Bounds on integral

Question

I am calculating Fourier coefficients for certain functions and have come across an integral of the form $$I=\int_0^{2\pi} \int_0^1 r^2e^{2\pi i r(m\cos\theta+n\sin\theta)}drd\theta,$$ where $m,n\in\mathbb{Z}$. This can be simplified to $$I=\int_0^{2\pi} \frac{-i +e^{2\pi i (m\cos\theta+n\sin\theta)}(i+2\pi(m\cos\theta+n\sin\theta)-2\pi^2 i (m\cos\theta+n\sin\theta)^2) }{4\pi^3 (m\cos\theta+n\sin\theta)^3}d\theta.$$ I am not interested in actually calculating the integral, but instead bounding $|I|$ as a function of $m,n\in\mathbb{Z}$. I am having difficulty locating good bounds in the literature. Does anyone have suggestions for where to look, or what bounds there may possibly be?

Answer 1

Write $m^2+n^2=a^2$ and note that $m\cos\theta+n\sin\theta=a\cos(\theta-\theta_a)$ for some fixed $\theta_a$ hence $$ I=\int_0^{2\pi} \int_0^1 r^2\mathrm e^{2\pi \mathrm i ar\cos\theta}\mathrm dr\mathrm d\theta. $$ Expand the exponential as a power series in $2\pi \mathrm i ar\cos\theta$, note that the odd powers of the cosine integrate to zero and integrate each remaining power of $r$. One gets $$ I=\sum_{k\geqslant0}(-1)^k\frac{(2\pi a)^{2k}}{(2k)!}\frac{4W_{2k}}{2k+3}, $$ where each $W_{2k}$ is a Wallis integral. Plugging in the value of each $W_{2k}$ yields $$ I=2\pi\sum_{k\geqslant0}(-1)^k\frac{(\pi a)^{2k}}{(k!)^2}\frac1{2k+3}, $$ that is, $(\pi a)^3I=2\pi G(\pi a)$ with $$ G(x)=\sum_{k\geqslant0}(-1)^k\frac{x^{2k+3}}{(k!)^2}\frac1{2k+3}. $$ Thus, $$ G'(x)=\sum_{k\geqslant0}(-1)^k\frac{x^{2k+2}}{(k!)^2}=x^2J_0(2x), $$ where $J_0$ is a Bessel function of the first kind. The asymptotic upper bound $|J_0(2x)|\leqslant1/\sqrt{\pi x}$ when $x\to+\infty$ yields $|G'(x)|\leqslant\sqrt{x^3/\pi}$ asymptotically, hence $|G(x)|\leqslant(2/5)\sqrt{x^5/\pi}$ asymptotically. Finally, when $a\to+\infty$, $$ |I|\stackrel{\mathrm{approx.}}{\leqslant}\frac4{5\sqrt{a}}=\frac45\frac1{\sqrt[4]{m^2+n^2}}. $$