I want to find the fourier transform of $e^{-x^2} = \int_{-\infty}^{\infty}e^{ikx-x^2}\,dx$ using contour integration.
I consider the rectangular contour $C$ with verticies $\pm R, \pm R + ik$
Then $f(z) = e^{ikz - z^2}$ is holomorphic on $C$ so $\int_C f(z) \,dz = 0$.
I tried parameterizing as such: $$C_1 : z = x \quad -R \leq x \leq R$$ $$C_2 : z = R + iky \quad 0 \leq y \leq 1$$ $$C_3 : z = -x + ik \quad R\leq x\leq R$$ $$C_4 : z = -R + ik(1-y) \quad 0 \leq y \leq 1$$
But then I find $\int_{C_1} = \int_{C_3}$ and $\int_{C_2} = \int_{C_4}$ which gives me $\int_{C_1}= -\int_{C_3}$
Which doesn't seem helpful. How should I approach this problem?