LEONHARD PAUL EULER (1707 – 1783)
Born

15 April 1707, Basel, Switzerland

Died

18 September 1783 (aged 76), St. Petersburg, Russia

Residence


Nationality


Fields


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Alma mater


Doctoral advisor

Leonhard Paul Euler (15 April, 1707 – 18
September, 1783) was a pioneering Swiss mathematician and physicist who spent most of his life in Russia and Germany. He wrote 886 books
in mathematics. Euler is the father of topology.
Euler made important discoveries in fields as
diverse as calculus and graph 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 renowned for his work in mechanics, optics, and astronomy
EULER CONTRIBUTIONS:
Euler worked in almost all areas of
mathematics: geometry, 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 quarto volumes. Euler's name is associated with
a large
number of topics.
MATHEMATICAL
NOTATION
Ø
f(x): 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.
Ø
Trigonometric
Functions: He also introduced the modern notation for the trigonometric
functions,
Ø
e,i,Σ,π : 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 did not originate with him.
ANALYSIS
Ø
Exponential
Expansion: Euler
is wellknown in analysis for
his frequent use and development of power series, the expression of functions as
sums of infinitely many terms,
Ø Power Series Expansions: Euler discovered the power series expansions for e and the inverse tangent function.
Ø
Complex Exponential
Function: He also
defined the exponential function for complex numbers, and discovered its
relation to the trigonometric functions. For any real number
Ø
Euler created the theory of hyper geometric series,
qseries,
hyperbolic trigonometric functions and the
analytic theory of continued fractions.
NUMBER THEORY
Ø
Prime Number: He found a formula for prime number. Ie. X^{2}X41=one
prime number. If we substitute the value X=1 to 40 then we get the prime
number.
Ø
Mersenne Prime: By 1772 Euler had proved that 2^{31} − 1
= 2,147,483,647 is a Mersenne prime. It
may have remained the largest known prime until 1867.
Ø
Totient Function:
He also invented the totient function
φ(n) which is the number of positive integers less than 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.
Ø
Euler's interest in number theory can be traced to the influence of Christian Goldbach,
his friend in the St. Petersburg Academy. He proved that the sum of the reciprocals of the primes diverges.
Euler proved Newton's identities,
Fermat's little
theorem, Fermat's
theorem on sums of two squares,
GEOMETRY
Ü
Euler line , Euler's circle
GRAPH THEORY
Seven Bridges of
Königsberg: In 1736, Euler solved 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 which 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: there is no Eulerian circuit. This solution is
considered to be the first theorem of graph theory, specifically of planar graph theory.
V − E + F = 2 : Euler also discovered the formula V − E + F = 2 relating the number of
vertices, edges, and faces of a convex polyhedron and hence of a planer graph.
PHYSICS
AND ASTRONOMY
 EulerBernoulli Beam Equation: Euler helped develop the EulerBernoulli beam equation, which became a cornerstone of engineering.
 His work in astronomy was recognized by a number of Paris Academy Prizes over the course of his career. His calculations also contributed to the development of accurate longitude tables. In addition, Euler made important contributions in optics.
APPLIED MATHEMATICS:
·
F_{5}: He
found the symbol of F_{5 }ie. F_{5
}=4,294,967,297 = 641 X 6 700 417
·
Music: 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.
Good think and thanks to help my school work
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