Fourier Series

• Fourier said you can express any function as a infinite linear combination of sin functions

$$F(x)=\sum _{n=0}^{\infty }{c}_n\sin \left(\frac{n\pi x}{L}\right)$$

• In “function space” the $\sin$ functions are orthogonal
• Using the Kronecker delta, we can rewrite

$$\underset{0}{\overset{L}{\int }}\sin \left(\frac{n_1\pi x}{L}\right)\sin \left(\frac{n_2\pi x}{L}\right)dx$$

as

$$\frac{L}{2}{\delta }_{n_1,n_2}$$

We use this property to solve this integrals necessary to calculate the coefficients.