1- Almost all first order systems are easier to solve numerically using computer systems (matlab, maple, etc). Yes, it takes some working out by hand first, but the compiling time is much less. Even wolfram limits computation time, so use these notes to your advantage.

2- Nearly all moving (rates of change) problems in real life don't...

Converting Higher Differential Equations into First Order Systems (Examples and Notes)]]>

Hopefully, in this small example, I will demonstrate there is nothing to be afraid of in this type of construction...

De-mystifying universal mapping properties: an example-quotient groups.]]>

$$

\begin{array}{|c|c|c|c|}\hline

\textbf{Type}...

Quantum Computing Series: Different Kinds of Measurement Operators]]>

We are given a trinomial of the form $ax^{2}+bx+c$, and asked to factor it into a product of two dissimilar binomials $(fx+u)(gx+v)$. The method that follows assumes $a,b,c$ have no common factor; if they do, you must factor out the greatest common factor before proceeding.

The method is as follows:

- Write $ax^{2}+bx+c$ as $(ax+\underline{\phantom{45}}\,)(ax+\underline{\phantom{45}}\,)$.
- Examine the factor pairs of the product $ac$, and...

AC Method for Factoring Trinomials]]>

Quantum Computing Series: Positive Operators are Hermitian]]>

$$\begin{array}{|c|c|c|c|c|c|} \hline

\textbf{Name} &\textbf{Matrix} &A^{\dagger}A=I? &A=A^{\dagger}? &\textbf{E-values}...

Quantum Computing Series: Useful Information Concerning Matrices Often Used]]>

1. I am decidedly NOT a fan of what I call cookie-cutter, follow-the-recipe, paint-by-numbers approach to labs. This is not experimental science. In real experimental science, you don't even usually have the hypothesis given to you! Essentially nothing is given to you. However...

Thoughts on Physics Labs]]>

What is the Riemann Hypothesis?]]>

The direct product of two groups $(G,\ast)$ and $(H,\ast')$ is defined to be the set:

$G \times H = \{(g,h): g \in G, h \in H\}$

together with the binary operation:

$(g,h)\star(g',h') = (g\ast g',h\ast' h')$ for all $g,g' \in G$ and...

The Universal Property of the Direct Product in Groups]]>

http://mathhelpboards.com/challenge-questions-puzzles-28/logarithm-integral-5273.html#post24028

... it has beenshown in an elementary way that is ...

$\displaystyle\int_{0}^{\infty}\frac{\ln x}{x^{2}+a^{2}}\ dx = \frac{\pi\ \ln a}{2\ a}\ (1)$

The integral (1) isvery useful to shed light on the behavior of the logarithm functionin the complex field. Let's suppose we want to solve the integralusing the methods of complex analysis, which means to integrate...

Behavior of the logarithm function in the complex domain ...]]>

Throughout, I will use the notation

\(\displaystyle F(w) = \mathfrak{L}(f) = \int_{0}^{\infty} e^{-wx}f(x)\, dx\)

and

\(\displaystyle f(w) = \mathfrak{L}^{-1}(F)\)...

Laplace Transforms (proofs of)]]>

Preliminaries:

---------------

The following series expansions for the trigonometric functions will be used throughout, where\(\displaystyle B_k\) and...

A trigonometric approach to infinite series (involving Zeta and Dirichlet Beta functions)]]>

http://www1.tmtv.ne.jp/~koyama/recentpapers/ei.pdf

The authors have a number of other on-line papers concerning the same...

The Multiple Sine function]]>

Derivatives and Integrals of the Hurwitz Zeta function]]>

I thought it might be nice to collect my notes and present my findings here. I thought I would present a small portion once a week, so that others can have a chance to give feedback and add their own insights.

My own personal discovery of the Pell numbers began with a simple desire to...

The Pell Sequence]]>

In this set of lectures we are going to explore the nice idea of analytic continuation and regularization of divergent series and integrals. Don't get panic ,the idea is so simple that you are actually using it without knowing. I'll try to make the tutorials as simple as possible so people with basic knowledge of complex variables will understand it. Basic knowledge of calculus will actually suffice. Even a high school student will be able to understand the concepts once...

Analytic continuation and Regularization simplified.]]>

$\displaystyle S = \sum_{n=1}^{\infty} \tan^{-1}\ \frac{\sqrt{3}}{n^{2} + n + 3}\ (1)$

... and that has been performed using the general identity...

$\displaystyle \sum_{n=1}^{\infty} \tan^{-1}\ \frac{c}{n^{2} + n + c^{2}} = \tan^{- 1} c\ (2)$

... so that is $\displaystyle S = \tan^{- 1} \sqrt{3} = \frac{\pi}{3}$. Scope of this thread is to find a general procedure to construct series of inverse functions.

Let...

Series of inverse functions..]]>

- The lower-most nodes are the elementariest functions to which the root of a general quintic can be extracted.
- In any sub-graph of the above, the lowermost elements are the form reduced from the topmost elements via Tschirnhausen...

Introduction to Theory of Quintics]]>

Sines, Cosines, and infinitely nested radicals]]>

\(\displaystyle (01) \quad \psi_{m \ge 1}(z) = (-1)^{m+1}m!\, \sum_{k=0}^{\infty} \frac{1}{(k+z)^{m+1}}\)

We will have frequent need of the reflection formula, which is obtained by repeated differentiation of the reflection formula for the Digamma function:

\(\displaystyle (02) \quad \psi_0(z) - \psi_0(1-z) = -\pi\cot \pi z \, \Rightarrow\)

\(\displaystyle (03) \quad...\)

Evaluations of the higher order Polygamma functions]]>

The Barnes' G-Function, and related higher functions]]>

For non-negative integers \(\displaystyle m\) and \(\displaystyle n\), where...

Nielsen Polylogarithms and the Generalized Logsine Integral]]>

First off, let's start with a (my apologies) formal definition:

We need 2 groups to start with: we shall call these groups (for reasons that hopefully will be clearer later on) $N$ and $H$.

The goal is to build a "bigger group" out of $N$ and $H$, but one in which $N$ and $H$ "interact"...for the time being, I will just say...

Semi-direct products: a gentle introduction]]>

\(\displaystyle \Gamma i_m(q)=\int_0^qx^m\log\Gamma(x)\,dx\)

Where \(\displaystyle 0 < q \le 1\). The special case where \(\displaystyle q=1\)...

A generalized Clausen Function, and associated loggamma integrals]]>

This thread is dedicated to discuss about Hardy-Littlewood's estimate of the \(\displaystyle N_0(T)\), i.e., the number of critical zeros of the Riemann zeta function with imaginary part smaller than \(\displaystyle T + 1\). The final result of Hardy-Littlewood estimate shows that there are infinitely many zeros of zeta that lies in the critical line.

This thread would be continued in more than one post, each of them showing different lemmas. The estimated number of post...

Hardy & Littlewood's Result]]>

-------------------

A wide variety of logarithmic...

Logarithmic Integrals, Polylogarithms, and associated functions]]>

To that end, this is not really a tutorial, but rather a random and somewhat arbitrary collection of related results that I've evaluated over the years, and found to be quite useful. Who know's, eh? With a bit of gentle coaxing I...

Inverse Sine/Tangent Integrals and related functions]]>

As students of calculus, we are taught to find the volumes of solids of rotation obtained by revolving given regions about horizontal and vertical axes of rotation. But, what if the axis of rotation is neither horizontal nor vertical? Please consider the following diagram:

View attachment 1398

We wish to revolve the region shaded in green about the line $y=mx+b$. Using the disk method, where the radius of a disk is $r$ and its thickness is $du$, we may write...

Solid of revolution about an oblique axis of rotation]]>

Rings are depicted in a ring, fields in an octagon, and algebraically closed fields in a rectangle. Objects in dashed rings are just sets (usually not even closed under addition!).

Key:

$\mathbb{Z}$: the ring of integers $\{..., -2, -1, 0, 1, ...\}$...

Flavors of Numbers]]>

I thought it might be instructive to work the problem in general terms. Please refer to the following diagram...

Folding to make boxes]]>

Let's let $A_i$ be the area of the base, and $r_i$ be some linear measure of its shape. We could then state:

\(\displaystyle A_i=kr_i^2\) where \(\displaystyle 0<k\) is the constant of proportionality.

This stems from the fact that...

On the volumes of pyramids]]>

Let one line be $\displaystyle y_1=m_1x+b_1$ and the other line be $\displaystyle y_2=m_2x+b_2$.

Now,we know the angle of inclination of a line is found from:

$\displaystyle m=tan(\theta)\,\therefore\,\theta=\tan^{-1}(m)$

Let $\displaystyle \theta_1$ be the angle of inclination of $\displaystyle y_1$ and $\displaystyle \theta_2$ be the...

Perpendicular lines and the product of their slopes]]>

att

jefferson alexander vitola

Comments and questions should be posted here...

L-R Circuit Simple]]>

att

jefferson alexander vitola

att

jefferson alexander vitola

Comments and questions should be posted here...

Inexact Differential Equation with 4 Degree]]>

I know that is a exercise easy to solve in the most general integrated particularly variable by for variable, but my goal and the main theme is do by different numerical methods,,,

att

jefferson alexander vitola

...

Numerical Approximation of Double Integral, Two Ways]]>

First, we need to compute the definite integral:

\(\displaystyle \int_{-h}^h Ax^3+Bx^2+Cx+D\,dx\)

Applying the FTOC, we find:

\(\displaystyle \int_{-h}^h...\)

Deriving the first and second order Newton-Cotes formulas]]>

$$\int_{0}^{\infty} \frac{\ln x}{x^{2}+ a^{2}}\ d x\ (1)$$

Scope of this note is to illustrate a general procedure to engage integrals like (1) in elementary way, i.e. without use comnplex analysis tecniques. The preliminary is the evaluation of the following indefinite integral, which doesn't appear in most 'Integration Manuals'...

$$\int x^{m}\ \ln^{n} x\ d x\ (2)$$

... where m and n are non negative...

Integrals with natural logarithm...]]>

... and the unanimous answer has been '... it doesn't exist any closed formula to find x as function of y...'. In my opinion the proposed problem is a good opportunity to use the following solving procedure to find the inverse of an analytic...

A general way to find the inverse functions...]]>

Type | $g(x)$ | $y_p(x)$ |

(I) | $p_n(x)=a_nx^n+\cdots+a_1x+a_0$ | $x^sP_n(x)=x^s\left(A_nx^n+\cdots+A_1x+A_0 \right)$ |

(II) | $ae^{\alpha x}$ | $x^sAe^{\alpha x}$ |

Justifying the Method of Undetermined Coefficients]]>

1. Circle

2. Ellipse

3. Parabola

4. Hyperbola,

you can see that there is a progression here: increasing angle $\alpha$ that the intersecting plane makes with the horizontal. To be clear about this, $\alpha$ is

It's also true that for an...

Connection Between the Angle of a Conic Section and the Energy of the Corresponding Orbit]]>

The staff here at MHB has decided, after some consideration, that it would be beneficial to our site if a few small changes are made in the way tutorial topics are posted and in the way comments/questions regarding these topics are handled.

A tutorial topic should be attempted to the best of the author's ability to be composed with textbook quality writing style, with all points clearly explained and properly formatted using $\LaTeX$. In topics with multiple posts, the flow...

Guidelines for posting and responding to tutorial topics]]>

This was my initial reply:

Let:

$L$ = distance upstream to the safehouse

$W$ = width of the river

$v_s$ = swimming speed

$v_r$ = running speed

All...

Using Snell's Law to determine the fastest escape route]]>

... Up until this point, one might wonder whether

or not the entire aspect of limits isn't too easy. Well, let's us consider

a rather simple, but non-linear function such as...

A Method for Proving Some Non-Linear Limits]]>

$$\int \frac{dx}{1+x^{4}}.$$

Answer: It's perhaps a not-so-well-known fact that, although $x^{2}+y^{2}$ does not factor over the reals, $x^{4}+y^{4}$ does. In fact,

$$x^{4}+y^{4}= \left(x^{2}+ \sqrt{2} \, xy+y^{2} \right) \left(x^{2}- \sqrt{2} \, xy+y^{2} \right).$$

Hence, we have

$$\int \frac{dx}{1+x^{4}}=\int \frac{dx}{\left(x^{2}+ \sqrt{2} \, x+1 \right) \left(x^{2}- \sqrt{2} \, x+1 \right)}.$$

Next, you can use partial fractions to pull them apart. That is, assume that you...

The Antiderivative of 1/(1+x^4).]]>

$$x= \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}.$$

Let's assume for the sake of this thread that $a,b,c$ are all real.

Now, your mantra when solving equations should be that

Method for Checking the Solutions to a Quadratic]]>

Suppose q divides n - 1 and \(\displaystyle q > \sqrt{n}-1\). If there exists an \(\displaystyle a\) such that \(\displaystyle a^{n-1} = 1 \pmod{n}\) and \(\displaystyle \mathrm{gcd}(a^{(n-1)/q} - 1, n)\) then n is a certified prime.

There is a generalization of...

Primality Tests]]>

Thank you Professor Bales, for sharing this with us.

Comments and questions pertaining to this tutorial may be posted in this topic.]]>

A closed conducting circle of wire is lying the plane of the page (or screen), and there is a magnetic $\mathbf{B}$ field pointing to the left, in the plane of the page (or screen). Hence, there is initially zero magnetic flux through the area of the circle. Then we rotate the loop about a...

Confusing Issue with Lenz's Law]]>

We have many great tutorial topics which are located in other sub-forums here, and for your convenience, we shall provide links to these topics here:

- Algebra Do's and Don't's
- Two methods for deriving the quadratic formula that I was not taught in school...

Tutorial topics located in other sub-forums]]>

For a proof that the above definition of the conic section is equivalent to the focus-directrix definition we will use, see

Conic sections]]>

The goal of this post is to arrive at the correct sign information for the potential due to a point charge...

Sign Problems in Electricity: potential, potential energy, force, field, and work.]]>

I invite anyone with any techniques of their own to add to this topic to give our readers as comprehensive a list of tips/tricks as possible.

Typically...

Factoring Quadratics]]>

$\displaystyle I= \int_{0}^{\infty} \frac{1 - \cos x}{x^{2}}\ e^{-x}\ dx$ (1)

I don't mask the fact that I like to follow 'unconventional' ways and in that situation I was perfecly coherent. Combining the series expansion...

$\displaystyle \frac{1 - \cos x}{x^{2}} = \sum_{n=0}^{\infty} (-1)^{n}\ \frac{x^{2 n}}{(2n+2)!}$ (3)

... and the well known result...

$\displaystyle...

A not very advanced integration technique...]]>

the horizontal. It starts from rest at the point $x_{0}$, where $x$ is measured positively down the plane. Find the

position $x=x(t)$.

Answer. We use conservation of energy, including rotational kinetic energy, to obtain the differential equation of motion.

Let $v$ be the velocity of the ball down the inclined plane.

Let $y$ be the height of the ball at time...

Useful Derivation for Labs Involving Rolling Balls Down an Inclined Plane.]]>

$\displaystyle (x_1,y_1),\,(x_2,y_2),\,(x_3,y_3)$

and we wish to find the area of the triangle whose vertices are at these points.

We may let the base

Let's begin with the familiar formula for the area

$\displaystyle A=\frac{1}{2}bh$...

Finding the area of a triangle formed by 3 points in the plane]]>

Method 1:

In the

Extend a line segment from the point to the line such that the segment intersects perpendicularly with...

Finding the distance between a point and a line]]>