De Moivre's Formula Research Articles

Combinatorial identities concerning trigonometric functions and Fibonacci/Lucas numbers

<abstract><p>In this work, by means of the generating function method and the De Moivre's formula, we derive some interesting combinatorial identities concerning trigonometric functions and Fibonacci/Lucas numbers. One of them confirms the formula proposed recently by Svinin (2022).</p></abstract>

Euler and De Moivre’s Formulas for Fundamental Matrices of Commutative Quaternions

In this study, Euler and De Moivre's formulas for fundamental matrices of commutative quaternions are obtained. Simple and effective methods are provided to find the powers and roots of these matrices with the aid of De Moivre's formula obtained from the fundamental matrices of commutative quaternions. Moreover, our results are supported by pseudo-codes and some examples. ...

Multiple-S-Shaped Critical Manifold and Jump Phenomena in Low Frequency Forced Vibration with Amplitude Modulation

Motivated by the forced harmonic vibration of complex mechanical systems, we analyze the dynamics involving different waves in a double-well potential oscillator coupling amplitude modulation control of low frequency. The combination of amplitude modulation factor significantly enriches the dynamical behaviors on the formation of multiple-S-shaped manifold and multiple jumping phenomena that alternate between epochs of slow and fast motion. We can conduct bifurcation analysis to identify two harmonic vibrations. One is that the singular orbit makes multiple jumps to a fast trajectory segment from one attracting equilibrium to another as the expression of slow variable by using the DeMoivre formula. With the increase of tuning frequency, the system exhibits relaxation-type oscillations whose small amplitude oscillations are produced by nonlinear local cycles together with a distinct large amplitude cycle oscillation accounting for the Melnikov threshold values. The tuning frequency may not only affect the asymptotic expressions for the solution curves near fold singularities but also allow for the large amplitude orbit vibrations near fold-cycle singularities. Numerical analysis for computing critical manifolds and their intersections is used to detect the dynamical features in this paper.

De Moivre's formula: are Sands' H-functions the same as Chao's?

It has often been said that there is no clear extension of the Courant-Snyder theory to coupled dynamics. In particular, one never sees an extension of the symplectic de Moivre type formula beyond one degree of freedom. In the process of analysing the existence and meaning of such a formula, I discovered quite accidentally that Sands' formalism [1], if expressed in terms of Ripken-like lattice functions germane to the full three dimensional oscillator, gives the exact same result as the more accurate Chao theory [2]. This semi-serious paper displays this elegant connection. It is based in the lattice functions of Ripken [3] as extended by Forest [4]. We review Forest's derivation from the point of view de Moivre's formula extended to three degrees of freedom.

An Algebraic Remark on the Fourier Series of a Trigonometric Polynomial

In a previous note [1], we showed how to linearize trigonometric polynomials using complex numbers, Euler's formula, and DeMoivre's formula. Here we explain why this gives, in fact, the Fourier series of these trigonometric polynomials.

DeMoivre's Formula to the Rescue

DeMoivre's Formula to the Rescue

Euler's Formula and De Moivre's Formula for Quaternions

Natural generalizations of Euler's formula and De Moivre's formula for quaternions are derived.

De Moivre's formula for quaternions

Euler's formula and De Moivre's formula for complex numbers are generalized for quaternions. De Moivre's formula implies that there are uncountably many unit quaternions satisfying x n = 1 for n ≥ 3.

A Direct Proof of De Moivre's Formula

A Direct Proof of De Moivre's Formula

A Direct Proof of De Moivre's Formula

A Direct Proof of De Moivre's Formula

On a Method of obtaining De Moivre's formula in the simplest terms

When we consider in a table of mortality the consecutive terms which express the number living at each age, we remark that the differences between those numbers are almost constant in the middle ages of life; they change afterwards, remaining however uniform for another series of terms; so that the table can be considered as a series of decreasing arithmetical progressions. Let c be the difference between lx and lx+1; and let us suppose this difference constant for n years. We shall then have for the value of the life annuity during that time.