Wikipedia: Leonhard Euler

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Leonhard Euler (1707 - 1783) was a mathematician, physicist and economist. 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 times. He dominated the eighteenth century and deduced many consequences of the then new calculus. He was blind for the last seventeen years of his live.

He is the physicist, who with [Daniel Bernoulli]?, established the law that the torque on a thin elastic beam is proportional to a measure of the 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 a set of laws of motion in fluid dynamics from Newton's laws of motion that state:

The force acting on a small element of a fluid is equal to the rate of change of its momentum.
The torque acting on a small element of a fluid is equal to the rate of change of its angular momentum.

In mathematics, he established his fame early on by solving a long-standing problem:

: 1/1² + 1/2² + 1/3² + 1/4² + ... = π² / 6

He also showed that for all real numbers x,

: e^ix = 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 polynomials. The most remarkable formula in the world is an easy consequence.

He defined the constant gamma?:

: γ = lim_n->∞ ( 1+ (1/2) + (1/3) + (1/4) ... + (1/n) - log(n) )

Its value is approximately 0.5772156 and it is still unknown whether it is rational or irrational, let alone algebraic or transcendental.

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

He made important contributions to number theory as well as to the theory of differential equations.

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.

External Links:

[Euler biography]

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