Leonhard Euler

Leonhard Euler. [1]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 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.)

Euler also made immense contributions to the language of modern mathematics. He was the first writer to define a function and write it as \( f(x) \); he was the first to use summation notation using the Greek letter \( \Sigma \); 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 \( e\) 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 \[ \frac1{0!} + \frac1{1!} + \frac1{2!} + \cdots = \sum_{k=0}^{\infty} \frac1{k!} \] and the value of the limit \[ \lim_{n\to\infty} \left( 1+\frac1{n}\right)^n \] and the number \( a\) with the property that \[ \int_1^a \frac1{x} \, dx = 1 \] were all the same number, and called that number \( e \). He also gave an approximation to \( 23 \) decimal places. He was also the first to prove that \( e \) is irrational, by proving that its continued fraction is \[ e = [2;1,2,1,1,4,1,1,6,1,1,8,1,\cdots] = 2+\frac1{1+\frac1{2+\frac1{1+\frac1{1+\frac1{4+\frac1{\cdots}}}}}}. \] Since rational numbers have finite continued fractions, this gives a proof that \( e\) is irrational.

For more on the history of \(e\), see the discovery of the number e.

In his analysis of the Riemann zeta function (see below), Euler analyzed the harmonic numbers \[ H_n = 1 + \frac12 + \cdots + \frac1{n}. \] In a 1731 paper, he showed that \[ \lim_{n\to\infty} (H_n - \ln(n)) \] exists and equals the sum of the conditionally convergent series \[ \sum_{n=2}^{\infty} (-1)^n \frac{\zeta(n)}{n}. \] The constant is now known as \( \gamma \), 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. \[ \int_0^{\infty} \frac{\ln(x)}{e^x} \, dx = -\gamma. \]

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 \( z=i\theta \) to the series \[ e^z = 1+z+\frac{z^2}{2!}+\cdots= \sum_{k=0}^{\infty} \frac{z^k}{k!} \] gives \[ \begin{align} e^{i\theta} &= \left( 1-\frac{\theta^2}{2!}+\frac{\theta^4}{4!}-\cdots\right) +i\left(\frac{\theta}{1!}-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}-\cdots\right) \\ &= \cos \theta+i\sin \theta. \end{align} \] This result is fundamental in modern complex analysis, and has many applications for trigonometry as well. The special case \( \theta = \pi \) gives \[ e^{i\pi} +1 = 0, \] which is often cited as the most beautiful formula in all of mathematics.

For more, see Euler's formula.

Euler developed the Euler product formula for the Riemann zeta function, \[ \zeta(s) = \sum_{n=1}^{\infty} \frac1{n^s} = \prod_{p \text{ prime}} \left( 1-\frac1{p^s} \right)^{-1}. \] He used this to show that the sum of the reciprocals of the primes diverges, and used a similar (and easier) argument to show that there were infinitely many primes.

Euler defined the totient function \(\phi(n) \) in 1736, and proved several facts about it, including the product formula \[ \frac{\phi(n)}{n} = \prod_{\stackrel{p|n}{p \text{ prime}}} \left( 1- \frac1{p} \right) \] and Euler's theorem \[ a^{\phi(n)} \equiv 1 \pmod n \] for all integers \( a \) coprime to \( n \). This function is of paramount importance in number theory and cryptography (as in the RSA cryptosystem).

Euler studied the problem of determining whether a number is a quadratic residue modulo a prime \( p \), and proved in 1748 the following simple criterion: \[ \left( \frac{a}{p} \right) = 1 \Leftrightarrow a^{\frac{p-1}{2}} \equiv 1 \pmod p . \] Here \( \left( \frac{a}{p} \right) \) is the Legendre symbol.

This result is used to prove the theorem of quadratic reciprocity.

Euler showed that the infinite product \[ \prod_{k=1}^{\infty} (1-x^k) \] equals \[ \sum_{m=-\infty}^{\infty} (-1)^m x^{m(3m-1)/2} \] and used this to derive a recursive formula for the partition function \( p(n) \): \[ p(n) = p(n-1)+p(n-2)-p(n-5)-p(n-7)+p(n-12)\cdots, \] 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 \(19^\text{th}\) century.)

The motivation for his exploration was the famous Seven Bridges of Konigsberg problem.

Euler showed that for convex polyhedra, \[ V-E+F = 2, \] where \( V,E,F\) 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 in 1765 that the orthocenter, circumcenter, and centroid of a triangle are collinear (they all lie on one line). This line, called the Euler line, has many other beautiful properties, detailed in the wiki: Euler line.

Euler proved a theorem characterizing all even perfect numbers. He showed that they were of the form \( \frac{q(q+1)}2 \), where \( q\) 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.

[1] Image from https://upload.wikimedia.org/wikipedia/commons/d/d7/Leonhard_Euler.jpg under the creative Commons licensing for reuse and modification.