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Investigating Some Euler Sums

So, why only odd powers? Mostly because the even powers were solved by Leonard Euler in the 18th century. Since the “mathematical toolbox” at that time did not contain the required tools, he needed 6 years to prove the validity of his deductions. Now, however, we have much more powerful tools available, as I have…

10 Math Things We All Learnt Wrong At School

The title is admittedly clickbait. Or a joke. Or a provocation. It depends on with whom you speak, or who reads it with which expectation. Well, I cannot influence any of that. I can only tell how I mean it, namely as an entertaining collection of simple truths which later on turn out to be…

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…

How Bayesian Inference Works in the Context of Science

Confessions of a moderate Bayesian part 3 Read part 1: How to Get Started with Bayesian Statistics Read part 2: Frequentist Probability vs Bayesian Probability Bayesian statistics by and for non-statisticians https://www.cafepress.com/physicsforums.13280237 Background One of the things that I like about Bayesian statistics is that it rather closely matches the way that I think about…

Exploring Frequentist Probability vs Bayesian Probability

Confessions of a moderate Bayesian, part 2 Read Part 1: Confessions of a moderate Bayesian, part 1 Bayesian statistics by and for non-statisticians https://www.cafepress.com/physicsforums.13280237 Background One of the continuous and occasionally contentious debates surrounding Bayesian statistics is the interpretation of probability. For anyone who is familiar with my posts on this forum, I am not…

How to Get Started with Bayesian Statistics

Confessions of a moderate Bayesian, part 1 Bayesian statistics by and for non-statisticians https://www.cafepress.com/physicsforums.13265286 Background I am a statistics enthusiast, although I am not a statistician. My training is in biomedical engineering, and I have been heavily involved in the research and development of novel medical imaging technologies for the bulk of my career. Due…

How to Solve Second-Order Partial Derivatives

Introduction A frequent concern among students is how to carry out higher order partial derivatives where a change of variables and the chain rule are involved.  There is often uncertainty about exactly what the “rules” are.  This tutorial aims to clarify how the higher-order partial derivatives are formed in this case. Note that in general…

The Analytic Continuation of the Lerch and the Zeta Functions

Introduction In this brief Insight article the analytic continuations of the Lerch Transcendent and Riemann Zeta Functions are achieved via the Euler’s Series Transformation (zeta) and a generalization thereof, the E-process (Lerch). Dirichlet Series is mentioned as a steppingstone. The continuations are given but not shown to be convergent by any means, though if you…

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