Leonhard Euler
| This is an AoPSWiki Word of the Week for July 17-July 23 |
Leonhard Euler (1707-1735, pronounced Oiler) was a famous Swiss mathematician. He made numerous contributions to many fields of mathematics and science. Euler is often considered to be one of the greatest mathematicians of all time, along with Isaac Newton, Archimedes, and Carl Friedrich Gauss.
Contents
Biography
Euler was born on April 15, 1707 in Basel, Switzerland. Euler's parents were Paul Euler, a pastor of the Reformed Church, and Marguerite Brucker, a pastor's daughter. He had two young sisters, named Anna Maria and Maria Magdalena. At the age of thirteen he enrolled at the University of Basel.
On January 7, 1734, he married Katharina Gsell. The young couple had thirteen children, only five of whom survived childhood.
After suffering a near-fatal fever in 1735, Euler became nearly blind in his right eye. Soon after his return to Russia in 1766, he became almost completely blind in his left eye. Despite his horrible eyesight, Euler continued his prolific research.
Even as an old man, Euler was famous for being able to calculate difficult arithmetic quickly in his head.
On September 18, 1783, Euler passed away in St. Petersburg, Russia after suffering a brain hemorrhage. He was buried in the Alexander Nevsky Monastery.
Contributions to Mathematics
- In 1735, Euler showed that
where 
is the zeta function. - Euler's polyhedral formula states that for any convex polyhedron, with
, 
and 
denoting the number of vertices, edges, and faces, respectively, 
. - Euler's totient theorem is a theorem related to Euler's totient function, or the sum of all numbers relatively prime to a number and less than it.
- Euler's identity is a famous identity of complex numbers which states that
for all 
.
where 
is the 
, 
and 
denoting the number of 
.
for all 
.Functions
Euler was the first to use the notation 
, and also was the first to call such structures functions. This led to great advancements in calculus and algebra.
Analysis
Euler contributed greatly to the expansion of a branch of mathematics called analysis. His achievements often involved power series. He is also credited with discovering Euler's constant, denoted as 
. Euler discovered that
![\[e^x = \sum_{n=0}^\infty {x^n \over n!} = \lim_{n \to \infty}\left(\frac{1}{0!} + \frac{x}{1!} + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!}\right)\]](http://latex.artofproblemsolving.com/2/6/7/2672ae05e431f9d0b9dac5120ff26f9633f029fc.png)
He also discovered the power series for the arctengent, which is
![\[\lim_{n \to \infty}\left(\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots + \frac{1}{n^2}\right) = \frac{\pi ^2}{6}\]](http://latex.artofproblemsolving.com/2/0/1/20124709031501d26d3af9bf6472f938412b2d8c.png)
Physics
Euler helped the wave theory of light become the dominant idea for most of the nineteenth century. He also made several other contributions to optics, such as disagreeing with Isaac Newton's theory of light.
In addition, he made several important discoveries in astronomy and mechanics.
