Orthogonalité des Sinus et Cosinus

Si $m\ne n$ : $$\boxed{{\int }_{-L}^L\cos \left(\frac{n\pi x}{L}\right)\cos \left(\frac{m\pi x}{L}\right)dx=0}$$ $$\boxed{{\int }_{-L}^L\sin \left(\frac{n\pi x}{L}\right)\sin \left(\frac{m\pi x}{L}\right)dx=2{\int }_0^L\sin \left(\frac{n\pi x}{L}\right)\sin \left(\frac{m\pi x}{L}\right)dx=0}$$ Si $m=n$ : $$\boxed{{\int }_{-L}^L\cos \left(\frac{n\pi x}{L}\right)\cos \left(\frac{m\pi x}{L}\right)dx=L}$$ $$\boxed{{\int }_{-L}^L\sin \left(\frac{n\pi x}{L}\right)\sin \left(\frac{m\pi x}{L}\right)dx=2{\int }_0^L\sin \left(\frac{n\pi x}{L}\right)\sin \left(\frac{m\pi x}{L}\right)dx=L}$$