# Convolution of a Gaussian with a Complex-Centered Delta Distribution

I'm working with convolution operations and encountered a theoretical question that I hope someone here can help clarify.

Consider a Gaussian function $$g(x)$$ and a delta distribution $$\delta(x - (a+bi))$$ where $$a,b > 0$$ and $$i$$ is the imaginary unit. The delta distribution is centered at a complex location $$(a+bi)$$.

My question is: What is the result of the convolution between $$g(x)$$ and $$\delta(x - (a+bi))$$? Specifically, is the result a Gaussian function shifted along the complex plane, positioned on the line $$\text{Im} = b$$, centered at $$\text{Re} = a$$?

In mathematical terms, given: $$g(x) = \exp\left(-\frac{x^2}{2\sigma^2}\right)$$ and $$\delta(x - (a+bi)),$$ what is $$g(x) * \delta(x - (a+bi))$$? Is it correct to interpret the convolution as yielding a Gaussian centered at $$a$$ on the line $$\text{Im} = b$$ in the complex plane? How is this operation defined and interpreted when the delta distribution is centered at a complex number?

This is nothing but the distribution of the random variable $$a+ib+X$$ concentrated on a horizontal line of the complex plane, or if you prefer, the random variable $$(a+X,b)$$ of $$R^2.$$