### About

Date: 2014-07-10### Summary

Spherical harmonics are made easier to grasp by relating them to the Fourier series on a square.### Contents

# Spherical harmonics and Fourier series¶

During physik321 Theoretische Physik 2 I had a bit of a hard time to understand what spherical harmonics are in a graphic way. If you have some intuition for Fourier series, I can give you a graphic way to understand the purpose of spherical harmonics.

Any square integrable function that is defined on a compact interval can be expanded into a Fourier series. That means that it can be written as a sum of various sine and cosine functions. If you just look at the and (where ) you see that they are higher frequencies or overtones.

You can also do this with functions of two dimensions. Say you have defined on a square. Then you can express this as a Fourier series in and in by forming , and so on. I have plotted the first few terms here:

That was created in *Mathematica 9* with:

```
GraphicsGrid[
Table[DensityPlot[Cos[a x] Cos[b y], {x, -Pi, Pi}, {y, -Pi, Pi},
PlotRange -> Full], {a, 0, 3}, {b, 0, 3}
]
]
```

These functions of and that are plotted above serve as a basis of the Hilbert space of square integrable functions . You will need to include the terms with and the mixed ones as well.

You could also extend this and replace and with and to get complex valued functions. The principle stays the same, except that you can now expand a complex function into such a series.

Spherical harmonics are the same idea, just on a sphere! Think of a function that is defined on a sphere, like the temperature on the surface of the earth. Then you can expand this temperature into a sort of Fourier series. The simple and now become the functions where and take the role of and before.

Since they are complex functions, one can plot the absolute value, the real or imaginary part. Here is the absolute value

That was generated with:

```
GraphicsGrid[
Table[
SphericalPlot3D[
Abs[SphericalHarmonicY[l, m, theta, phi]],
{theta, 0, Pi}, {phi, 0, 2 Pi}, PlotRange -> Full
], {l, 0, 3}, {m, -l, l}
]
]
```

The real part:

Generated with:

```
GraphicsGrid[
Table[
SphericalPlot3D[
Re[SphericalHarmonicY[l, m, theta, phi]],
{theta, 0, Pi}, {phi, 0, 2 Pi}, PlotRange -> Full
], {l, 0, 3}, {m, -l, l}
]
]
```

And the imaginary part last:

Generated with:

```
GraphicsGrid[
Table[
SphericalPlot3D[
Im[SphericalHarmonicY[l, m, theta, phi]],
{theta, 0, Pi}, {phi, 0, 2 Pi}, PlotRange -> Full
], {l, 0, 3}, {m, -l, l}
]
]
```