= Quartic decay sectors for a contour integral
{title2=$\operatorname{Re}(\zeta^4)<0$}
For a <contour integral> with factor $e^{\zeta^4/4+z\zeta}$, the quartic term dominates the linear term on rays at large radius. Writing $\zeta=re^{i\theta}$ gives <real part> $r^4\cos(4\theta)/4$ for the leading exponent. There are four decay sectors, centered at $\theta=\pi/4+j\pi/2$ and each of angular width $\pi/4$. On a closed subsector the integrand and all its $z$ <derivatives> decay faster than any exponential in $r$, locally uniformly in $z$. <Contours> joining two such sectors therefore allow <differentiation under the integral sign> and have vanishing infinite endpoint terms. Distinct geometric <contours> still need an independence proof; their initial moments often give a <Vandermonde matrix> calculation.
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