Spherical Harmonics Axisymmetry

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Thread starter Maher
Start date Feb 23, 2019
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Harmonics Spherical Spherical harmonics

In summary, spherical harmonics are mathematical functions used to describe waves in a spherical environment. They are utilized in various fields, such as physics, mathematics, and engineering, to represent solutions to differential equations. Axisymmetry refers to symmetry around an axis, and in the context of spherical harmonics, it means that the functions are symmetric around the polar axis of a sphere. Spherical harmonics are used to model the behavior of waves in axisymmetric systems and have applications in physics, mathematics, and engineering. The calculation of spherical harmonics axisymmetry involves solving the Helmholtz equation and utilizing specific mathematical formulas expressed as trigonometric functions and associated Legendre polynomials.

Feb 23, 2019

Maher

I'm expanding a function in spherical harmonics. I want to conserve axisymmetry of the function. what harmonics would respect that? Should I only include m=0 terms?

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Feb 23, 2019

Orodruin

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As long as you put your spherical coordinates such that the z-axis is the symmetry axis, yes.

1. What are spherical harmonics?

Spherical harmonics are a set of mathematical functions used to represent the angular dependence of a function on a sphere. They are commonly used in physics and engineering to describe the behavior of physical systems that exhibit spherical symmetry.

2. What is axisymmetry in relation to spherical harmonics?

Axisymmetry refers to the property of a system where it is symmetric about an axis. In the context of spherical harmonics, axisymmetry means that the function being described has the same value at every point along a fixed axis, regardless of rotation about that axis.

3. How are spherical harmonics used in scientific research?

Spherical harmonics are used in a variety of scientific fields, including physics, chemistry, and geophysics. They are commonly used to solve differential equations, describe the behavior of electromagnetic fields, and model the shape of molecules.

4. Can spherical harmonics be used to describe non-spherical systems?

Yes, spherical harmonics can be used to describe non-spherical systems by expanding the function in terms of spherical harmonics and then truncating the series at a certain order. This allows for an approximation of the non-spherical system using spherical harmonics.

5. Are there any real-world applications of spherical harmonics axisymmetry?

Yes, spherical harmonics axisymmetry has many real-world applications. For example, it is used in satellite imaging to analyze the Earth's gravitational field, in nuclear magnetic resonance to determine molecular structure, and in computer graphics to create realistic lighting effects.