Numerical Tribute to Achievements of Euler

Abstract

This paper aims to make a tribute to one of the world's brightest personalities, the mathematical physicist Leonhard Euler (1707-1783). Our purpose is to expose, in a concise and entertaining way by utilizing modern computational techniques, a better understanding of the influence of Euler; also, to establish the remarkable significance in  the history of science. A first analysis was done with the series that defines Euler and Bernoulli numbers and polynomials of Bernoulli and Euler; an additional result is the characterization of the functions that lead to the Euler-Mascheroni constant. In hydrodynamics it is also feasible to evaluate graphically the connection between sizes in diameter and the exit angle of the height of Euler for turbo-machines. In differential equations of Cauchy-Euler solutions in the cases of distinct real roots and complex roots are generated. Furthermore we report the generation of the Fourier series and the Fourier transform calculated by using direct Matlab commands. In calculus of variations, it is conceivable to obtain plots from a problem of the Euler Lagrange equations. Finally, Euler totient function is graphically analyzed. We seek the following benefits: The association between Euler numbers and generating functions is more effectively understood, also the nature of the Euler constant and the Euler function  The Fourier transform is more comprehensible; also, it can be established that Matlab is an alternative way to study the conceptual development of achievements of physics and mathematics. We show that the abundance of Euler work and the history of mathematics, acquire better knowledge with the application of numerical tools