Introduction to the Analysis of Infinities

Leonhard Euler's Introductio in analysin infinitorum (1748) was a foundational work in mathematical analysis. It established the concept of a function, relating variables and utilizing infinitesimals and infinite quantities. Euler's work advanced analytic geometry and trigonometry, treating trigonometric functions as numerical ratios and connecting them to complex numbers. He also introduced many modern mathematical notations, including (\Sigma) for summation and (e) for the base of natural logarithms.1

Euler's contributions extended to infinite series, where he solved problems that had previously stumped mathematicians. He found the sum of the series (\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \dots) to be (\frac{\pi^2}{6}). He also discovered a fundamental property of the zeta function, linking sums over integers to products over prime numbers, which provided deeper insight into the infinitude of primes.2