Kronecker delta

The ${\delta }_{n,m}$ is known as the Kronecker delta. Makes life easier.

$${\delta }_{nm}=\{\begin{array}{ll}0& n\ne m\\ 1& n=m\end{array}$$

Integration Interlude

• If we take the integral of two of these orthogonal $\sin$ functions:

$$\underset{0}{\overset{L}{\int }}\sin \left(\frac{n_1\pi x}{L}\right)\sin \left(\frac{n_2\pi x}{L}\right)dx$$ $$\{\begin{array}{l}{n}_1\ne n_2\text{ this integral is 0 }\\ n_1=n_2\text{ this integral is not L/2}\end{array}$$

Wow! this is the Kronecker delta

$${\delta }_{n,m}=\{\begin{array}{l}0n\ne m\\ 1n=m\end{array}$$