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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

The Analytic Continuation of the Lerch Transcendent and the Riemann Zeta Function 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 are mentioned as a steppingstone. The continuations are given…

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 outlined. Another reference text, I cite Theory and Applications of Infinite Series by Knopp. As for original content, I only have hope that the method of using the sets…

Mathematician Mary Somerville Features in Google Doodle

The Google Doodle for 2 February 2020 celebrated Mary Somerville, the Scottish polymath and science writer, and with Caroline Herschel, the joint first ever woman to be made an honorary member of the Royal Astronomical Society. Born in Jedburgh, Scotland, in 1780, Somerville received little formal education compared to her brothers. Largely self-taught, she pursued…

Explore the Fascinating Sums of Odd Powers of 1/n

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 an expansion of two of my previous insights (Further Sums Found Through Fourier Series, Using the Fourier Series To Find Some Interesting Sums). You…

SOHCAHTOA: Seemingly Simple, Conceivably Complex

Preface My first experience with derivatives was seeing how they are obtained from the usual definition $$f'(x)=\underset{\text{$\Delta $x}\to 0}{\text{Lim}}\frac{f (\text{$x+\Delta $x})-f (x)}{\text{$\Delta $x}}.$$ I accepted the binomial theorem derivation in the case of polynomials and the small angle explanation in the case of sines and cosines  until my math instructor asserted, without justification, that the…

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 $$\nabla \phi(x,y,z) = \vec F(x,y,z)$$ $$\nabla \times \vec F(x,y,z) = \vec 0$$ Extended explanation Suppose we are given a vector field ##\vec F(x,y,z)=\langle f(x,y,z),g(x,y,z),h(x,y,z)\rangle## that has a potential function ##\phi## and we wish to recover the potential…

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 M, and B are constants and [itex]M\neq 0[/itex]. Equations [tex]Mx+B=0[/tex] Extended explanation Since [itex]M\neq 0[/itex] the solution is given by [tex]x=-B/M\;.[/tex] The variable x does not have to be a number. For example, x…

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