Two functions and are said to be orthogonal if and only if
and is nonzero if and only if
The term orthogonal means at right angles to implying an analogy between functions and vectors. The orthogonal set of
functions may be considered to play the role of unit or basis vectors, and the integration process fills the role of the scalar or
Examples of orthogonal functions
It is not difficult to show that the following collections of functions are orthogonal
- are orthogonal to .
Orthogonality of the cosines
We consider the integral of the product of two cosine functions
which follows from the evenness of cosine.
There are three cases to consider
Applying the appropriate double angle identity, we may write
By the evenness of cosine, we may write
Employing the identity relating the product of cosines to a difference of cosines
owing to the fact that for all integers .
Hence, the set of cosines is orthogonal.
Orthogonality of the sines
By the fact that the product of odd functions is even we may write
Employing the double angle formula for sine
Employing the identity relating the product of sines to a difference of cosines, we may write