Monthly Archives: April 2013

Asymptotic Expansion of an integral

Problem: find the asymptotic expansion of the following integral: $$I_n=\int_0^1\exp(x^n)dx$$ as $n \to \infty$. This expansion, while at first pedestrian-seeming, turns out to have a very interesting set of coefficients. You can expand the exponential in a Taylor series quite accurately: $$\exp{\left ( x^n \right )} = 1 + x^n + \frac12 x^{2 n} + […]

Inverse Laplace Transform IV

$$\hat{f}(s)=\frac{e^{-a\sqrt {s(s+r_0)}}}{\sqrt {s(s+r_0)}}$$ This one is interesting because of the presence of two branch points: one at $s=0$ and the other at $s=-r_0$. We will consider the complex integral $$\oint_C ds \frac{e^{-a \sqrt{s (s+r_0)}}}{\sqrt{s (s+r_0)}} e^{s t}$$ where $C$ is the following contour pictured below: This one’s a bit odd because we are removing the […]

Volume of three cylinders

This one is not a M.SE question that I answered, but rather a response to a challenge from another M.SE user. The problem is to find the volume of three orthogonal, intersecting cylinders: $$\begin{align}x^2+y^2&=1\\x^2+z^2&=1\\ y^2+z^2&=1\end{align}$$ The intersection region is pictured below: Problems like these are notoriously difficult because they are difficult to visualize properly. I […]

A sum involving hyperbolic functions II

Another sum took the form $$\sum_{n=1}^{\infty}\frac{\sinh\pi}{\cosh(2n\pi)-\cosh\pi}$$ For some reason, I couldn’t get the residue theorem to help me here. I took a different apporach instead. I begin with the following result (+): $$\sum_{k=1}^{\infty} e^{-k t} \sin{k x} = \frac{1}{2} \frac{\sin{x}}{\cosh{t}-\cos{x}}$$ I will prove this result below; it is a simple geometrical sum. In any case, […]

A sum involving hyperbolic functions I

We had a user who habitually posted difficult sums and integrals. This annoyed many of the more experienced users because he showed little interest in anything other than seeing someone else write a complete solution. I didn’t mind – these were nifty and challenging. One problem is an evaluation of a sum: $$\sum_{n=1}^{\infty}\frac{\cosh(2nx)}{\cosh(4nx)-\cosh(2x)}=\frac1{4\sinh^2(x)}$$ We use […]

Integrals with Log, Part IV

The problem, as originally stated, was to prove that $$\int_0^{\infty} dx \frac{x}{x^2+a^2}\log{\left(\frac{x+1}{x-1}\right)} = \pi \arctan{\frac{1}{a}}$$ This is a challenging integral with unexpected twists and turns in its evaluation. Ultimately, though, it all works out, albeit not the way the OP expected: there is a nonzero imaginary part, which is why he put the argument of […]

An integral over a nested function

This was a case study in a brilliant problem posted by a very inexperienced M.SE user. There were 2 answers offered to this problem: mine and another. The other was very wrong, its wrongness blindingly so; yet, the user chose to accept this wrong answer. This act of course in M.SE has the effect of […]

Integrals with Log, Part III

This one is an example of teamwork across the M.SE user base. I came up with a substitution and realization into a nifty sum, but got a bit stuck. Another power user, known as @marvis at the time, took over and evaluated the sum to get the correct result. This one was an extremely difficult […]

Integrals with Log, Part II

Evaluate $$\int_0^1 dx\: \frac{1+x}{1-x^3} \ln\left(\frac{1}{x}\right)$$ This one is relatively simple and wouldn’t normally make it to this blog, but the result is easily generalizable to a very cool, highly nontrivial result. To start, make a substitution $x=e^{-y}$: $$\begin{align}\int_0^1 dx\: \frac{1+x}{1-x^3} \ln\left(\frac{1}{x}\right) &= \int_0^{\infty} dy \: y \,e^{-y} \frac{1+e^{-y}}{1-e^{-3 y}}\\ &= \int_0^{\infty} dy \: y (e^{-y}+e^{-2 […]

Integrals with Log, Part I

Someone posted this gorgeous integral to see if there was a closed form for the following. $$\int_0^{\pi/3} dx \: \ln^2\left(\frac{\sin x }{\sin (x+\pi/3)}\right)$$ It turns out that this integral takes on a very simple form amenable to analysis via residues. Let $u = \sin{x}/\sin{(x+\pi/3)}$. We may then find that (+) $$\tan{x} = \frac{(\sqrt{3}/2)u}{1-(u/2)}$$ A little […]

Inverse Laplace Transform, Part III

We now try to attack the following inverse LT: $$ \frac{\exp\left(\frac{x}{2}\sqrt{(U/D)^2+4s/D}\right)}{s\sqrt{(U/D)^2+4s/D}}$$ I could not find any tables that had this paired with its ILT. It is also strange in that the branch point of the function is not at zero, as it was in the other 2 cases. Nevertheless, as will be seen below, the […]

Inverse Laplace Transform, Part II

It is well-known that the function $f(s)=s^{-1/2}$ is invariant to Laplace and Fourier transforms (to within a scale factor). Proving that this is the case, at least for Laplace transforms, is far from trivial. I outline the computation below; the previous computation will serve as a guide. This integral may be attacked with the residue […]

Inverse Laplace transform, Part I

There are a lot of people interested in inverting Laplace transforms that have branch cuts in the complex plane. Some of these are known; at least one that I have done I could not find anywhere and looks like an original computation as far as I can tell. In any case, let’s start with one […]

An explanation

For the past 4 months, I have immersed myself in the Math.stackexchange community.  In those 4 moths, I have rediscovered a lot of my lost mathematics and have met some very interesting, like minded people.  Stack Exchange is a question-and-answer community site, and has an incredible array of tools for archiving knowledge and visualizing performance […]