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I want to calculate a convolution of a discontinuous function $f$ with a continuous function $g$. For example

$$(f*g)(t) = \left(\dfrac{t+a}{t^2-b^2}\right) * \left(\dfrac{t-c+id}{(t-c)^2+d^2}\right)$$

where $a$, $b$, $c$, $d$ are real constants.

Because $f(t)=\dfrac{t+a}{t^2-b^2}$ has vertical asymptotes at $t=\pm b$, one way to evaluate the convolution is to make a substitution in the convolution integrand. Is this the easiest way to solve the convolution? If so, are there any specific substitutions that are ideal for this kind of problem?

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  • $\begingroup$ If the convolution integral is understood in the principal value sense, partial fraction decomposition reduces it to a sum of integrals of the form $$\operatorname {v. \! p.} \int_{\mathbb R} \frac {dt} {t (t - z)} = -\pi i \operatorname* {Res}_{t = 0} \frac 1 {t (t - z)} \operatorname {sgn} \operatorname {Im} z$$ with $z \not \in \mathbb R$. $\endgroup$ – Maxim 2 days ago

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