Does $\int\_{0}^{T} \frac{1}{T}\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{\left(t-t\_{0}\right)^{2}}{2\sigma^{2}}}e^{i\omega t}dt$ have a closed-form solution?



Integral involving a gaussian function
If an error function is allowed:


$$\frac{e^{\frac{1}{2} i \omega \left(i \sigma ^2 \omega +2 t\_0 \right)} \left(\text{erf}\left(\frac{i \sigma ^2 \omega + t\_0 }{\sqrt{2}
 \sigma }\right)-\text{erf}\left(\frac{i \sigma ^2 \omega +t\_0 -\text{T}}{\sqrt{2} \sigma }\right)\right)}{2 \text{T}}$$


For 


$$\int\_0^T \frac{1}{T} \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{\left(t-t\_{0}\right)^{2}}{2\sigma^{2}}} \cos \left( \omega t \right)\ dt$$


the answer is more complicated:


$$\frac{e^{-\frac{1}{2} \omega \left(\sigma ^2 \omega +2 i t\_0 \right)}
 \left(-\text{erf}\left(\frac{-t\_0 +i \sigma ^2 \omega }{\sqrt{2} \sigma
 }\right)+e^{2 i t\_0 \omega } \left(\text{erf}\left(\frac{\text{t0}+i \sigma ^2
 \omega }{\sqrt{2} \sigma }\right)+\text{erf}\left(\frac{-i \sigma ^2 \omega
 -\text{t0}+\text{T}}{\sqrt{2} \sigma }\right)\right)+\text{erf}\left(\frac{i \sigma
 ^2 \omega -t\_0+\text{T}}{\sqrt{2} \sigma }\right)\right)}{4 \text{T}}$$


Otherwise I don't see how one can get a closed-form solution.


