  The Harish-Chandra--Howe local character expansion expresses the characters
of reductive, $p$-adic groups in terms of Fourier transforms of nilpotent
orbital integrals on their Lie algebras, and Murnaghan--Kirillov theory
expresses many characters of reductive, $p$-adic groups in terms of Fourier
transforms of semisimple orbital integrals (also on their Lie algebras). In
many cases, the evaluation of these Fourier transforms seems intractable; but,
for $\operatorname{SL}_2$, the nilpotent orbital integrals have already been
computed. In this paper, we use a variant of Huntsinger's integral formula, and
the theory of $p$-adic special functions, to compute semisimple orbital
integrals.
