  We prove the curious identity in the sense of formal power series: \[
\int_{-\infty}^{\infty}[y^m] \exp\left(-\frac{t^2}2
+\sum_{j\ge3}\frac{(it)^j}{j!}\, y^{j-2}\right)\mathrm{d} t =
\int_{-\infty}^{\infty}[y^m] \exp\left(-\frac{t^2}2+
\sum_{j\ge3}\frac{(it)^j}{j}\, y^{j-2}\right)\mathrm{d} t, \] for
$m=0,1,\dots$, where $[y^m]f(y)$ denotes the coefficient of $y^m$ in the Taylor
expansion of $f$. The generality of this identity from the perspective of
saddle-point method is also examined.
