Normalizing wavefunction obtained from Lorentzian wave packet

[XProtocol]

Homework Statement

1. Consider a wave packet with $$ A(k) = \frac {N}{k^2 + \alpha^2} $$
where $\alpha$ is some positive constant. This is Lorentzian wave packet.

a. Find the form of $\psi(x)$ (You may have to use your knowledge of contour integrals in Complex analysis)
b. Find normalization N such that $$ \int_{-\infty}^{\infty} dx\left | \psi(x) \right |^2 = 1 $$

Relevant Equations

Wave obtained by superposing waves with different amplitude $A(k)$ depending on k is given by:
$$\psi(x) = \int_{-\infty}^{\infty} A(k)e^{ikx}dk $$

Part a: Using the above equation. I got
$$\psi(x) = \int_{-\infty}^{\infty} \frac{Ne^{ikx}}{k^2 + \alpha^2}dk $$

So basically I needed to solve above integral to get the wave function. To solve it, I used Jordan's Lemma & Cauchy Residue Theorem.
And obtained $$\psi(x) = \frac {N \pi e^{-x\alpha}}{\alpha} $$

For part b, I need to find value of N by solving the equation: $$ \frac {\pi ^2 N^2}{\alpha^2} \int_{-\infty}^{\infty} e^{-2x\alpha} dx = 1 $$
Which I obtained as $$|\psi(x)|^2 = \frac {N^2\pi^2}{\alpha^2} e^{-2x\alpha} $$
Now, this integral is not converging. So either my wavefunction is incorrect or I'm doing something wrong in part b. A friend of mine also got same $\psi(x)$ due to which I'm really confused as to where could we be going wrong.

 

[vela]

You didn't evaluate the contour integral correctly. You should have two contours, one for $x>0$ and one for $x<0$.

 

[XProtocol]

I'm not able to understand why it will have two contours and not 1 i.e $x = -\infty$ to $x=+\infty$. Can you please explain

 

[vela]

Reread what Jordan’s lemma says. It mentions two contours.

https://en.wikipedia.org/wiki/Jordan's_lemma