Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 39C / b / ii / Solution

If the initial field is also an even function, then $A(k)$ is real and even. Provided $A(k_0)\ne 0$, the leading term becomesIts large zeros satisfyThereforeIf $A(k_0)=0$, the leading stationary-phase coefficient vanishes and a higher-order term determines the zeros.