  The Isgur-Wise function vastly reduces the weak-decay form factors of hadrons
containing one heavy quark. In this paper, we extract the Isgur-Wise functions
from the instantaneous Bethe-Salpeter method, and give the numerical results
for the $B_c$ decays to charmonium where the final states include $1S$, $1P$,
$2S$ and $2P$. The overlapping integral of the wave functions for the initial
and final states is the Isgur-Wise function, as the heavy quark effective
theory does. In the case of accurate calculation, describing form factors need
to introduce more relativistic corrections which are the overlapping integrals
with the relative momentum between the quark and antiquark to Isgur-Wise
function. The relativistic corrections to Isgur-Wise function provide greater
contributions especially involving the excited state, and therefore are
necessary to be adopted.
