ProofWiki:Mathematicians/Leonhard Paul Euler
From ProofWiki
Swiss mathematician and physicist who pioneered much of the foundation of modern mathematics.
- Introduced much of the notation which is used today, including e and the modern notation for trigonometric functions.
- Proved Fermat's Little Theorem.
- In 1783, on the basis of considerable numerical evidence, conjectured the Law of Quadratic Reciprocity, which was eventually proven by Gauss in 1798.
- Proved the converse of the result known to Euclid, that if $2^p - 1$ is prime, then $2^{p-1} \left({2^p - 1}\right)$ is perfect. That is, Euler proved that if $n$ is an even perfect number, then $n$ is of the form $2^{p-1} \left({2^p - 1}\right)$ where $p$ is prime. The results together are known as the Theorem of Even Perfect Numbers.
Contents |
Nationality
Swiss
History
- Born: 15 April 1707, Basel, Switzerland
- Died: 18 Sept 1783, St Petersburg, Russia
Theorems and Problems
Geometry
Analysis and Calculus
- Euler-Maclaurin Summation Formula (with Colin Maclaurin)
- Euler Formula for Sine Function
- Often credited with solving the Basel Problem, but it is believed that this was in fact solved by Nicolaus I Bernoulli.
- Euler's Reflection Formula
Complex Analysis
Number Theory
- Euler's Criterion
- Theorem of Even Perfect Numbers
- Euler's Theorem
- Euler-Binet Formula (with Jacques Philippe Marie Binet) (also known as Binet's Formula)
- Eulerian Integer (also known as Eisenstein Integer for Ferdinand Eisenstein)
Graph Theory
Definitions
Analysis and Calculus
- Euler's Number
- Euler-Mascheroni Constant (with Lorenzo Mascheroni)
- Cauchy-Euler Equation (with Augustin Louis Cauchy)
Number Theory
Graph Theory
... and the list goes on.
Books and Papers
- 1736: Solutio problematis ad geometriam situs pertinentis (The solution of a problem relating to the geometry of position) in which was given the Handshake Lemma and solution to the Bridges of Königsberg problem, possibly the first ever paper in graph theory.
- 1736-37: Mechanica
- 1739: Tentamen Novae Theoriae Musicae
- 1740: Methodus Inveniendi Lineas Curvas
- 1748: Introductio in Analysin Infinitorum
- 1755: Institutiones Calculi Differentialis
- 1765: Theoria Motus Corporum Solidorum
- 1768-70: Institutionum Calculi Integralis
Also see
- John J. O'Connor and Edmund F. Robertson: "Leonhard Paul Euler": MacTutor History of Mathematics archive
- Eric Temple Bell: Men of Mathematics (1937): Chapter $\text{IX}$
- George F. Simmons: Calculus Gems (1992): Chapter $\text{A}.21$
