I am trying to evaluate the following integral $$ I = \int_0^t s^{2\alpha - 1} \exp\left(\frac{i \sqrt{2} \left(t^{2 \alpha + 1} - s^{2 \alpha + 1}\right)}{ 2 \alpha + 1}\right)\mbox{d}{s} $$ where $0 \le \alpha \le 1$.
I am not quite sure how to solve this. Any help would be much appreciated.
Set $s^{2\alpha+1}=w$. Then $$ I = \int_0^t s^{2\alpha - 1} \exp\left(\frac{i \sqrt{2} \left(t^{2 \alpha + 1} - s^{2 \alpha + 1}\right)}{ 2 \alpha + 1}\right)\mbox{d}{s}=(1+2\alpha)^{-1}\exp\left(\frac{i\sqrt{2}t^{2\alpha+1}}{2\alpha+1}\right)\int_0^{t^{2\alpha+1}}\frac{dw}{w^{\frac{1}{2\alpha+1}}} \exp\left(\frac{-i\sqrt{2}w}{2\alpha+1}\right) $$ $$ =(1+2\alpha)^{-1}\exp\left(\frac{i\sqrt{2}t^{2\alpha+1}}{2\alpha+1}\right)\times$$ $$\left[2^{-\frac{\alpha}{2 \alpha+1}} t^{2\alpha} \left(\frac{i t^{2 \alpha+1}}{2 \alpha+1}\right)^{\frac{1}{2 \alpha+1}-1} \left(\Gamma \left(\frac{2 \alpha}{2 \alpha+1}\right)-\Gamma \left(\frac{2 \alpha}{2 \alpha+1},\frac{i \sqrt{2} t^{2 \alpha+1}}{2 \alpha+1}\right)\right)\right]\ . $$ I checked with Mathematica and it seems to work.