  Let $f$ be a fixed self-contragradient Hecke-Maass form for $SL(3,\mathbb
Z)$, and $u$ an even Hecke-Maass form for $SL(2,\mathbb Z)$ with Laplace
eigenvalue $1/4+k^2$, $k>0$. A subconvexity bound
$O\big(k^{4/3+\varepsilon}\big)$ in the eigenvalue aspect is proved for the
central value at $s=1/2$ of the Rankin-Selberg $L$-function $L(s,f\times u)$.
Meanwhile, a subconvexity bound $O\big((1+|t|)^{2/3+\varepsilon}\big)$ in the
$t$ aspect is proved for $L(1/2+it,f)$. These bounds improved corresponding
subconvexity bounds proved by Xiaoqing Li (Annals of Mathematics, 2011). The
main technique in the proof, other than those used by Li, is an $n$th-order
asymptotic expansion of a weighted stationary phase integral, for arbitrary
$n\geq1$. This asymptotic expansion sharpened the classical result for $n=1$ by
Huxley.
