Convolution of functions in the time domain is equivalent to multiplication in the frequency domain. However, I am interested in multiplication of functions in the time domain, which is convolution in the frequency domain.
For example, consider the constant function $f(t) = 1$. I would like to evaluate $f(t)^{2}$, but in the frequency domain. Using the convolution theorem, this is equivalent to $\frac{1}{2\pi i}F(s) * F(s)$. Since $F(s) = \frac{1}{s}$, this can be written as (disregarding the constant factor) $$ \large{\int_{-\infty}^{s} F(\zeta)F(s-\zeta)d\zeta} = \large{\int_{-\infty}^{s} \frac{d\zeta}{\zeta (s-\zeta)}}$$
Using partial fraction expansion, the rational function becomes $$ \large{ \frac{1}{s\zeta} + \frac{1}{s(s-\zeta)} } $$
So the integral reduces to $$ \large{ \frac{1}{s} \int_{-\infty}^{s} \frac{1}{\zeta}} + \frac{1}{s-\zeta} d\zeta = \frac{1}{s}(\log(\zeta) + \log(s-\zeta)) |_{-\infty}^{s}$$
I am not sure how to evaluate this integral. Are my limits incorrect since the variable is complex, or is there something else I missed?