This report introduces spherical harmonics, functions defined on the surface of a sphere that play a central role in mathematical analysis, especially in problems with spherical symmetry. They appear in many fields, such as 3D representation within computer graphics, simulation light behaviour and angular momentum within quantum mechanics.

We begin by developing the theory from first principles. We look at what homogeneous harmonics polynomials are and explain how spherical harmonics arise by restricting these polynomials to the unit sphere. Using this we discuss properties such as orthogonality and dimension. We also discuss zonal harmonics, which are symmetric around a chosen axis.

In three dimensions, we solve Laplace’s equation in spherical coordinates to derive explicit formulas for spherical harmonics. Associated Legendre polynomials will play a key role here. This directly connects with angular momentum, which will also be looked at in this report. This report aims to give students an introduction on spherical harmonics and how they can be used.