Reciprocal lattice and Fourier series

[touqra]

First off, this is not a homework problem. I was reading Charles Kittel solid states book on Chapter 2, equation 3:

electron number density, n(x), expanded in a Fourier series:

$$n(x) = n_0 + \sum_{p} [C_p cos(\frac{2\pi p x}{a}) + S_p sin(\frac{2\pi p x}{a})]$$

From this expansion, wouldn't the density n(x = 0) be infinity ? since $$C_p$$ shouldn't be zero for the Fourier expansion to make sense.

 

[kanato]

Why shouldn't C_p be zero for it to make sense? You realize that the value of C_p depends on p, right? So there could be C_p values which are non-zero and others which are zero.

 

[Borthakur.gg]

Basically Fourier series repersentation is applied to functions which are bounded.The next thing is that you have to check for appropriate Dirichlet's conditions.

Thus at the very beginning, a peassumption for applying Fourier series expansion is that the function it represents is always finite.
More simply speaking, kanato is right.
O.K.?