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I have the following function:


where $\operatorname{sech}(x)=\frac{2}{e^x+e^{-x}}$ is the hyperbolic secant.

Clearly, the integral is a beast to evaluate. However, all I need is to prove the continuity of $I_n(a)$ at $n=2,4,6$; I don't need the closed form of $I_n(a)$. In particular, I am interested in the continuity of $I_n(a)$ in the neighborhood of $a$ around zero, i.e. $a\in[-\epsilon,\epsilon]$ for some small positive $\epsilon$ (I plotted it in that region and it "looks" continuous to me, however, I'd like a rigorous proof).

This seems like a simple problem, but I have no idea where to start. Someone suggested the Lebesgue Dominated Convergence Theorem, but unfortunately, I am shaky on the measure theory, having never taken a course on the subject. I read the wikipedia page, but have no idea how to use. Is there perhaps a simpler way? If not, can someone help?

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The book "Undergraduate Analysis" by Serge Lang has a chapter dedicated to problems like this. It is a kind of tool-box for questions regarding convergence of integrals, and it uses no measure theory. You can try having a look. For example in the present case I guess that you should prove that the integral is absolutely convergent uniformly with respect to $a$, which I think is true. –  Giuseppe Negro Feb 6 '13 at 1:33
(My previous suggestion is the integral version of a well-known technique in series of functions. Namely, a uniformly convergent series of continuous functions is continuous. The same applies to improper Riemann integrals, with essentially the same proof. In the present case, the integrand function depends continuously on $a$). –  Giuseppe Negro Feb 6 '13 at 1:36
Looks like my library owns this book and it also looks like it's designed for someone like me (the book description states that this book is "suitable for students who have had two years of calculus" which is the amount of calculus that I had). I'll retrieve it from the library tomorrow (it's closed right now) and will post here what I figure out. Thanks! :) –  M.B.M. Feb 6 '13 at 1:45
You're welcome, I hope this helps. IMHO that's a great book. The chapter I was referring to is XIII. I think that §3 contains all the techniques you need. –  Giuseppe Negro Feb 6 '13 at 2:27
I got the book and it's indeed very good -- I think it'll help me in the other problems. However, someone explained the Dominated Convergence Theorem to me, which I think yields a very simple solution to the problem (see my answer -- I hope it's correct). –  M.B.M. Feb 7 '13 at 18:08
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up vote 1 down vote accepted

The Dominated Convergence Theorem has been explained to me, and I think I can use it as follows to prove the continuity of $I_n(a)$.

Let $g(x)=x^6e^{-x^2}$. Clearly $g(x)$ is integrable, as $\int_{-\infty}^{\infty}|g(x)|dx$ is the sixth moment of the Gaussian distribution times a constant. Since $0<\operatorname{sech}(x)\leq1$, the integrand is dominated by $g(x)$, i.e. $|x^6e^{-x^2}\operatorname{sech}^n(ax)|\leq g(x)$.

To prove continuity, we can show that $\lim_{a\rightarrow a_0} I_n(a)=I_n(a_0)$ for the region $a_0\in[-\epsilon,\epsilon]$ in which we are interested. We can show this as follows:

$$\begin{array}{rcl}\lim_{a\rightarrow a_0}I_n(a)&=&\lim_{a\rightarrow a_0}\int_{-\infty}^{\infty}x^6e^{-x^2}\operatorname{sech}^n(ax)dx\\ &=&\int_{-\infty}^{\infty}x^6e^{-x^2}\lim_{a\rightarrow a_0}\operatorname{sech}^n(ax)dx\\ &=&\int_{-\infty}^{\infty}x^6e^{-x^2}\operatorname{sech}^n(a_0x)dx\\ &=&I_n(a_0) \end{array}$$

where the movement of limit inside the integral in the second equality is allowed since we have shown that the conditions for the Dominated Convergence Theorem hold. Thus, we proved that $I_n(a)$ is continuous on the entire real number domain for $n\geq 1$.

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Ok! This is indeed the "standard" solution. –  Giuseppe Negro Feb 7 '13 at 19:06
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