## Monthly Archives: April 2013

### 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 […]