### Reduction of Order For Recursions

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This is not meant as a full introduction to recursion relations but it should suffice for just about any level of the student.Most of us remember recursion…

### A Novel Technique of Calculating Unit Hypercube Integrals

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Introduction
In this insight article, we will build all the machinery necessary to evaluate unit hypercube integrals by a novel technique. We will first…

### The Extended Riemann Hypothesis and Ramanujan’s Sum

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Riemann Hypothesis and Ramanujan's Sum ExplanationRH: All non-trivial zeros of the Riemannian zeta-function lie on the critical line.
ERH: All…

### A Trick to Memorizing Trig Special Angle Values Table

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In calculus classes when you are asked to evaluate a trig function at a specific angle, it's 99.9% of the time at one of the so-called special angles we…

### An Introduction to Theorema Primum

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Introduction
Whilst no doubt most frequenters of "Physics Forums" will be familiar with Nicolaus Copernicus as the scientist who advanced the (at the…

### Python’s Sympy Module and the Cayley-Hamilton Theorem

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Two of my favorite areas of study are linear algebra and computer programming. In this article I combine these areas by using Python to confirm that a…

### Physical Applications of the “Tan Rule”

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Introduction
Every secondary school student who has encountered trigonometry in his/her Math syllabus will most likely have come across the sine, cosine,…

### Investigating Some Euler Sums

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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…

### Computing the Riemann Zeta Function Using Fourier Series

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Euler's amazing identity
The mathematician Leonard Euler developed some surprising mathematical formulas involving the number ##\pi##. The most famous…

### Geometry of Side-side-angle (SSA) and Impossible Triangles

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What is the ambiguous case?
In high school geometry, the idea of proofs is often first introduced to American students. A common task is to use a basic…

### How to Solve Second-Order Partial Derivatives

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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…

### The Analytic Continuation of the Lerch and the Zeta Functions

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Introduction
In this brief Insight article the analytic continuations of the Lerch Transcendent and Riemann Zeta Functions are achieved via the Euler's…

### A Path to Fractional Integral Representations of Some Special Functions

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Introduction
This bit is what new thing you can learn reading this:) As for original content, I only have hope that the method of using the sets
$$C_N^n:…

### Explore the Fascinating Sums of Odd Powers of 1/n

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The goal is to get a little bit closer to the values of the zeta function (ζ(s)) and the eta function (η(s)) for some odd values of s. This insight is…

### SOHCAHTOA: Seemingly Simple, Conceivably Complex

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Preface
My first experience with derivatives was seeing how they are obtained from the usual definition
$$f'(x)=\underset{\text{$\Delta $x}\to 0}{\text{Lim}}\frac{f…

### Demystifying Parameterization and Surface Integrals

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Introduction
This article will attempt to take the mystery out of setting up surface integrals. It will explain the basic ideas underlying surface integration…

### Demystifying the Chain Rule in Calculus

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Introduction
There are a number of posts on PF involving a general confusion over the multi-variable chain rule. The problem is often caused by…

### How to Make Units of Measurement Work for You

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How do we use units? You may see one of these speed limit signs, nearly every day. Even though neither of them displays units, drivers know they are implied.…

### A Journey to The Manifold SU(2): Representations

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Part 1
Representations
Image source: [24]
6. Some useful bases of ##\mathfrak{su}(2,\mathbb{C})##
Notations can differ from author…

### Learn Further Sums Found Through Fourier Series

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In an earlier insight, I looked at the Fourier series for some simple polynomials and what we could deduce from those series. There is a lot more to be…

### The Pantheon of Derivatives – Part V

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Important Theorems - biased, of course
Implicit Function Theorem [1]
Jacobi Matrix (Chain Rule).
Let ## (x_0,y_0 ) ## be a point in$$U_1…

### The Pantheon of Derivatives – Part IV

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Lie Derivatives
A Lie derivative is in general the differentiation of a tensor field along a vector field. This allows several applications…

### The Pantheon of Derivatives – Part III

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Some Topology
Whereas the terminology of vector fields, trajectories, and flows almost by itself suggests its origins and physical relevance,…

### The Pantheon of Derivatives – Part II

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Generalizations Beyond ##\mathbb{R}## and ##\mathbb{C}##
As mentioned in the section on complex functions (The Pantheon of Derivatives - Part…

### The Pantheon of Derivatives – 5 Part Series

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Differentiation in a Nutshell
I want to gather the various concepts in one place, to reveal the similarities between them, as they are often…

### Trick to Solving Integrals Involving Tangent and Secant

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This little trick is used for some integration problems involving trigonometric functions is probably well-known, but I only learned it yesterday. So…

### Using the Fourier Series To Find Some Interesting Sums

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Preliminaries
If f(x) is periodic with period 2p and f’(x) exists and is finite for -π<x<π, then f can be written as a Fourier series:
[itex]f(x)=\sum_{n=-\infty}^{\infty}a_{n}e^{inx}…

### Learn Partial Differentiation Without Tears

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Differentiation is usually taught quite well. Perhaps that's because it is the first introduction to calculus, which is considered a big step in a student's…

### Learn How to Solve the Cubic Equation for Dummies

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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,…

### Omissions in Mathematics Education: Gauge Integration

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The current (pure) mathematics curriculum at the university is well established. Most of the choices made are sensible. But still, there are some important…

### Explaining the General Brachistochrone Problem

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Consider a problem about the curve of fastest descent in the following generalized statement. Suppose that we have a Lagrangian system $$L(x,\dot x)=\frac{1}{2}g_{ij}(x)\dot…

### Some Misconceptions on Indefinite Integrals

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Integration is an incredibly useful technique taught in all calculus classes. Nevertheless, there are certain paradoxes involved with integration that…

### Mathematical Irrationality for Dummies

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On one of my restless wanderings around the Internet, I came upon a collection of proofs of the statement that the square root of 2 is an irrational…

### Learn Complex and Irrational Exponents for the Layman

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This Insight is part of our "Young Authors" series, where talented young students showcase their knowledge.
This is how both you and I learned…

### What Are Eigenvectors and Eigenvalues in Math?

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Two important concepts in Linear Algebra are eigenvectors and eigenvalues for a linear transformation that is represented by a square matrix. Besides…

### Introduction to Partial Fractions Decomposition

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Partial fractions decomposition is an algebraic technique that can be used to decompose (break down) a product of rational expressions into a sum…

### How to Solve Nonhomogeneous Linear ODEs using Annihilators

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My previous Insights article, Solving Homogeneous Linear ODEs using Annihilators, discussed several examples of homogeneous differential equations, equations…

### Solving Homogeneous Linear ODEs using Annihilators

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In this Insights article we'll look at a limited class of ordinary differential equations -- homogeneous linear ODES with constant coefficients. Although…

### Learn About Matrix Representations of Linear Transformations

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Let X and Y be finite-dimensional vector spaces. Let ##T:X\to Y## be a linear transformation. Let ##A=(e_1,\dots,e_n)## and ##B=(f_1,\dots,f_m)## be ordered…

### Informal Introduction to Cardinal Numbers

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Cardinal numbers
We will now give an informal introduction to cardinal numbers. We will later formalize this by using ordinal numbers. Informally, cardinal…

### Is There a Rigorous Proof Of 1 = 0.999…?

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Yes.First, we have not addressed what 0.999... actually means. So it's best first to describe what on earth the notation [tex]b_0.b_1b_2b_3...[/tex]…

### Errors in Probability: Continuous and Discrete Distributions

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1. Classifying as discrete, continuous, or mixed
These statements (or equivalents) can be found in authoritative-seeming websites:X "A random variable…

### Frequently Made Errors in Probability: Conditionals in Natural Language

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1. Turning a verbal condition into Algebra
An actual thread..."A study of auto accidents has found that 40% of all fatal accidents are…

### Some Conceptual Difficulties in the Roles of Variables and Constants

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1. Variables and Constants
When is a constant not a constant? When it varies.In the standard equation ##y=ax+b##, we are used to thinking of x and…

### A Continuous, Nowhere Differentiable Function: Part 2

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This is Part 2 of a series of articles in which the goal is to exhibit a continuous function that is nowhere differentiable and to explore some interesting…

### A Continuous, Nowhere Differentiable Function: Part 1

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When studying calculus, we learn that every differentiable function is continuous, but a continuous function need not be differentiable at every point.…