Leonhard Euler
Leonhard
Euler (15 April 1707 – 18
September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer who made important and influential discoveries in
many branches of mathematics like infinitesimal calculus and graph theory while also making pioneering contributions to
several branches such as topology and analytic number theory. He also introduced much of the modern mathematical terminology
and notation, particularly
for mathematical analysis, such as
the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory.
Euler was
one of the most eminent mathematicians of the 18th century and is held to be
one of the greatest in history. He is also widely considered to be the most
prolific mathematician of all time. His collected works fill 60 to 80 quartovolumes, more than anybody in the field. He spent most of
his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia.
A statement
attributed to Pierre-Simon Laplace expresses
Euler's influence on mathematics: "Read Euler, read Euler, he is the
master of us all."
Contributions to mathematics and physics
Euler worked in almost all areas of
mathematics, such as geometry, infinitesimal calculus, trigonometry, algebra,
and number theory,
as well as continuum physics, lunar theory and
other areas of physics.
He is a seminal figure in the history of mathematics; if printed, his works,
many of which are of fundamental interest, would occupy between 60 and 80 quartovolumes. Euler's
name is associated with a large
number of topics.
Euler
is the only mathematician to have two numbers named after him:
the important Euler's
number in calculus, e,
approximately equal to 2.71828, and the Euler–Mascheroni
constant γ (gamma)
sometimes referred to as just "Euler's constant", approximately equal
to 0.57721. It is not known whether γ is rational or irrational.
Mathematical notation
Euler
introduced and popularized several notational conventions through his numerous
and widely circulated textbooks. Most notably, he introduced the concept of
a function and
was the first to write f(x) to denote the function f applied
to the argument x. He also introduced the modern notation for
the trigonometric
functions, the letter e for the base
of the natural logarithm (now
also known as Euler's number),
the Greek letter Σ for
summations and the letter i to denote the imaginary unit. The
use of the Greek letter π to
denote the ratio
of a circle's circumference to its diameter was
also popularized by Euler, although it originated with Welsh mathematician William
Jones.
Analysis
The
development of infinitesimal
calculus was at the forefront of 18th-century
mathematical research, and the Bernoullis—family friends of Euler—were responsible for much
of the early progress in the field. Thanks to their influence, studying
calculus became the major focus of Euler's work. While some of Euler's proofs
are not acceptable by modern standards of mathematical
rigour (in
particular his reliance on the principle of the generality
of algebra), his ideas led to many great advances.
Euler is well known in analysis for
his frequent use and development of power series,
the expression of functions as sums of infinitely many terms, such as
Notably,
Euler directly proved the power series expansions for e and
the inverse tangent function.
(Indirect proof via the inverse power series technique was given by Newton and Leibniz between
1670 and 1680.) His daring use of power series enabled him to solve the
famous Basel problem in
1735 (he provided a more elaborate argument in 1741):
Euler
introduced the use of the exponential
function and logarithms in
analytic proofs. He discovered ways to express various logarithmic functions
using power series, and he successfully defined logarithms for negative
and complex numbers,
thus greatly expanding the scope of mathematical applications of logarithms. He
also defined the exponential function for complex numbers, and discovered its
relation to the trigonometric
functions. For any real number φ (taken
to be radians), Euler's formula states
that the complex
exponential function satisfies called "the most
remarkable formula in mathematics" by Richard P.
Feynman, for its single uses of the notions of
addition, multiplication, exponentiation, and equality, and the single uses of
the important constants 0, 1, e, i and π. In
1988, readers of the Mathematical
Intelligencer voted it "the Most Beautiful
Mathematical Formula Ever". In
total, Euler was responsible for three of the top five formulae in that poll.
In
addition, Euler elaborated the theory of higher transcendental
functions by introducing the gamma function and
introduced a new method for solving quartic equations. He also found a way to calculate integrals
with complex limits, foreshadowing the development of modern complex analysis.
He also invented the calculus
of variations including its best-known result,
the Euler–Lagrange
equation.
Euler
also pioneered the use of analytic methods to solve number theory problems. In
doing so, he united two disparate branches of mathematics and introduced a new
field of study, analytic
number theory. In breaking ground for this new field,
Euler created the theory of hypergeometric series, q-series, hyperbolic
trigonometric functions and the analytic theory of continued
fractions. For example, he proved the infinitude of
primes using the divergence of the harmonic
series, and he used analytic methods to gain some
understanding of the way prime numbers are
distributed. Euler's work in this area led to the development of the prime number
theorem.
Number theory
Euler's
interest in number theory can be traced to the influence of Christian
Goldbach, his friend in the St. Petersburg Academy. A
lot of Euler's early work on number theory was based on the works of Pierre de Fermat.
Euler developed some of Fermat's ideas and disproved some of his conjectures.
Euler
linked the nature of prime distribution with ideas in analysis. He proved
that the sum of the reciprocals of the
primes diverges. In doing so, he discovered the connection
between the Riemann
zeta function and the prime numbers; this is known as
the Euler product formula for the Riemann
zeta function.
Euler
proved Newton's
identities, Fermat's
little theorem, Fermat's
theorem on sums of two squares,
and he made distinct contributions to Lagrange's
four-square theorem. He also invented the totient function φ(n), the number of positive
integers less than or equal to the integer n that are coprime to n. Using properties of this
function, he generalized Fermat's little theorem to what is now known as Euler's theorem.
He contributed significantly to the theory of perfect numbers,
which had fascinated mathematicians since Euclid.
He proved that the relationship shown between perfect numbers and Mersenne primes earlier proved by Euclid was one-to-one,
a result otherwise known as the Euclid–Euler
theorem. Euler also conjectured the law of quadratic
reciprocity. The concept is regarded as a fundamental
theorem of number theory, and his ideas paved the way for the work of Carl Friedrich
Gauss. By
1772 Euler had proved that 2 − 1 = 2,147,483,647 is
a Mersenne prime. It may have remained the largest known
prime until 1867.
Graph theory
In
1735, Euler presented a solution to the problem known as the Seven
Bridges of Königsberg. The
city of Königsberg, Prussia was
set on the Pregel River, and included two large islands that were
connected to each other and the mainland by seven bridges. The problem is to
decide whether it is possible to follow a path that crosses each bridge exactly
once and returns to the starting point. It is not possible: there is no Eulerian circuit. This solution is considered to be the first
theorem of graph theory,
specifically of planar graph theory.
Euler
also discovered the formula relating the number of
vertices, edges and faces of a convex polyhedron, and
hence of a planar graph.
The constant in this formula is now known as the Euler
characteristicfor the graph (or other mathematical object),
and is related to the genus of
the object. The
study and generalization of this formula, specifically by Cauchy and L'Huilier, is
at the origin of topology.
Applied mathematics
Some
of Euler's greatest successes were in solving real-world problems analytically,
and in describing numerous applications of the Bernoulli numbers, Fourier series, Euler numbers,
the constants e and π,
continued fractions and integrals. He integrated Leibniz's differential
calculus with Newton's Method of
Fluxions, and developed tools that made it easier to
apply calculus to physical problems. He made great strides in improving
the numerical
approximation of integrals, inventing what are now
known as the Euler
approximations. The most notable of these approximations
are Euler's method and
the Euler–Maclaurin
formula. He also facilitated the use of differential
equations, in particular introducing the Euler–Mascheroni
constant:
One
of Euler's more unusual interests was the application of mathematical ideas in
music. In 1739 he wrote the Tentamen novae theoriae musicae, hoping
to eventually incorporate musical theory as
part of mathematics. This part of his work, however, did not receive wide
attention and was once described as too mathematical for musicians and too
musical for mathematicians.
Physics and astronomy
Euler helped develop the Euler–Bernoulli
beam equation, which became a cornerstone of engineering.
Aside from successfully applying his analytic tools to problems in classical
mechanics, Euler also applied these techniques to
celestial problems. His work in astronomy was recognized by a number of Paris
Academy Prizes over the course of his career. His accomplishments include
determining with great accuracy the orbits of comets and other celestial
bodies, understanding the nature of comets, and calculating the parallax of
the sun. His calculations also contributed to the development of accurate longitude tables.
In
addition, Euler made important contributions in optics.
He disagreed with Newton's corpuscular
theory of light in the Opticks, which was then the prevailing theory. His 1740s
papers on optics helped ensure that the wave theory of
lightproposed by Christiaan Huygens would
become the dominant mode of thought, at least until the development of
the quantum
theory of light.
Logic
Euler
is also credited with using closed curves to
illustrate syllogistic reasoning
(1768). These diagrams have become known as Euler diagrams.
An
Euler diagram is a diagrammatic means
of representing sets and
their relationships. Euler diagrams consist of simple closed curves (usually
circles) in the plane that depict sets.
Each Euler curve divides the plane into two regions or "zones": the
interior, which symbolically represents the elements of
the set, and the exterior, which represents all elements that are not members
of the set. The sizes or shapes of the curves are not important; the
significance of the diagram is in how they overlap. The spatial relationships
between the regions bounded by each curve (overlap, containment or neither)
corresponds to set-theoretic relationships (intersection, subset and disjointness). Curves whose interior zones do not intersect
represent disjoint sets.
Two curves whose interior zones intersect represent sets that have common
elements; the zone inside both curves represents the set of elements common to
both sets (the intersection of
the sets). A curve that is contained completely within the interior zone of
another represents a subset of
it. Euler diagrams were incorporated as part of instruction in set theory as
part of the new math movement
in the 1960s. Since then, they have also been adopted by other curriculum
fields such as reading.
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