Interesting way to compute $\pi$ and its consequences

A poster on SE wondered how he could prove the following: $$\sum_{j=2}^\infty \prod_{k=1}^j \frac{2 k}{j+k-1} = \pi$$ Well! I never would have thought of this on my own, but I’m not really here to think up stuff like this on my own. Rather, I am here to solve the problems of confused souls, and here […]

Cosine transform of sinc cubed

Some dude asked to evaluate the following integral: $$\int_0^{\infty} dt \, \frac{\sin^3{\pi t}}{(\pi t)^3} \cos{u t}$$ I propose to perform a direct computation using Cauchy’s theorem, i.e., extension into the complex plane. I will then verify the solution using the convolution theorem. DIRECT EVALUATION Rewrite the integral as $$\frac12 \int_{-\infty}^{\infty} dt \, \frac{\sin^3{\pi t}}{(\pi t)^3} […]

An integral that is much easier than it looks

This is one of those cases where some poor, inexperienced user puts up a tough-looking integral. In Math.SE, there is a large contingent that jumps on these and gets quite upset if the consensus is that it can’t be done. Thus, when such a user posted this integral for evaluation: $$I(a,b)=\int_0^1 dt \, t^{-3/2}(1-t)^{-1/2}\exp\left(-\frac{a^2}{t}-\frac{b^2}{1-t} \right)$$ […]

Integral with hyperbolic functions

The problem here is to evaluate the following integral: $$\displaystyle \int_{0}^{\infty} dx \frac{\cosh (ax) \cosh (bx)}{\cosh (\pi x)} $$ such that $|a|+|b| < \pi$. This one can be done a number of ways, including complex analysis. Atypical for me, I chose a different way, one which led to a surprising result. It should be noted […]

Radical simplification of Gamma function expression

This one popped up and I have no idea what motivated my idea that there was, in fact, a way to radically simplify the god-awful Gamma functions. But, well…you’ll see. The problem asks whether the expression $$\frac{\Gamma\left(\frac{1}{10}\right)}{\Gamma\left(\frac{2}{15}\right)\Gamma\left(\frac{7}{15}\right)}$$ has any chance of being simplified somehow. Recall that a gamma function is like a factorial, but defined […]

Unusual contour to evaluate a real integral

A poster asked how to evaluate the following integral $$\int_0^{\infty} dx \frac{\sin{a x}}{e^{2 \pi x}-1}$$ by extension into the complex plane and applying Cauchy’s theorem. Specifically, construct a rectangle $0\to R\to R+i\to i \to 0$ and integrate round it. For this contour, one must indent about the poles at $z=0$ and $z=1$. In that case, […]

Solving the heat equation using a Laplace transform

Someone posed the following on M.SE: Consider the heat equation on the half line $$u_t = ku_{xx},\quad x > 0,\, t > 0,\\ u(x,0) = 0, \,x \in\mathbb{R},\\ u(0,t) = \alpha(t),\, t > 0. $$ This is a problem illustrated in Fetter and Walecka, but I will illustrate my own solution below. This is derived […]

Closed form of a sum

Determine the closed form for $$\sum_{n=1}^{\infty} \frac{1}{2^n \left ( 1+ \sqrt[2^n]{2}\right)}$$ There were many high-rep users on M.SE asking how one could even assume such a closed-form exists. A quick glance at this shows that the sum should be bounded from above by $1/2$. Otherwise, anything goes, right? Actually, this is a problem that is […]

Focus of a ball lens

We had a question involving how a simple equation in geometrical optics is derived. Although very simple for experienced Optics people like myself, it is very difficult for those not schooled in the geometry of refraction to put together. In *Optical Design Fundamentals for Infrared Systems 2nd ed.*, Mr. Riedl writes: A sphere or ball […]

Integrals with Log, Part V

Evaluate $$\int_0^1 dx\, \frac{\ln(1+x^2)}{1+x^2}$$ This innocent-looking integral is actually kind of a bear. The way I attack it is to express the integral in terms of two separate integrals: one that may be evaluated by extension into the complex plane, and a familiar one easily evaluated. The integral in the complex plane will have its […]

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 […]