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'''Leonhard Euler''' ([[April 15]], [[1707]]  - [[September 18]], [[1783]]) (pronounced "oiler", not "yooler") was a [[Switzerland|Swiss]] [[mathematician]] and [[physicist]]. Leonhard Euler used the term "[[function]]" (first defined by [[Gottfried Leibniz|Leibniz]] - [[1694]]) to describe an [[expression]] involving various [[argument]]s; ie: y = F(''x''). Euler is credited with being one of the first to apply [[calculus]] to [[physics]]. 

Born and educated in [[Switzerland]], he worked as a professor of mathematics in [[Saint Petersburg]], later in  [[Berlin]], and then returned to Saint Petersburg. He is considered to be the most prolific mathematician of all time. He dominated the [[eighteenth century]] mathematics and deduced many consequences of the then new [[calculus]]. He was completely [[blind]] for the last seventeen years of his life, during which time he produced almost half of his total output.

Euler was deeply religious throughout his life. The widely told anecdote that Euler challenged [[Denis Diderot]] at the court of [[Catherine II of Russia|Catherine the Great]] with "''Sir, (a+b<sup>n</sup>)/n = x; hence God exists, reply!''" 
is however, false.

== Discoveries ==
Euler, with [[Daniel Bernoulli]], established the law that the [[torque]] on a thin elastic beam is [[proportional]] to a measure of the [[elasticity (physics)|elasticity]] of the material and the [[moment of inertia]] of a cross section, about an axis through the [[center of mass]] and perpendicular to the plane of the couple.

He also deduced the [[Euler equations]], a set of laws of motion in [[fluid dynamics]], directly  from [[Newton's laws of motion]].  These equations are formally identical to the [[Navier-Stokes equations]] with
zero [[viscosity]].  They are interesting chiefly because of the existence of
[[shock wave]]s.


In mathematics, he made important contributions to [[number theory]] as well as to the theory of [[differential equations]].
His contribution to analysis, for example, came through his synthesis of [[Gottfried Leibniz|Leibniz]]'s differential [[calculus]] with [[Isaac Newton|Newton]]'s method of [[fluxion]]s. 

He established his fame early on by solving a long-standing problem:

:<math>\sum_{n=1}^\infty \frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots = \frac{\pi^2}{6}</math>

He also showed that for all [[real number]]s ''x'',

:''e''<sup>''ix''</sup> = cos(''x'') + ''i'' sin(''x'')

This is [[Euler's formula]], which establishes the central role of the 
[[exponential function]]. In essence, all functions studied in elementary analysis are either variations of the exponential function or they are [[polynomial]]s.
[[The most remarkable formula in the world]] is an easy consequence.

In [[1735]], he defined the [[Euler-Mascheroni constant]] useful for [[differential equation]]s:

:<math>\gamma = \lim_{n \rightarrow \infty } \left( 1+ \frac{1}{2} + \frac{1}{3} + \frac{1}{4} ... + \frac{1}{n} - \log(n) \right)</math>

He is a co-discoverer of the [[Euler-Maclaurin formula]] which is an extremely useful tool for calculation
of difficult integrals, sums and series.


Euler wrote ''Tentamen novae theoriae musicae'' in 1739 which was an attempt to combine [[mathematics]] and [[music]]; a biography comments that the work was "for musicians too advanced in its mathematics and for mathematicians too musical". 

In economics, he showed that if each factor of production is paid the value of its marginal product, then (under constant returns to scale) the total income and output will be completely exhausted.

In [[geometry]] and [[algebraic topology]], there is a relationship called [[Euler's Formula]] which relates the number of edges, vertices, and faces of a [[simply connected]] polyhedron.  Given such a polyhedron, the sum of the vertices and the faces is always the number of edges plus two. i.e.: ''F'' - ''E'' + ''V'' = 2.  The theorem also applies to any planar graph.  For nonplanar graphs, there is a generalization:  If the graph can be embedded in a [[manifold]] ''M'', then ''F'' - ''E'' + ''V'' = &#967;(''M''), where &#967; is the [[Euler characteristic]] of the manifold, a constant which is invariant under continuous deformations.  The Euler characteristic of a simply-connected manifold such as a sphere or a plane is 2.  A generalization of Euler's formula for arbitrary planar graphs exists: ''F'' - ''E'' + ''V'' - ''C'' = 1, where ''C'' is the number of components in the graph.  

In [[1736]] Euler solved a problem known as the [[seven bridges of K�nigsberg]], publishing a paper ''Solutio problematis ad geometriam situs pertinentis'' which may be the earliest application of [[graph theory]] or [[topology]].

'''External Links:'''

* MacTutor biography of Euler: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Euler.html

'''See also:'''
* [[Mathematician]], [[Physicist]], [[Mathematical constants]], [[Complex number]], [[Euler's conjecture]], [[Euler number]]

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====References====
* Adapted from the [[Wikipedia]] article, "Leonhard_Euler" [http://en.wikipedia.org/wiki/Leonhard_Euler http://en.wikipedia.org/wiki/Leonhard_Euler], used under the [[GNU Free Documentation License]]