Introductio in analysin infinitorum (Latin: [1] Introduction to the Analysis of the Infinite) is a two-volume work by Leonhard Euler which lays the foundations of mathematical analysis. Written in Latin and published in 1748, the Introductio contains 18 chapters in the first part and 22 chapters in the second. It has Eneström numbers E101 and E102. [2] [3]
Chapter 1 is on the concepts of variables and functions. Chapter 4 introduces infinite series through rational functions.
According to Henk Bos,
Euler accomplished this feat by introducing exponentiation ax for arbitrary constant a in the positive real numbers. He noted that mapping x this way is not an algebraic function, but rather a transcendental function. For a > 1 these functions are monotonic increasing and form bijections of the real line with positive real numbers. Then each base a corresponds to an inverse function called the logarithm to base a, in chapter 6. In chapter 7, Euler introduces e as the number whose hyperbolic logarithm is 1. The reference here is to Gregoire de Saint-Vincent who performed a quadrature of the hyperbola y = 1/x through description of the hyperbolic logarithm. Section 122 labels the logarithm to base e the "natural or hyperbolic logarithm...since the quadrature of the hyperbola can be expressed through these logarithms". Here he also gives the exponential series:
Then in chapter 8 Euler is prepared to address the classical trigonometric functions as "transcendental quantities that arise from the circle." He uses the unit circle and presents Euler's formula. Chapter 9 considers trinomial factors in polynomials. Chapter 16 is concerned with partitions, a topic in number theory. Continued fractions are the topic of chapter 18.
Carl Benjamin Boyer's lectures at the 1950 International Congress of Mathematicians compared the influence of Euler's Introductio to that of Euclid's Elements , calling the Elements the foremost textbook of ancient times, and the Introductio "the foremost textbook of modern times". [5] Boyer also wrote:
The first translation into English was that by John D. Blanton, published in 1988. [6] The second, by Ian Bruce, is available online. [7] A list of the editions of Introductio has been assembled by V. Frederick Rickey. [8]
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number x, one has
The number e is a mathematical constant, approximately equal to 2.71828, that is the base of the natural logarithm.
Leonhard Euler was a Swiss mathematician, physicist, astronomer, geographer, logician, and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus. He introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory.
In mathematics, Euler's identity is the equality
In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. In other words, a transcendental function "transcends" algebra in that it cannot be expressed algebraically using a finite amount of terms.
In mathematics education, precalculus is a course, or a set of courses, that includes algebra and trigonometry at a level which is designed to prepare students for the study of calculus, thus the name precalculus. Schools often distinguish between algebra and trigonometry as two separate parts of the coursework.
A hyperbolic sector is a region of the Cartesian plane bounded by a hyperbola and two rays from the origin to it. For example, the two points (a, 1/a) and (b, 1/b) on the rectangular hyperbola xy = 1, or the corresponding region when this hyperbola is re-scaled and its orientation is altered by a rotation leaving the center at the origin, as with the unit hyperbola. A hyperbolic sector in standard position has a = 1 and b > 1.
In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of xy = 1 in Quadrant I of the Cartesian plane. The hyperbolic angle parametrises the unit hyperbola, which has hyperbolic functions as coordinates. In mathematics, hyperbolic angle is an invariant measure as it is preserved under hyperbolic rotation.
In mathematics, hyperbolic coordinates are a method of locating points in quadrant I of the Cartesian plane
In mathematics, particularly in geometry, quadrature is a historical process of drawing a square with the same area as a given plane figure or computing the numerical value of that area. A classical example is the quadrature of the circle . Quadrature problems served as one of the main sources of problems in the development of calculus. They introduce important topics in mathematical analysis.
Grégoire de Saint-Vincent - in Latin : Gregorius a Sancto Vincentio, in Dutch : Gregorius van St-Vincent - was a Flemish Jesuit and mathematician. He is remembered for his work on quadrature of the hyperbola.
The 18th-century Swiss mathematician Leonhard Euler (1707–1783) is among the most prolific and successful mathematicians in the history of the field. His seminal work had a profound impact in numerous areas of mathematics and he is widely credited for introducing and popularizing modern notation and terminology.
Elements of Algebra is an elementary mathematics textbook written by mathematician Leonhard Euler around 1765 in German. It was first published in Russian as "Universal Arithmetic", two volumes appearing in 1768-9 and in 1770 was printed from the original text. Elements of Algebra is one of the earliest books to set out algebra in the modern form we would recognize today, and is one of Euler's few writings, along with Letters to a German Princess, that are accessible to the general public. Written in numbered paragraphs as was common practice till the 19th century, Elements begins with the definition of mathematics and builds on the fundamental operations of arithmetic and number systems, and gradually moves towards more abstract topics.
The history of logarithms is the story of a correspondence between multiplication on the positive real numbers and addition on the real number line that was formalized in seventeenth century Europe and was widely used to simplify calculation until the advent of the digital computer. The Napierian logarithms were published first in 1614. E. W. Hobson called it "one of the very greatest scientific discoveries that the world has seen." Henry Briggs introduced common logarithms, which were easier to use. Tables of logarithms were published in many forms over four centuries. The idea of logarithms was also used to construct the slide rule, which became ubiquitous in science and engineering until the 1970s. A breakthrough generating the natural logarithm was the result of a search for an expression of area against a rectangular hyperbola, and required the assimilation of a new function into standard mathematics.
A timeline of calculus and mathematical analysis.
In exponentiation, the base is the number b in an expression of the form bn.
Most of the terms listed in Wikipedia glossaries are already defined and explained within Wikipedia itself. However, glossaries like this one are useful for looking up, comparing and reviewing large numbers of terms together. You can help enhance this page by adding new terms or writing definitions for existing ones.
