Trigonometric computations are used in nearly all fields of geometry, physics and engineering. The triangulation technique has the great importance, which allows measuring the distance in astronomy and geography, monitoring the satellite navigation system. Also trigonometry is used in such areas as music theory, acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medicine and so on. Daniel Bernoulli, Fourier and Cauchy made a huge contribution to the involvement of trigonometry and developed the idea of trigonometric series.
In the middle of the XVIII century, appeared the most important in its consequences "argument of a string". Euler in a controversy with d'Alembert suggested a more general definition of a function than previously assumed. In particular, he suggested that the function might be defined by a trigonometric series. In his writings, Euler used several representations of algebraic functions in a series of multiple arguments to trigonometric functions. Euler was not engaged with the general theory of trigonometric series and he did not investigate the convergence of the series, but he got some important results. In particular, he led the expansion of integer powers of sine and cosine.
In the XIX-XX centuries, there were some rapid advances in the theory of trigonometric series and related areas of mathematics: harmonic analysis, the theory of random processes, encoding audio, video, and other. Daniel Bernoulli expressed confidence that any (continuous) function on a given interval can be represented by a trigonometric series. Discussions continued until 1807, when Fourier published his theory of representations of arbitrary piecewise-analytic functions by trigonometric series.
Fourier Statement was not strict in the modern sense, but it contained a study of the convergence of the majority of the series has received. Fourier suggested special decomposition for functions defined on the real line and the non-periodic, called as Fourier integral.
The versatility and effectiveness of a Fourier analysis method have made a great impression on the scientific world. Earlier trigonometric series are used in mathematical physics, mainly for the study of periodic processes (vibrations of the string, celestial mechanics, the motion of a pendulum, and so on). The work of Fourier investigated processes of a different kind (heat transfer) and the trigonometric series helped to gain valuable practical results. From this point, trigonometric series and integrals become a powerful tool for the analysis of a variety of functions. Poisson and Cauchy continued and deepened Fourier results.
Daniel Bernoulli made a significant contribution in mathematics. He worked on a Riccati equation, which is commonly found in a variety of mechanics problems. Bernoulli calculated the limit of the expression (1 + 1 / n) n. It is now the well-known number of e that is the base of natural logarithms. Bernoulli studied the theory of series, a variety of special functions, probability theory. He introduced the concept of moral expectation, which then was widely used by Laplace, Poisson. In gambling, a moral expectation of the loss exceeds the moral expectation of winning. Subsequently, the concept of moral expectations has not received further development. Bernoulli proposed to solve the problem of probabilistic methods of differential calculus, counting unit "infinitesimal" compared with other "big numbers", appearing in the problem.
Just as the rest of the family, Daniel has contributed to the development of mathematics. He calculated the base of natural logarithms, which is used in the present day terms. Continuing to work in line with the ideas of his father, he paid attention to the theory of series, probability theory, enthusiastically studied the function. Daniel Bernoulli was the first to suggest that the gas pressure affects the thermal motion of the smallest particles.
The contribution of the scientist in hydrodynamics is very important, where he figured out the equation of motion of an ideal incompressible fluid (today known as the postulate of the law of conservation of energy). Bernoulli's equation made possible to explain many issues of fluid mechanics, and to this day, it is used in the calculation of the parameters of the pumps and other devices.
In total, more than 50 years were dedicated to the study of oscillation. At the beginning of his observations, he analyzed the small oscillations of goods. Over time, a mathematician, moved to the study of string vibrations then announced the definition of simple harmonic motion. In later times, this term has become known as the principle of superposition. In terms of molecular theory, Daniel explained Boyle's law, and without it is also impossible to imagine modern physics - the position is included in the school curriculum. Additionally, a brilliant researcher greatly enriched the theory of gases, contributing to the discoveries made by his father - Johann Bernoulli and his brother.
Daniel Bernoulli is considered the founder of mathematical physics with Euler, with whom he did researches and was friendly throughout life. Writings of the junior member of Bernoulli are translated into most world languages in higher education in the departments of physics and mathematics.
The most important researches of Daniel Bernoulli belonged to pure mathematics, mechanics, probability theory and physiology. “Hydrodynamica, sive de viribus et motibus fluidorum. Argentorati 1738.” is considered the main Bernoulli’s work and the most complicated.
Most of all, Daniel Bernoulli is famous for works in the field of mathematical physics and the theory of differential equations. He is considered, along with D'Alembert and Euler, the founder of mathematical physics (Ball, 1972).
He thoroughly enriched the kinetic theory of gases, hydrodynamics and aerodynamics, theory of elasticity, and so on. He first made a statement that the cause of the gas pressure is the thermal motion of the molecules. In his classic "Hydrodynamics", he derived the equation of the steady flow of an incompressible fluid (Bernoulli's equation), the underlying dynamics of liquids and gases. In terms of molecular theory, he explained the Boyle's law (Dauben, 2000).
Bernoulli created one of the first formulations of the law of conservation of energy (kinetic energy), and (together with Euler) first formulation of the law of conservation of angular momentum (1746). For many years, he studied and mathematically modeled elastic vibrations, introduced the concept of harmonic oscillation, given the principle of superposition of vibrations.
In mathematics, he published a series of studies on the probability theory, the theory of series and differential equations. He was the first to apply mathematical analysis to the problems of probability theory (1768); previously only combinatorial approach was used. Bernoulli also promoted mathematical statistics, examined a number of practical problems using probabilistic methods (Heyde & Seneta, 2001).
A number of works are devoted to the problems of the hesitation of some mechanical and physical systems with a finite or an infinite number of degrees of freedom. In the work of the vibrating string (1755), he first applied solutions of differential equations with partial derivatives of trigonometric series. Bernoulli considered the cylindrical function of the first kind, expressed in the form of an infinite power series. He got results in the development of numerical methods for algebraic equations and more general by using returnable series. He laid the foundations of the theory of differential equations with partial derivatives. He developed the theory of probability, for the first time using it infinitesimal calculus (Gillispie, 1980).
Joseph Fourier, who was born in Auxerre, France, came from a modest family. The Analytical theory of heat was presented in 1822. The work was threefold: one mathematical and two physical. From a mathematical point of view, Fourier proved that any function of a variable, whether continuous or discontinuous, could be expanded in a series of sines of multiple variable. The theory of the dimensions of the homogeneous equation was among the findings of the physical work, according to which the equation can be formally correct only if the dimensions on both sides of the equation are the same. Another significant contribution to the development of Fourier physics was a proposal of its own differential equations in partial derivatives for the heat. To this day, every student of mathematical physics knows this equation (May, 1973).
Cauchy wrote about 800 papers. This happened not only because of hard work of Cauchy and genius of his mind, but also attention to his work from his contemporaries. In the rich scientific heritage of Cauchy, there are different types of work in different fields of mathematics. In them, he presented the results of his own research, and the results of the didactic activity - excellent textbooks of mathematical analysis, which became a model of scientific thinking for the next generation of mathematicians (Belhoste, 1991).
Cauchy gave the first clear definition of the basic concepts of mathematical analysis - limit, continuity of a function, the convergence of a number, etc. He established the precise conditions for the convergence of the Taylor series for this function, and drew a distinction between the convergence of the series in general, and its convergence to the function. Introduced the concept of the radius of convergence of a power series, gave a definition of the integral as a limit of sums, proved the existence of integrals of continuous functions.
Cauchy found the expression of an analytic function as an integral over the contour (Cauchy's integral) and deduced from this view the expansion of the power series. Thus, he developed the theory of functions of complex variable: using the contour integral, he found the expansion of the function into a power series, determined the radius of convergence of this series, developed the theory of residues, as well as its application to the analysis of various issues, etc. In the theory of differential equations, Cauchy first raised the general problem of finding solutions of differential equations with given initial conditions (called the Cauchy problem), gave way to the integration of partial differential equations of the first order.
Cauchy was also engaged in geometry, algebra (symmetric polynomials, properties of determinants), and number theory (Fermat's theorem on polygonal numbers, the law of reciprocity). He owns a study on trigonometry, mechanics, theory of elasticity, optics, astronomy. Cauchy was a member of the Royal Society of London, the St. Petersburg Academy of Sciences and a number of other academies in Europe (Bell, 1953).
References
Arago, F. (2010). Biographies of distinguished scientific men. S.l.: Nabu Press.
Ball, W. W. (1972). A short Account of the history of mathematics.
Belhoste, B. (1991). Augustin-Louis Cauchy: A biography. New York: Springer-Verlag.
Bell, E. T. (1953). Men of mathematics. Harmondsworth: Penguin.
Dauben, J. W. (2000). The History of mathematics from antiquity to the present: A selective annotated bibliography. Providence: American Mathematical Society.
Gillispie, C. C. (1980). Dictionary of scientific biography. New York: Scribner's Sons.
Heyde, C. C., & Seneta, E. (2001). Statisticians of the centuries. New York: Springer.
May, K. O. (1973). Bibliography and research manual of the history of mathematics. Toronto: University of Toronto Press.
