Monthly Archives: May 2013

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 […]