We produce an estimate for the K-Bessel function \(K_{r + i t}(y)\) with positive, real argument y and of large complex order \(r+it\) where r is bounded and \(t = y \sin \theta \) for a fixed parameter \(0\le \theta \le \pi /2\) or \(t= y \cosh \mu \) for a fixed parameter \(\mu >0\). In particular, we compute the dominant term of the asymptotic expansion of \(K_{r + i t}(y)\) as \(y \rightarrow \infty \). When t and y are close (or equal), we also give a uniform estimate. As an application of these estimates, we give bounds on the weight-zero (real-analytic) Eisenstein series \(E_0^{(j)}(z, r+it)\) for each inequivalent cusp \(\kappa _j\) when \(1/2 \le r \le 3/2\).

我们为K-贝塞尔函数\（K_ {r + it}（y）\）产生一个估计，该估计具有正实实数y且具有大复数阶\（r + it \），其中r是有界且\（t = y为固定参数\（0 \ le \ theta \ le \ pi / 2 \）或\（t = y \ cosh \ mu \）为固定参数\（\ mu> 0 \）。特别地，我们将\（K_ {r + it}（y）\）的渐近展开的主导项计算为\（y \ rightarrow \ infty \）。当t和y接近（或相等），我们也给出一个统一的估计。作为这些估计的应用，我们为每个不等价的尖点\（\ kappa _j \ ）给出了零权重（实解析）爱森斯坦级数\（E_0 ^ {（j）}（z，r + it）\）的界限。 ）时\（1/2 \ le r \ le 3/2 \）。