# How to solve this complex Gaussian integral? [duplicate]

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?

• Possible duplicate: physics.stackexchange.com/q/368186/2451 – Qmechanic May 21 '18 at 7:58
• Year, I've read that but I do not understand it so well :/ – Armani42 May 21 '18 at 8:02
• Please be more specific as to what you do not understand about the answers to the other question since otherwise we don't really know what to elaborate on – ACuriousMind May 21 '18 at 9:46
• So I want to know if it is ok to solve the integral using a fresnel integral? Or do I halos have to integrate in 4 steps like in the other thread? – Armani42 May 21 '18 at 16:40