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Computing the Riemann Zeta Function Using Fourier Series

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…

Rindler Motion in Special Relativity: Hyperbolic Trajectories

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…

Lenses and Pinholes: What Does “In Focus” Mean?

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 for…

Learn About Quantum Amplitudes, Probabilities and EPR

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…