Ytukyg

I would like to directly derive the probability density function (PDF) for a Chi-squared distribution with $k$ degrees of freedom using characteristic functions.

If $X_{1}, X_{2}, dots, X_{k}$ are independent, standard normal random variables, then $$ Y = sum_{i=1}^{k} X_{i}^{2} $$ and $Y$ is chi-squared distributed with $k$ degrees of freedom. The PDF for $Y$ when $k = 1$ is given by $$ f(x, 1) = frac{1}{sqrt{2 pi x}} e^{-frac{1}{2} x} $$ and its respective characteristic function is $$ varphi_{Y_{1}} (omega) = int_{-infty}^{infty} f(x, 1) e^{iwx} dx = (1 - i2 omega)^{-frac{1}{2}} text{,}$$ where $i$ is the imaginary number. For a general $k$ degrees of freedom, $Y$'s characteristic function is given by $$varphi_{Y} (omega) = (1 - i2 omega)^{-frac{k}{2}} text{.}$$ Is it possible to explicitly derive $f(x, k)$ using the inverse Fourier transform, where $$ f(x, k) = frac{1}{2 pi} int_{-infty}^{infty} (1 - i2 omega)^{-frac{k}{2}} e^{-iwx} domega ?$$
I have had no success with this approach, but I'm probably missing something very obvious.

Failing that, is it possible to derive a formula for $k=2$, where $$f(x, 2) = frac{1}{2 pi} int_{-infty}^{infty} (1 - i2 omega)^{-1} e^{-iwx} domega ?$$ This would at least allow me derive the PDF inductively. In addition, I am aware that $f(x, 1)$ can be represented as the Gamma distribution $text{Gamma}(x, frac{1}{2}, frac{1}{2})$ and the sum of independent Gamma random variables is known to be Gamma distributed, therefore, for this example, $$ Y sim text{Gamma} ( cdot, frac{k}{2}, frac{1}{2}) text{.} $$