(Main question) I have an integral involving complex variables (but integrated on the real domain) that I need to evaluate numerically, but I am interested in knowing if it can be reduced to any given known form for which we have approximation schemes better than quadrature, which is what I am currently doing.
Here is the integral:
$$\int_{0}^{T} \left(c - iu \mathrm{e}^{-\eta(T - t)}\right)^{\alpha}dt$$
Notes on bounds on the parameters:
- $\alpha \in (0,1) \cup (1,2)$
- $c, \eta, T > 0$
(Secondary question) I also suspect its conjugate is
$$\int_{0}^{T} \left(c + iu \mathrm{e}^{-\eta(T - t)}\right)^{\alpha}dt$$
But I also don't know how to prove it, because although the integrand is the conjugate, the same doesn't necessarily apply after integration, but my numerical experiments seem to indicate that it is.