There is a function $f(x)$ whose domain is $\left\{ x \mid x\in [0,\infty), x \in \mathbb{R} \right\}$, and what is the Fourier transform of it? Is it
$$ \tilde{f}(k) = \int_{0}^{\infty}e^{-ikx}f(x) \, {\rm d} x ?$$
And how about the inverse Fourier transform? What I'm trying to do is to solve the heat diffusion equation $$ \frac{\partial T}{\partial t}=K\triangledown^2 T $$ in cylindrical coordinates. Assume that the function is independent of $\theta$ and $z$, so I only need to deal with PDE with 2 dimension, and turn into the form of $$ \frac{\partial T}{\partial t}=K\left(\frac{\partial^2 T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}\right). $$ Also, the initial condition is $$ T(r,0)= \begin{cases} T_2 & 0 \leq r \leq R\\ T_1 & r \geq R\\ \end{cases}. $$ I'm trying to solve this equation via Fourier transform, but $r$ is always greater than $0$; thus the domain of the function is $[0,\infty)$. I'm not sure if this function can be transformed. That's why I have this question.