Let (M, g) be a two-dimensional Riemannian manifold of finite diameter with a conical singularity. Under the assumption that the metric near the cone point C is rotationally invariant, but not necessarily flat, we give an explicit formula for the coefficient \(b_{1/2}(C)\) in the heat trace expansion \(\operatorname {tr}(\operatorname {exp}(-t\Delta _g))\sim _{t\searrow 0} (4\pi t)^{-1}\sum _{j=0}^\infty a_j(M) t^j+\sum _{j=0}^\infty b_{j/2}(C)t^{j/2}+\sum _{j=0}^\infty c_{j/2}(C) t^{j/2} \log t\) . In the case that the Gaussian curvature K of (M, g) satisfies \(|K(p)|\rightarrow \infty \) as \(p\rightarrow C\) , we show that \(b_{1/2}(C)\) varies irrationally under constant rescalings of the distance circles near the cone point. This is a sharp contrast to the behavior of \(b_0(C)\) and of those coefficients \(b_j(C)\) which appear in certain known formulas in the case of orbifold cone points or corners of geodesic polygons.
