθdφ = clm for each m and l.

## 5. How are spherical harmonics used in different scientific fields?

Spherical harmonics have a wide range of applications in science, including in quantum mechanics for describing the behavior of electrons in atoms, in electromagnetics for modeling the radiation pattern of antennas, and in geophysics for studying the Earth's magnetic field. They are also used in computer graphics and image processing for creating and manipulating 3D shapes and textures.

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