I have a complex Gaussian Integral for my QFT course to solve without looking at the integral table but I don't know how I should do it
The integral is: $$\int_{-\infty}^{+\infty} \exp\left(\frac{i\cdot x^2}{2}\right) \mathrm{d}x = \left(1+i\right) \sqrt{\pi}.$$
My suggestion would be to take:
$$\exp\left(\frac{i\cdot x^2}{2}\right) = \cos{\left(\frac{x^2}{2}\right)} + i \sin{\left(\frac{x^2}{2}\right)}\,,$$
and solve it as a Fresnel integral. Is that a right Approach or would you do something else?
