Rigel Kent

Around 1640 Fermat documented this method in a awkwardly titled paper: De aequationum localium transmutatione et emendatione ad multimodam curvilineorum inter se vel cum rectilineis comparationem (On the transformation and alteration of local equations for the purpose of variously comparing curvilinear figures among themselves or to rectilinear figures, to which is attached the use of geometric proportions in squaring an infinite number of parabolas and hyperbolas). The ideas presented in the paper was dated probably 20 years ago but it was only written in 1658, two years after the publication of John Wallis's Arithmetica Infinitorum (1656). But it was not published or circulated until 1679. , 30 years before Newton and Leibniz officially “invented" calculus, Fermat managed to convinced himself that $$\int_0^k x^n\,dx= \frac{k^{n+1}}{n+1} = L(n+1)$$ Fermat's trick was to slice the area under the curve of \(y = x^n\) into infinitely many irregular intervals so that the

In 1720, Christian Goldbach's experiments with infinite series was published in the German journal Acta Eruditorum , he titled his paper Specimen methodi ad summas serierum and proceeded to furnish a number of worked examples The following examples are given in Goldbach's 1720 paper: \begin{align}\tfrac{1}{2}+\tfrac{1}{6}+\tfrac{1}{12}+\tfrac{1}{20}+\ldots\\ \tfrac{2}{64}+\tfrac{11}{544}+\tfrac{26}{1972}+\tfrac{47}{5104}+\ldots \\ \tfrac{1}{2}+\tfrac{1}{4}+\tfrac{1}{8}+\tfrac{1}{16}+\ldots \\ \tfrac{1}{3}+\tfrac{1}{81}+\tfrac{1}{19683}+\tfrac{1}{43046721} + \ldots\end{align}whose general terms are \(\frac{1}{x^2+x}\), \(\frac{3x^2-1}{9x^4+36x^3+33x^2-6x-8}\), \(\frac{1}{2x}\), and \(\frac{1}{2x^x-x}\). of his finite difference technique for relocating the general term of a given sequence. The general principle is that in the general case, the \(n\)-th term, \(a_n\), of any number sequence $$a_1, a_2, a_3, \ldots$$ can be expressed by using an infinite series involving

Anybody with one-to-two years of basic calculus exposure can easily work out the following definite integral of \(x^n\): $$\int_{0}^k x^n \,dx = \frac{k^{n+1}}{n+1} = {\rm L}(n + 1)$$ where \({\rm L}(n) = {k^n}/{n}\). This is a mid-17th-century result known to both Newton and Leibniz , but their results were not written in modern notation In his 1669 paper “De Analysi per Aequationes Numero Terminorum Infinitas" (Of analysis by equations of an infinite number of terms), Newton wrote: Let the base \(AB\) of any curve \(AD\) have \(BD\) for its perpendicular ordinate; and call \(AB = x\), \(BD = y\), and let \(a, b, c,\) etc be given quantities, and \(m\) and \(n\) whole numbers, then following rule (Rule I) can be used to compute the quadrature of simple curves. If \(ax^{m/n}=y\), then$$\frac{an}{m+n}x^{\frac{m+n}{n}} = {\rm area} \, ABD$$Newton then proceeded to give a number of examples. He wrote: The thing will be evident by an example. If \(x^2 = y\), th

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Euler's formula $$e^{i \Omega} = \sin \left(A- \Omega \right) + i \sin \Omega,$$ where \(A = \tfrac{1}{2}\pi\), is an interesting formula because we can use it express various types of sins in our complex world, and it says that sins of various degrees are ultimately imaginary. Here is the proof. First we suppose \(\Omega = \tfrac{1}{2}\pi \omega\), since $$e^{i\pi \omega/2} = e^{(i\pi/2)\omega}=\left(e^{i\pi/2}\right)^\omega $$we must have $$\sin \left(A - \tfrac{1}{2}\pi \omega \right) + i \sin \tfrac{1}{2}\pi \omega = \left[\sin \left(A - \tfrac{1}{2}\pi \right) + i \sin \tfrac{1}{2}\pi\right]^{\omega}=i^\omega$$ and if we start with \(\Omega = -\tfrac{1}{2}\pi \omega\), the following is true: $$\sin \left(A + \tfrac{1}{2}\pi \omega \right) - i \sin \tfrac{1}{2}\pi \omega = \left[\sin \left(A + \tfrac{1}{2}\pi \right) - i \sin \tfrac{1}{2}\pi\right]^{\omega} = (-i)^\omega$$ Since \(\sin \left(A+ \tfrac{1}{2}\pi \omega \right)=\sin \left(A - \tfrac{1}{2}\pi \omega \righ