Problems for 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 A_{n} such that if we define basis functions e_{n} = A_{n} sin(nθ) then
.
Solution
(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.