For questions about the Kronecker-delta, which is a function of two variables (usually non-negative integers).
In mathematics, the Kronecker delta is a function of two variables
$$\delta _{{ij}}={\begin{cases}0&{\text{if }}i\neq j,\\1&{\text{if }}i=j.\end{cases}}$$
In linear-algebra, the values of Kronecker delta $(\delta_{ij})_{i,j = 1,\dots,n}$ form an $n \times n$ identity matrix $I$.
It satisfies the shifting property $$\sum_{i=-\infty}^\infty a_i \delta_{ij} =a_j, \quad j \in \Bbb{Z}.$$
It has the contour integral representation $$\delta_{mn}=\frac{1}{2\pi}\oint_{|z|=1} z^{m-n-1} \, {\rm d}z, \quad m,n \in \Bbb{Z}.$$
It has an exponential-sum representation $$\delta _{nm}={\frac {1}{N}}\sum _{k=1}^{N}e^{2\pi i{\frac {k}{N}}(n-m)}, \quad m,n \in \Bbb{Z},$$ which can be proved by the formula for the finite geometric-series.
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