fourier transform calculate

Question

$\phi(x)=\int_{\infty}^\infty (\frac{N}{k^2+\alpha^2})e^{ixk} dk$

I think this integral is related to the Fourier series. There are many obstacles in the calculation process.

This calculation is like what is in physics. help me please.

Answer 1

Well, when we use Euler's formula:

$$e^{x\text{k}i}=\cos\left(x\text{k}\right)+\sin\left(x\text{k}\right)i\tag1$$

So, we get:

$$\mathscr{I}\left(x\right):=\int_{-\infty}^\infty\left(\frac{\text{N}}{\alpha^2+\text{k}^2}\right)\cdot e^{x\text{k}i}\space\text{d}\text{k}=\int_{-\infty}^\infty\frac{\text{N}\cdot\cos\left(x\text{k}\right)+\text{N}\cdot\sin\left(x\text{k}\right)i}{\alpha^2+\text{k}^2}\space\text{d}\text{k}\tag2$$

Now, we can seperate the real and imaginary part:

• Real part: $$\Re\left(\mathscr{I}\left(x\right)\right)=\text{N}\cdot\int_{-\infty}^\infty\frac{\cos\left(x\text{k}\right)}{\alpha^2+\text{k}^2}\space\text{d}\text{k}\tag3$$
• Imaginary part: $$\Im\left(\mathscr{I}\left(x\right)\right)=\text{N}\cdot\int_{-\infty}^\infty\frac{\sin\left(x\text{k}\right)}{\alpha^2+\text{k}^2}\space\text{d}\text{k}\tag4$$