A Quadratic Maximum?

Question

What does the following mean?

Context: Laplace integrals

Consider the integral $$\int_a^b f(t) \exp(x\phi(t))\,\,\,dt$$ where $\phi(t) $ achieves its maximum at some $t=c\in (a,b)$. Assume that the maximum is quadratic.

What could a "quadratic maximum" mean?

Thank you.

Answer 1

Since $\phi(t)$ reaches a maximum at $c$, assuming that $\phi''(c)\lt0$, near $c$ we have $$ \begin{align} \phi(t) &=\phi(c)+\phi'(c)(t-c)+\frac12\phi''(t-c)(t-c)^2+O(t-c)^3\\ &=\phi(c)+\frac12\phi''(t-c)(t-c)^2+O(t-c)^3 \end{align} $$ Thus, $\phi$ behaves like a quadratic function near $c$.

Therefore, we get the stationary phase approximation: $$ \begin{align} \int_a^bf(t)e^{x\phi(t)}\,\mathrm{d}t &\stackrel.=e^{x\phi(c)}\int_a^bf(t)e^{\frac12x\phi''(c)(t-c)^2}\,\mathrm{d}t\\ &\stackrel.=e^{x\phi(c)}f(c)\sqrt{\frac{\vphantom{\lambda}2\pi}{-x\phi''(c)}} \end{align} $$