Perturbative heavy quark fragmentation function throughO(αs2)

We derive the initial condition for the perturbative fragmentation function of a heavy quark through order $\mathcal{O}\left({\alpha }_s^2\right)$ in the $\overline{\mathrm{MS}}$ scheme. This initial condition is useful for computing heavy quark (or lepton, in case of QED) energy distributions from calculations with massless partons. In addition, the initial condition at $\mathcal{O}\left({\alpha }_s^2\right)$ can be used to resum collinear logarithms $\ln {(Q}^2{/m}^2)$ in heavy quark energy spectrum with next-to-next-to-leading logarithmic accuracy by solving the DGLAP equation.