# How Euler Changed Analysis

Nowadays, the idea of function pervades mathematics, and math students readily recognize the notation f(x) as representing a function. But it took centuries for mathematicians to go from the use of algebraic expressions for describing certain curves to the general notion of formulas (or functions) as stand-alone objects of considerable mathematical interest in themselves. Leonhard Euler (1707-1783) played a fundamental role in making the function one of the central objects of mathematics.

On Aug. 8, an audience of Euler enthusiasts at the MAA's Carriage House Conference Center heard mathematician and historian Rüdiger Thiele of the University of Leipzig speak about Euler's work on functions. Thiele's lecture, titled "How Euler Changed Analysis," focused on Euler's efforts to broaden and apply the notion of a function in a variety of mathematical contexts.

The word "function" comes from the Latin functio, which means a performance or an activity. Thiele traced how, through the work of Gottfried Leibniz (1646-1716), Guillaume de L'Hôpital (1661-1704), Euler's boyhood teacher Johann Bernoulli (1667-1748), and Euler himself, the idea that a curve exists if it can be constructed by some geometric or mechanical process evolved into the concept that a curve exists if it can be described by an analytical expression, whether a polynomial, an infinite power series, or some other form.

Thiele described Euler's impact on analysis by noting that, "Euler gave the function its analytical meaning." Functions became not only tools but also objects themselves, he said. Euler also introduced the now-familiar f(x) notation for a function, using it first for a linear relation.

Thiele's discussion also touched on Euler's use of functions in solving differential equations, particularly those linked to physical phenomena such as waves and vibrating strings and ballistic trajectories. The general worldview at the time was that the uses of a function were very narrow, and they were almost irrelevant in fields such as physics. Euler's work showed that functions have a broad range of applications that span the mathematical spectrum and are useful for solving physics problems.

Rüdiger Thiele studied mathematics and physics at the Martin Luther University of Halle-Wittenberg in Halle, Germany. He has been a faculty member at the Karl Sudhoff Institute for the History of Medicine and the Natural Sciences at the University of Leipzig since 1986. In 2004, he won the MAA's Lester R. Ford award for his article "Hilbert's Twenty-Fourth Problem," which appeared in the January 2003 issue of The American Mathematical Monthly.