I hope someone can tell me something about the error term in the formula calculating the oscillatory integral like $\int_a^b g(x)e(f(x))d x$. Specially, the exact formula on page 114 of M. Huxley's great book-(Area, lattice points, and exponential sums, London Mathematical Society Monographs, New Series, 13. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1996), or lemma 5.1.2 and lemma 5.1.3 in his book.
I just learned that in one of Huxley's papers (On stationary phase integrals ) a theorem states that-
let $b>a, f^{(2)}\in C[a,b],g^{(3)}\in C[a,b]$. If $|f^{(2)}|\le \frac{B^2}{M^2} \cdot T,|f^{(3)}|\le \frac{B^3}{M^3} \cdot T,$ $|f^{(4)}|\le \frac{B^4}{M^4} \cdot T$ , and $|g|\le U,|g^{(1)}|\le \frac{B}{N} \cdot U,$ $|g^{(2)}|\le \frac{B^2}{N^2} \cdot U$ , and $|f^{(2)}|\ge \frac{T}{B^2M^2},T\gg B$ for some constants $M\ge b-a,B\ge 1,T,N,U>0$, one has $$\int_a^b g(x)e(f(x))d x=\frac{g(x_0)e(f(x_0)+1/8)}{\sqrt{f^{(2)}(x_0)}}+\frac{g(b)e(f(b))}{2\pi i f^{(1)}(b)}-\frac{g(a)e(f(a))}{2\pi i f^{(1)}(a)}+O[\frac{B^4M^4U}{T^2}(\frac{1}{(x_0-a)^3}+\frac{1}{(b-x_0)^3})]+O[\frac{B^{13}MU}{T^{3/2}}(1+\frac{M}{B^4N})^2],$$where $f^{(1)}(x_0)=0,a<x_0<b.$
So if somebody has this book, could you tell me something about the exact formula in my question? Much obliged.
Great thanks if you can send the electric version to my emailbox [email protected].
