This Integral is giving me headaches: $$I(x|T):=\int_0^T \int_0^t \tau^{-1/2} (T-\tau)^{-1/2} \exp(\frac{ix^2}{2(T-\tau)})d\tau dt$$ where $x$ and $T$ are positive reals.
Can it be evaluated in terms of elementary functions? Can it be evaluated in terms of special functions (Bessel, Gamma, Error, ...)?
Sadly, I can't provide starting points, because I have not found any promising ones.
Edit 1:
So, I've tried a couple things now. First of all, taylor-expanding the Exponential function and switching summation and the integration is not possible in this case: $$\int_0^T \int_0^t \tau^{-1/2} (T-\tau)^{-1/2} \exp(\frac{ix^2}{2(T-\tau)})d\tau dt \neq \sum_{n=0}^{\infty}(\frac{ix^2}{2})^n \frac{1}{n!} \int_0^T \int_0^t \tau^{-1/2} (T-\tau)^{-(n+1/2)}d\tau dt$$
Next, I simplified the Integral a bit, so that I'm now dealing only with:
$$\int_0^1 \int_0^t \tau^{-1/2} (1-\tau)^{-1/2} \exp(i\frac{b}{1-\tau})d\tau dt=\int_0^1t^{-1/2}(1-t)^{1/2}\exp(i\frac{b}{1-t})dt=:J$$ where $b$ is a positive real.
As pointed out in the comments, this being an convolution integral, it could be written as an inverse Laplace transformation:
$$J(b)=\mathcal{L}^{-1}\{\mathcal{L}[t^{-1/2}]\cdot\mathcal{L}[(t)^{1/2}\exp(i\frac{b}{t})]\}(1)$$
And while I can find sources on $\mathcal{L}[(t)^{1/2}\exp(\frac{b}{t})]$ (in the DLMF, for example), an expression for $\mathcal{L}[(t)^{1/2}\exp(i\frac{b}{t})]$ seems hard to come by. (Mathematica provides the following:$$-((1+i) (-1)^{7/8} \pi \left| b\right| ^{3/4} \left(\text{ber}_{-\frac{3}{2}}\left(2 \sqrt{s} \sqrt{\left| b\right| }\right)-i \text{ber}_{\frac{3}{2}}\left(2 \sqrt{s} \sqrt{\left| b\right| }\right)+i \text{bei}_{-\frac{3}{2}}\left(2 \sqrt{s} \sqrt{\left| b\right| }\right)+\text{bei}_{\frac{3}{2}}\left(2 \sqrt{s} \sqrt{\left| b\right| }\right)\right))\frac{1}{\sqrt{2} s^{3/4}}$$ where bei and ber are some Kelvin functions I never heard of.)
Could it be possible to compute: $$J'(b)=\mathcal{L}^{-1}\{\mathcal{L}[t^{-1/2}]\cdot\mathcal{L}[(t)^{1/2}\exp(\frac{b}{t})]\}(1)$$ and use analytic continuation to get $J(b)$?
Also, A difficult one-variable exponential integral and Find the Green's Function and solution of a heat equation on the half line seem relevant.
