### 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 […]

### 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 […]

### 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 […]