Evaluating the limit of a certain definite integral

Solution 1:

Consider what happens on an interval $[n\pi-\frac\pi2,n\pi+\frac\pi2]$. Let $x=\tan(z)$ and $u=x/\sqrt{\epsilon}$, then $$ \begin{align} \lim_{\epsilon\to0}\frac1{\sqrt{\epsilon}}\int_{n\pi-\frac\pi2}^{n\pi+\frac\pi2}e^{-\tan^2(z)/\epsilon}\,\mathrm{d}z &=\lim_{\epsilon\to0}\frac1{\sqrt{\epsilon}}\int_{-\infty}^\infty e^{-x^2/\epsilon}\frac{\mathrm{d}x}{1+x^2}\\ &=\lim_{\epsilon\to0}\int_{-\infty}^\infty e^{-u^2}\frac{\mathrm{d}u}{1+\epsilon u^2}\\ &=\int_{-\infty}^\infty e^{-u^2}\,\mathrm{d}u\\[6pt] &=\sqrt\pi\tag{1} \end{align} $$ For all $\epsilon\gt0$, we have $$ \frac1{\sqrt\epsilon}e^{-\tan^2(z)/\epsilon}\le\frac{|\cot(z)|}{\sqrt{2e}}\tag{2} $$ Thus, for any $\lambda\gt0$, Dominated Convergence says $$ \lim_{\epsilon\to0}\frac1{\sqrt{\epsilon}}\int_{n\pi-\frac\pi2}^{n\pi+\frac\pi2}\big[|z-n\pi|\ge\lambda\big]e^{-\tan^2(z)/\epsilon}\,\mathrm{d}z=0\tag{3} $$ where $[\cdot]$ are Iverson Brackets. Combining $(1)$ and $(3)$ gives $$ \lim_{\epsilon\to0}\frac1{\sqrt{\epsilon}}\int_{n\pi-\frac\pi2}^{n\pi+\frac\pi2}\big[|z-n\pi|\lt\lambda\big]e^{-\tan^2(z)/\epsilon}\,\mathrm{d}z=\sqrt\pi\tag{4} $$ Limits $(3)$ and $(4)$ tell us that $\frac1{\sqrt\epsilon}e^{-\tan^2(z)/\epsilon}$ is an approximation of $$ \sqrt\pi\sum_{n\in\mathbb{Z}}\delta(z-n\pi)\tag{5} $$ where $\delta(z)$ is the Dirac delta function.

Thus, $$ \bbox[5px,border:2px solid #C0A000]{\lim_{\epsilon\to0}\frac1{\sqrt{\epsilon}}\int_0^xz\,e^{-\tan^2(z)/\epsilon}\,\mathrm{d}z =\left\{\begin{array}{} \displaystyle\pi^{3/2}\,\frac{\lfloor x/\pi\rfloor^2+\lfloor x/\pi\rfloor}2&\text{if }x\not\in\pi\mathbb{Z}\\ \displaystyle\pi^{-1/2}\,\frac{x^2}2&\text{if }x\in\pi\mathbb{Z} \end{array}\right.}\tag{6} $$ The plot would look something like

Solution 2:

NOTE:

I wanted to give a special thanks to @robjon for his insightful comments.

We first observe that $\lim_{\epsilon\to 0}e^{-\tan z/\epsilon}=0$ unless $z=\ell \pi$, $\ell$ an integer. Therefore, all of the "action" of the integration will take place over intervals around $\ell \pi$. So, let's first see what is happening for $0<x<\pi/2$.

In the spirit of Laplace's Method, we have for $0<z<\pi/2$, $\tan^z =z^2+O(z^4)$ and thus for $0<x<\pi/2$

$$\begin{align} \epsilon^{-1/2}\int_0^xze^{-\tan^2z/\epsilon}dz&\sim\epsilon^{-1/2}\int_0^xze^{-z^2/\epsilon}dz\\\\ &=\epsilon^{-1/2}\left.\left(-\epsilon^{-z^2/\epsilon}\right)\right|_{z=0}^{z=x}\\\\ &=\epsilon^{1/2}\left(1-e^{-x^2/\epsilon}\right) \end{align}$$

which clearly goes to zero as $\epsilon\to 0$.

Next, we observe that the integration around singularities of the tangent function pose no challenge. Thus, for a general $(L-1)\pi<x<L\pi$, and $\delta >0$ we can write

$$\begin{align} \epsilon^{-1/2}\int_0^x ze^{-\tan^2z/\epsilon}dz&=\epsilon^{-1/2}\sum_{\ell=0}^{L-2}\left(\int_{\ell \pi+\delta}^{(\ell+1)\pi-\delta}ze^{-\tan^2z/\epsilon}dz+\int_{(\ell+1)\pi-\delta}^{(\ell+1)\pi+\delta}ze^{-\tan^2z/\epsilon}dz\right)\\\\ &+\epsilon^{-1/2}\int_{(L-1)\pi+\delta}^{x}ze^{-\tan^2z/\epsilon}dz \tag 1\\\\ \end{align}$$

We observe that in $(1)$ the only integrals that will contribute in the limit as $\epsilon \to 0$ are those around integer multiples of $\pi$. Thus, we have for $(L-1)\pi<x<L\pi$ and $\delta>0$

$$\begin{align} \lim_{\epsilon \to 0}\epsilon^{-1/2}\int_0^x ze^{-\tan^2z/\epsilon}dz&=\lim_{\epsilon \to 0} \epsilon^{-1/2}\sum_{\ell=0}^{L-2}\left(\int_{(\ell+1)\pi-\delta}^{(\ell+1)\pi+\delta}ze^{-\tan^2z/\epsilon}dz\right) \tag 2\\\\ \end{align}$$

We proceed to evaluate the integrals in $(2)$. To that end we have

$$\begin{align} \epsilon^{-1/2}\int_{(\ell+1)\pi-\delta}^{(\ell+1)\pi+\delta}ze^{-\tan^2z/\epsilon}dz &=\epsilon^{-1/2}\left(\int_{-\delta}^{\delta}ze^{-\tan^2z/\epsilon}dz+(\ell +1)\pi\int_{-\delta}^{\delta}e^{-\tan^2z/\epsilon}dz\right)\\\\ &=(\ell +1)\pi\epsilon^{-1/2}\int_{-\delta}^{\delta}e^{-\tan^2z/\epsilon}dz\\\\ &\sim (\ell +1)\pi\epsilon^{-1/2}\int_{-\delta}^{\delta}e^{-z^2/\epsilon}dz\\\\ &= (\ell +1)\pi\int_{-\delta/\epsilon^{1/2}}^{\delta/\epsilon^{1/2}}e^{-z^2}dz\\\\ &\to (\ell +1)\pi^{3/2} \end{align}$$

Summing over $\ell$ we find for $(L-1)\pi<x<L\pi$

$$\lim_{\epsilon \to 0}\epsilon^{-1/2}\int_0^xze^{-\tan^2z/\epsilon}dz=\frac{L(L-1)\pi^{3/2}}{2}$$

One final note concerns the case in which $x=L\pi$. For that case, we see that we need to add one more integral, namely

$$\begin{align} \lim_{\epsilon\to 0}\epsilon^{-1/2}\int_{L\pi-\delta}^{L\pi}ze^{-\tan^z/\epsilon}&=L\pi\int_{-\infty}^0e^{-z^2}dz\\\\ &=\frac12 L\pi^{3/2} \end{align}$$

Thus, for $x=L\pi$ we have

$$\lim_{\epsilon \to 0}\epsilon^{-1/2}\int_0^xze^{-\tan^2z/\epsilon}dz=\frac{L^2\pi^{3/2}}{2}$$

Putting it all together we have

$$\lim_{\epsilon \to 0}\epsilon^{-1/2}\int_0^xze^{-\tan^2z/\epsilon}dz= \begin{cases} \frac{L(L-1)\pi^{3/2}}{2},&(L-1)\pi<x<L\pi\\\\ \frac{L^2\pi^{3/2}}{2},&x=L\pi \end{cases} $$