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Euler’s amazing identity The mathematician Leonard Euler developed some surprising mathematical formulas involving the number ##\pi##. The most famous equation is ##e^{i \pi} = -1##, which is one of the most important equations in modern mathematics, but unfortunately, it wasn’t invented by Euler.Something that is original with Euler is this amazing identity: Equation 1: ##1…
Our destination In our last article, Hyperbolic Trajectories, we derived some facts about the trajectory of a rocket that is undergoing constant (proper) acceleration. In this article, we will explore what these facts mean for the passengers on board a rocket undergoing Rindler motion. The goal is to come up with a coordinate system that…
Introduction: Why Rindler Motion? When students learn relativity, it’s usually taught using inertial (constant velocity) motion. There are lots of reasons for this, but mainly it’s because it’s the easiest kind of motion for deriving the results of relativity, and historically, thinking about inertial motion is what led to Einstein’s theory. An unfortunate side-effect of…
Introduction In introductory physics, the optics unit often teaches about virtual and real images, focal lengths, indexes of refraction, etc. Some questions that are sometimes glossed over in the rush to present the mathematical formulas and definitions are: What does it mean for an image to form at a particular location? What does it mean…
This little trick is used for some integration problems involving trigonometric functions is probably well-known, but I only learned it yesterday. So under the assumption that others don’t know it, I will share my insight. When you take calculus, one of the most mysterious integrals is this one involving the secant: ∫ sec(θ) dθ = log(sec(θ)…
This is a little note about quantum amplitudes. Even though quantum probabilities seem very mysterious, with weird interference effects and seemingly nonlocal effects, the mathematics of quantum amplitudes is completely straightforward. (The amplitude squared gives the probability.) As a matter of fact, the rules for computing amplitudes are almost the same as the classical rules…
Everybody learns the “quadratic formula” for solving equations of the form [itex]A x^2 + B x + C = 0[/itex], even though you don’t really need such a formula, because you can solve for [itex]x[/itex] through the technique of “completing the square”. What you need a formula for is the solution to the cubic equation:…