Introductio in analysin infinitorum (Latin: 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.
Carl 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". Boyer also wrote:
The first translation into English was that by John D. Blanton, published in 1988. The second, by Ian Bruce, is available online. A list of the editions of Introductio has been assembled by V. Frederick Rickey.
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.