I am doing some quantum field theory (physics) calculations with the fermion mass term, and it is making the calculation much more challenging.
Say I have
$$F(t,m) = \int_0^1 dx \frac{1}{1-x}\frac{1}{(t ~(1-x)+m(1-x)^2)^\epsilon}\frac{1}{t}$$
where $\epsilon$ is the dimensional regularization parameter, which is deemed to be small.
I would like to take a Laplace transformation $$\mathcal{L}[F(t,m)]=\tilde{F}(s,m)$$ and this seems much more difficult than computing $\tilde{F}(s,0)$, which (at least using mathematica) one can trivially compute as
$$\tilde{F}(s,0) = \mathcal{L}\left[\frac{1}{t^{1+\epsilon}}\right]\int_0^1dx\frac{1}{(1-x)^{1+\epsilon}}\\ =s^\epsilon \Gamma[-\epsilon]\int_0^1dx\frac{1}{(1-x)^{1+\epsilon}}\,.$$
However, things are a lot more complicated now that the $\epsilon$ exponentiated denominator entangles $\tau$ and $x$ for the $m\neq 0$ calculation. Of course, one of the complication stems from $x$ being integrated, but even an unintegrated case such as
$$\mathcal{L}[G(t,m)]$$ where $$G(t,m)=\frac{1}{t}\frac{1}{(t+m)^\epsilon}$$ seems quite nontrivial.
Any insight on how to proceed or advice on useful properties/tricks of Laplace transformation that may be helpful in this calculation?
Thanks a lot!