Important non-elementary functions: Delta function

3.18 Important non-elementary functions: Delta function

The \(\delta\) function for real numbers \(x\) corresponds to the Kronecker \(\delta_{ij}\) for integer numbers \(i\), \(j\). It is defined by

\[\delta(x)=\left\{\begin{array}{cl}0&x\neq 0\\\infty&x=0\end{array}\right. \quad \mbox{and} \quad \int_{-\epsilon}^{+\epsilon}\delta(x)dx = 1\]

The major property of the \(\delta\) function is it’s projection property within an integral for all reasonable functions \(f(x)\)

\[f(x_0)= \int_{x_0 - \epsilon}^{x_0 + \epsilon} f(x) \delta(x-x_0) dx\]

Exactly spoken, the \(\delta\) function is not a function but a distribution, i.e. there are many representation of this function. All of them need a limiting process. Some important representation of the \(\delta\) function are

\[\delta(x)= \lim_{\epsilon \to 0} \left\{\begin{array}{cl}0&|x|\gt\frac{\epsilon}{2}\\ \frac{1}{\epsilon}&|x|\leq \frac{\epsilon}{2}\end{array}\right.\]

\[\delta(x)= \lim_{\sigma \to 0} \frac{1}{\sqrt{2\pi}\sigma} e^{\frac{-x^2}{2\sigma^2}}\]

\[\delta(x)= \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{ikx}dk\]

For basis vectors \(e_n\) with integer index \(n\) the Kronecker \(\delta\) is used to define the orthonormality relation

\[\left\langle e_n|e_m\right\rangle = \delta_{n,m}\]

One very important example is

\[\delta_{n,m}= \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{ik(n-m)}dk\]

For basis vectors \(f_k(x)\) with continuous index \(k\) orthonormality is defined by the \(\delta\) function

\[\left\langle f_k(x)|f_l(x)\right\rangle = \int f^*_k(x) f_l(x) dx = \delta(k-l)\]

e.g. for the Fourier functions we have

\[\left\langle e^{ikx}|e^{ilx}\right\rangle = \frac{1}{2 \pi} \int_{-\infty}^{+\infty} e^{-ikx} e^{ilx} dx = \delta(k-l)\]