What is the Fourier transformation $y=exp(-λ|x|)$
Using the identity: $y ̃(k)=∫_{-∞}^{∞}\exp(ikx)y(x)\,dx$
$y ̃(k)=∫_{-∞}^{∞}\exp(ikx) exp(-λ|x|)\,dx$
$y ̃(k)=∫_{0}^{∞}\exp(ix(k-λ))+∫_{-\infty}^{0}\exp(ix(k+λ))$
I do not know how to continue this as when i try to do it i can't define $exp(\infty)$, is there a identity that i can use
$f(x)=e^{-a|x|}$, then $$\sqrt{2\pi} F(k)=\int_{-\infty}^{\infty} e^{-a|x|} e^{-ikx} dx=\int_{-\infty}^{0} e^{ax} e^{-ikx} dx+ \int_{0}^{\infty} e^{-ax} e^{-ikx}$$ $$\implies \sqrt{2\pi} F(k)=\frac{1}{a-ik}-\frac{1}{-a-ik}=\frac{2a}{a^2+k^2}$$