### Posts

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

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The Analytic Continuation of the Lerch Transcendent and the Riemann Zeta FunctionIntroduction
In this brief Insight article the analytic continuations…

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

0. Introduction
As for the reference material I have used the text Special Functions by Askey, Andrews, and Roy which covers much of the theorems here…

### SOHCAHTOA: Seemingly Simple, Conceivably Complex

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…

### How to Find Potential Functions? A 10 Minute Introduction

Definition/Summary
Given a vector field ##\vec F(x,y,z)## that has a potential function, how do you find it?
Equations
$$\nabla \phi(x,y,z) = \vec F(x,y,z)$$…

### What is a Linear Equation? A 5 Minute Introduction

Definition/Summary
A first order polynomial equation in one variable, its general form is [itex]Mx+B=0[/itex] where x is the variable. The quantities…

### What are Significant Figures? A 5 Minute Introduction

Definition/Summary
Significant figures (commonly called "sig figs") are the number of figures (digits) included when rounding-off a number.For example,…

### How Most Proofs Are Structured and How to Write Them

... or the answer to: "I have no idea where to start!"
Proofs in mathematics are what mathematics is all about. They are subject to entire books, created…

### What is a Fibre Bundle? A 5 Minute Introduction

Definition/Summary
Intuitively speaking, a fibre bundle is space E which 'locally looks like' a product space B×F, but globally may have a different…

### What is a Real Number? A 5 Minute Introduction

Definition/Summary
The real numbers are most commonly encountered number system, familiar to the layman via the number line, and as the number system…

### What is a Parabola? A 5 Minute Introduction

Definition/Summary
A parabola has many definitions, a classical one being, "A Parabola is the locus of all points equidistant from a given point (called…

### What Is a Limit of a Function? A 5 Minute Introduction

Definition/Summary
Limits are a mathematical tool which is used to define the 'limiting value' of a function i.e. the value a function seems to approach…

### What is a Tangent Line? A 5 Minute Introduction

Definition/Summary
The tangent to a curve in a plane at a particular point has the same Gradient as the curve has at that point.More generally, the…

### Lie Algebras: A Walkthrough The Representations

Part III: Representations
10. Sums and Products.
Frobenius began in ##1896## to generalize Weber's group characters and soon investigated…

### Lie Algebras: A Walkthrough The Structures

Part II: Structures5. Decompositions.Lie algebra theory is to a large extend the classification of the semisimple Lie algebras…

### Lie Algebras: A Walkthrough The Basics

Part I: Basics
1. Introduction.
This article is meant to provide a quick reference guide to Lie algebras: the terminology, important theorems,…

### How to Self Study Abstract Algebra

There are three big parts of mathematics: geometry, analysis and algebra. In this insight I will try to give a roadmap towards learning basic abstract…

### An Interesting Ramsey Theory Riddle

Ramsey theory has its origins in a very nice riddle
Consider a party of 6 people. Any two of these 6 will either be meeting each other for the first time…

### How to Self Study Intermediate Analysis

If you wish to follow this guide, then you should know how to do analysis on ##\mathbb{R}## and ##\mathbb{R}^n##. See my previous insight if you wish to…

### Intro to the Millennium Prize Problems

IntroductionIn this Insight, I will go over the background information for the Millennium Prize problems and briefly describe three of them. A future…

### A Forward on Real Numbers and Real Analysis

It is important to realize that in standard mathematics, we attempt to characterize everything in terms of sets. This means that notions such as natural…

### Matrix Representations of Linear Transformations

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…

### Why Do People Say That 1 And .999 Are Equal?

Why do people say 1 and 0.999... are equal? Aren't they two different numbers?No, they really are the same number, though this is often very counterintuitive…

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

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]…

### The History and Concept of the Number 0

The goal of this FAQ is to clear up the concept of 0 and specifically the operations that are allowed with 0.The best way to start this FAQ is to look…

### How I Overcame Learning Challenges That I Faced Studying Science

Introduction
For the past few days, during my summer break, I have been intensively self-studying mathematics (namely number theory) for several hours…