PX275 - H2 - calculating coefficients

\frac{1}{L} \int_{- L / 2}^{L / 2} \exp (\pm 2 π i (n - m) x / L) d x = δ_{n m}

multiplying the complex fourier series by $\exp (- i 2 π m x / L)$ and integrating:

\begin{aligned} \int_{- L / 2}^{L / 2} f (x) \exp (- i 2 π m x / L) d x & = \int_{- L / 2}^{L / 2} \sum_{n = - \infty}^{\infty} C_{n} \exp (i 2 π (n - m) x / L) d x \\ = L C_{m} \end{aligned}

∴ C_{m} = \frac{1}{L} \int_{- L / 2}^{L / 2} f (x) \exp (- 2 i π m / L) d x

this can be used as two-directional transformation:
- forward: coefficients from a function
- backward: function from coefficients