# Are the spherical harmonics orthogonal?

## Are the spherical harmonics orthogonal?

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. These functions form an orthogonal system, and are thus basic to the expansion of a general function on the sphere as alluded to above.

Are spherical harmonics normalized?

The spherical harmonics are orthogonal and normalized, so the square integral of the two new functions will just give 12(1+1)=1.

### How are spherical harmonics written as trigonometric functions?

These satisfy the constraint thatl2 N and ¡l•m•l; thus, there are 2l¯1 basis functions of orderl. The orderldetermines the frequency of the basis functions over the sphere. The spherical harmonics may be written either as trigonometric functions of the spherical coordinatesµand

How to calculate the Legendre of a spherical harmonic?

The associated Legendre functions are obtained by differentiating the Legendre polynomial m times and multiplying by (1 − x2)m / 2, Pmℓ(x): = (1 − x2)m / 2 dm dxmPℓ(x). By convention the label m on the functions is indicated as a superscript. This label is not to be confused with a power: Pmℓ is not Pℓ raised to the power m.

#### Why are spherical harmonics important in group theory?

Further, spherical harmonics are basis functions for SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3).

How are spherical harmonics related to Fourier series?

SPHERICAL harmonics are a frequency-space basis for representing functions deﬁned over the sphere. They are the spherical analogue of the 1D Fourier series. Spherical harmonics arise in many physical problems ranging from the computation of atomic electron conﬁgurations to the representation of gravitational and magnetic ﬁelds of planetary bodies.