7
$\begingroup$

I recently started to read about spherical harmonics. I have a question about this spherical harmonics basis function equation which is mentioned in this StupidSH article:

enter image description here

What are those lm() and Re() used for?

The article can be found here: http://www.ppsloan.org/publications/StupidSH36.pdf

$\endgroup$
4
$\begingroup$

The notation Re() and Im() refer to the real and imaginary parts of a complex number. Mathematicians and physicists are accustomed to using spherical harmonics (and Fourier transforms too) that are complex-valued, due to the factor $e^{im\phi}$. You would then also have complex coefficients, in general, in the spherical harmonic expansion of a (real or complex) function.

Using Euler's formula, $e^{im\phi} = \cos(m\phi) + i\sin(m\phi)$. So that factor encodes cosine and sine waves (that oscillate $m$ times as you move around the equator of the sphere) in its real and imaginary parts, respectively.

When we know we're going to be working strictly with real-valued functions, it may be more convenient to use real-valued variants of the spherical harmonics, where $e^{im\phi}$ is replaced by either $\cos(m\phi)$ or $\sin(m\phi)$, and using real coefficients instead of complex ones. We trade a single complex coefficient for two real coefficients, so we haven't lost any information or flexibility; it's essentially a change of basis. This formulation just explicitly ensures that everything always comes out real.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.