in below link, (formula (34)-(40)) there are some definition of Dirac delta function in terms of other functions such as Airy function, Bessel function of the first kind, Laguerre polynomial,....
http://mathworld.wolfram.com/DeltaFunction.html
Is there any definition of the Dirac delta function in terms of the sech (secant hyperbolic) or cosh (cosine hyperbolic) functions?
Please say your references. Thanks
On
The delta "function" $\delta(x)$ is supposed to be zero as $x$ gets large in either direction, so basing one on $\cosh(x)$ is hard because $\cosh(x) \to +\infty$ as $x$ gets large in either direction. That makes $\operatorname{sech} (x)$ a good candidate. Since $\int_{-\infty}^{+\infty} \operatorname{sech} x dx=\pi$, you can use $\lim_{k \to \infty}\frac k \pi \operatorname{sech} (kx)$ as a delta "function".
Whenever $f$ is a nonnegative function with integral 1, one has approximation by convolution. Specifically, define $f_c(x) = f(x/c)/c$. Then $\lim_{c \to 0^+} g * f_c = g$ for mildly restricted $g$. (For example, $L^p$ for $1 \leq p < \infty$ is enough.) In particular you can take $f(x)=\text{sech}(x)/\pi$.