AbstractWe produce an estimate for the K-Bessel function $$K_{r + i t}(y)$$
K
r
+
i
t
(
y
)
with positive, real argument y and of large complex order $$r+it$$
r
+
i
t
where r is bounded and $$t = y \sin \theta $$
t
=
y
sin
θ
for a fixed parameter $$0\le \theta \le \pi /2$$
0
≤
θ
≤
π
/
2
or $$t= y \cosh \mu $$
t
=
y
cosh
μ
for a fixed parameter $$\mu >0$$
μ
>
0
. In particular, we compute the dominant term of the asymptotic expansion of $$K_{r + i t}(y)$$
K
r
+
i
t
(
y
)
as $$y \rightarrow \infty $$
y
→
∞
. 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 $$E_0^{(j)}(z, r+it)$$
E
0
(
j
)
(
z
,
r
+
i
t
)
for each inequivalent cusp $$\kappa _j$$
κ
j
when $$1/2 \le r \le 3/2$$
1
/
2
≤
r
≤
3
/
2
.
A uniform estimate for rough paths