Fourier Series Problem - Representing an Even Function

maple
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In quantum mechanics, a free particle in an infinite potential well has the wave function (ie. overlap <x/phi>). Its eigenfunctions take the form:

(2/a)^1/2 * sin(n*pi*x/a), n is ofcourse an integer.

My question is that do all eigenfunctions form a basis? And if so how can you represent an even function with eigenfunctions which are clearly odd- my understanding of the Fourier Series is that its equals a function by representing it as an infinite sum of both sin and cos terms, and if the function is even, the coefficients of the sin terms are zero.

Eg. why can I represent:

cos (pi*x/a) = Infinite Sum (A(subscript)n * sin(n*pi*x/a)).

Any assistance would be much appreciated.
 
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You cannot represent any function with sines in the manner you hypothesize. I would guess the answer is that the Eigenfunctions form a basis of the space of the set of solutions of the wave equation. Not all functions are solutions.
 

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