I am currently reading a book by Lukacs on characteristic functions.
He states that the Kolmogorov canonical representation, given by
$$\log \phi(\omega)=i\omega c+\int_{-\infty}^{\infty}\Big(\exp(i\omega x)-1-{i\omega x}\Big)\frac{dK(x)}{x^2}$$
where
$$K(x) =\int_{-\infty}^x(1+y^2)\;dF(y)$$
and
$$c = a +\int_{-\infty}^\infty y\;dF(y)$$
can be deduced from the Lévy-Khinchine canonical representation given by
$$\log \phi(\omega)=i\omega a+\int_{-\infty}^{\infty}\Big(\exp(i\omega x)-1-\frac{i\omega x}{1+x^2}\Big)\frac{1+x^2}{x^2}\;dF(x)$$
I cannot seem to wrap my head around this one. Can anyone clear up what is happening here? Preferebely in a step-by-step manner.