calculus, complex numbers, number theory, graph theory, and geometry, many of which bear his name. (A common joke about Euler is that to avoid having too many mathematical concepts named after him, the convention is to give them the name of the first person after Euler to have discovered them.)Leonhard Euler (1707-1783) was a Swiss mathematician and physicist who made fundamental contributions to countless areas of mathematics. He studied and inspired fundamental concepts in
Euler also made immense contributions to the language of modern mathematics. He was the first writer to define a function and write it as ; he was the first to use summation notation using the Greek letter ; he introduced the standard notation for trigonometric functions; and many more.
What follows is a list of some of the highlights of Euler's work. It will necessarily be incomplete, as Euler led a long and extremely prolific mathematical life.
Euler was not the first to identify as an interesting mathematical constant (depending on what one takes "discover" to mean, the constant was discovered by either Napier in the early 1600s or Bernoulli in the late 1600s), but he did make several important contributions to its study. In one of his most famous books, Introductio in Analysin infinitorum (1748), he showed that the infinite sum and the value of the limit and the number with the property that were all the same number, and called that number . He also gave an approximation to decimal places. He was also the first to prove that is irrational, by proving that its continued fraction is Since rational numbers have finite continued fractions, this gives a proof that is irrational.
For more on the history of , see the discovery of the number e.
In his analysis of the Riemann zeta function (see below), Euler analyzed the harmonic numbers In a 1731 paper, he showed that exists and equals the sum of the conditionally convergent series The constant is now known as , due to its connection with the gamma function, and is usually called the Euler-Mascheroni constant. It arises in sums involving the zeta function and integrals involving logarithms and exponentials, e.g.
Euler often manipulated infinite series in ways that were not rigorous by modern standards, but his intuition generally steered him to correct results. These manipulations could often be justified after the fact by modern mathematicians.
As a pioneering user of Taylor series, Euler noticed that plugging in to the series gives This result is fundamental in modern complex analysis, and has many applications for trigonometry as well. The special case gives which is often cited as the most beautiful formula in all of mathematics.
For more, see Euler's formula.
Euler showed that the infinite product equals and used this to derive a recursive formula for the partition function : where the numbers that appear are precisely the pentagonal numbers. This is Euler's pentagonal number theorem.
Among Euler's contributions to graph theory is the notion of an Eulerian path. This is a path that goes through each edge of the graph exactly once. If it starts and ends at the same vertex, it is called an Eulerian circuit.
Euler proved in 1736 that if an Eulerian circuit exists, every vertex has even degree, and stated without proof the converse that a connected graph with all vertices of even degree contains an Eulerian circuit. (This was proved in the late century.)
The motivation for his exploration was the famous Seven Bridges of Konigsberg problem.
Euler showed that for convex polyhedra, where are the number of vertices, edges, and faces, respectively. Using similar definitions, he showed that the same was true for a connected planar graph (planar means that the edges can be drawn on a piece of paper without crossing each other). In modern language, we say that the right side of the equation is the Euler characteristic of the object. Generalizations of this idea (including, most notably, the genus) were among the earliest foundational problems in topology.
Euler proved a theorem characterizing all even perfect numbers. He showed that they were of the form , where is a Mersenne prime. The fact that all such numbers were perfect was known as far back as Euclid's Elements, but Euler's result was new. The proof is quite elementary and is in the wiki on perfect numbers.
 Image from https://upload.wikimedia.org/wikipedia/commons/d/d7/Leonhard_Euler.jpg under the creative Commons licensing for reuse and modification.