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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

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The method of stationary phase is covered in many standard books: e.g. De Bruijn's Asymptotic methods in analysis; Titchmarsh's Theory of the Riemann zeta function; Stein's books on harmonic analysis; Montgomery's 10 lectures on harmonic analysis and analytic number theory; Graham and Kolesnik's book on exponential sums ... . I hope you can find one of these books in your library. – Lucia Nov 17 '13 at 20:52
@Lucia Thanks. I was informed by Brad Rodgers the copy can be downloaded in a website ( Much obliged for timely reply, good day. – H.Flip Nov 18 '13 at 3:52

You can download this book here

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