# delta function on a complex number

for a real number we know that

$$f(a)= \int_{-\infty}^{\infty}dx \delta (x-a)f(x)$$

but what happens for $$\int_{-\infty}^{\infty}dx \delta (x-2i)f(x)$$ ?

is this equal to $f(2i)$ or equal to $0$ , of course $i= \sqrt -1$ a complex number :)

if i use the generalize funtion approach $\delta (ix)= \frac{sinh(nx)}{x \pi}$ but in the limit $n \to \infty$ this limit makes no sense

• I'm personally only familiar with the definition over the reals, but if this has the usual properties of $\delta$ functions then this integral should be zero since $x \neq 2i$. – Spencer Feb 9 '14 at 11:19
• I think the crux of the matter is that the first equation is in fact how we define $\delta(x-a)$, rather than being a consequence of the context. It's therefore left to us to define the second integral in whichever way makes sense in light of our specific needs. – Jonathan Y. Feb 9 '14 at 17:53
• I'm not actually familiar with that property of real numbers, does anybody have a handy link discussing the properties of the delta function? – JacksonFitzsimmons Jun 29 '15 at 22:02

The integral $\int\limits_{-\infty}^{\infty}\delta(x-2i)\,f(x)\,dx$ doesn't make sense, but the integral illustrated in (1) below does make sense with the variable substitution $s=x+2\,i$ given certain assumptions on $f$ (e.g. complex analytic) .

(1) $\quad\int\limits_{-\infty+2\,i}^{\infty+2\,i}\delta(s-2i)\,f(s)\,ds=\int\limits_{-\infty}^{\infty}\delta(x)\,f(x+2\,i)\,dx=f(2\,i)$

An integral of $\delta(s)$ for $s\in\mathbb{C}$ makes sense if-and-only-if it can be mapped to an integral of $\delta(x)$ for $x\in\mathbb{R}$. This can be done for integrals along lines both parallel and perpendicular to the real axis which is illustrated in the following answer I posted to a similar question.

Delta function with imaginary argument

• I have removed my comments. At least the added attention has woken up the OP. – robjohn Sep 4 '18 at 17:00

Depending on the application you can generalize the dirac delta distribution using its cauchy representation to get a reasonable analytic continuation on the complex plane. $$f(z)=\oint_{C}\frac{f(\zeta)}{z-\zeta}$$ Here's a short paper that uses this perscription with a short description.look at equations 4.7 and 4.8 and the paragraph preceding them: http://arxiv.org/pdf/1212.2132v1.pdf

One precise way to define the $\delta$-function is as a unit point measure on $\mathbb{R}$, $${\delta}(x-a)dx=d{\mu_a}(x)$$ for real $a$. Given this, we want to extend it to have a meaning for non-real $a$. This looks problematic. Secondly, in your example, you have to require some analyticity to give a meaning to $f(2i)$