Edward F. Redish
Consider the space of functions of the form
Let's create an orthonormal basis for this space.
(a) First, demonstrate that the functions sin nθ form an orthonormal set; that is, prove that
(b) Next, find constants An such that if we define basis functions en = An sin(nθ) then
(a) This is most easily done by making the complex expansion of the sin terms and observing that the integral of exp(inθ) vanishes if n is not zero.
So we get
Now if we do the integreal of the product of the sines we get
If we assume that n and m are positive then only two of the terms can be non-zero and we get
This means that the functions are orthogonal.
(b) To make them orthonormal, we define the basis vectors:
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Last revision 15. November, 2005.