Leonhard Euler Research Paper

Leonhard Euler (1707-1783) Switzerland

Euler may be the most influential mathematician who ever lived (though some would make him second to Euclid); he ranks #77 on Michael Hart's famous list of the Most Influential Persons in History. His colleagues called him "Analysis Incarnate." Laplace, famous for denying credit to fellow mathematicians, once said "Read Euler: he is our master in everything." His notations and methods in many areas are in use to this day. Euler was the most prolific mathematician in history and is often judged to be the best algorist of all time. (The ranking #4 may seem too low for this supreme mathematician, but Gauss succeeded at proving several theorems which had stumped Euler.)

Just as Archimedes extended
Euclid's geometry to marvelous heights, so Euler took marvelous advantage of the analysis of Newton and Leibniz: He gave the world modern trigonometry, pioneered (along with Lagrange) the calculus of variations, generalized and proved the Newton-Giraud formulae, etc. He was also supreme at discrete mathematics, inventing graph theory. He also invented the concept of generating functions; for example, letting p(n) denote the number of partitions of n, Euler found the lovely equation: Sn p(n) xn = 1 / ?k (1 - xk)

Euler was also a major figure in number theory: He proved that the sum of the reciprocals of primes less than x is approx.
(ln ln x), invented the totient function and used it to generalize Fermat's Little Theorem, found both the largest then-known prime and the largest then-known perfect number, proved e to be irrational, proved that all even perfect numbers must have the Mersenne number form that Euclid had discovered 2000 years earlier, and much more. Euler was also first to prove several interesting theorems of geometry, including facts about the 9-point Feuerbach circle; relationships among a triangle's altitudes, medians, and circumscribing and inscribing circles; and an expression for a tetrahedron's area in terms of its sides. Euler was first to explore topology, proving theorems about the Euler characteristic, and the famous Euler's Polyhedral Theorem, F+V = E+2 (although it may have been discovered by Déscartes and first proved rigorously by Jordan). Although noted as the first great "pure mathematician," Euler engineered a system of pumps, wrote on philosophy, and made important contributions to music theory, acoustics, optics, celestial motions and mechanics. He extended Newton's Laws of Motion to rotating rigid bodies; and developed the Euler-Bernoulli beam equation. On a lighter note, Euler constructed a particularly "magical" magic
square.

Euler combined his brilliance with phenomenal concentration.
He developed the first method to estimate the Moon's orbit (the three-body problem which had stumped Newton), and he settled an arithmetic dispute involving 50 terms in a long convergent series. Both these feats were accomplished when he was totally blind. (About this he said "Now I will have less distraction.") François Arago said that "Euler calculated without apparent effort, as men breathe, or as eagles sustain themselves in the
wind."

Four of the most important constant symbols in mathematics (p, e, i = v-1, and ? = 0.57721566...) were all introduced or popularized by Euler, along with operators like S. He did important work with Riemann's zeta function ?(s) = ? k-s (although it was not then known by that name); he anticipated the concept of analytic continuation by showing ?(-1) = 1+2+3+4+... = -1/12. As a young student of the Bernoulli family, Euler discovered the striking identity ?(2) = p2/6 This catapulted Euler to instant fame, since the left-side infinite sum (1 + 1/4 + 1/9 + 1/16 + ...) was a famous problem of the time. Among many other famous and important identities, Euler proved the Pentagonal Number Theorem (a beautiful result which has inspired a variety of discoveries), and the Euler Product Formula ?(s) = ?(1-p-s)-1 where the right-side product is taken over all primes p. His most famous identity (which Richard Feynman called an "almost astounding ... jewel") unifies the trigonometric and exponential functions: ei x = cos x + i sin x. (The particular instance ei p+1 = 0 connects the four most important constants almost wondrously.)

Some of Euler's greatest formulae can be combined into curious-looking formulae for p: p2 = - log2(-1) = 6 ?p?Prime(1-p-2)-1/2