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#2**+3 **

Leonhard Euler (1707 - 1783) was a Swiss mathematician, physicist, astronomer, logician, and engineer.

Euler's contributions aren't just limited to one field of mathematics; his important and influential discoveries impacted many branches of science and math such as: physics, astronomy, logic, engineering, infinitesimal calculus, graph theory, topology, analytic number theory, music theory, fluid dynamics, and mechanics.

**Contributions:**

**Mathematical Notation**

He introduced many mathematical notations, such as \(f(x)\) for function, \(e\) for the base of natural logarithm, \(\pi\) for the ratio of circumference to diameter of a circle, and \(i\) for \(\sqrt{-1}\).

**Complex Analysis**

Perhaps his most famous formula of all, is what's known as the Euler's formula, which states for any real number \({\displaystyle \varphi },\) the complex exponential function satisfies \({\displaystyle e^{i\varphi }=\cos \varphi +i\sin}\).

The Euler's identity is a special case of this formula,

\({\displaystyle e^{i\pi }+1=0\,.}\)

This identity takes the five most important numbers in mathematics, and puts them together in this beautiful relationship.

**Applied Mathematics**

Euler facilitated the use of differential equations, the Euler–Mascheroni constant:

\({\displaystyle \gamma =\lim _{n\rightarrow \infty }\left(1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+\cdots +{\frac {1}{n}}-\ln(n)\right).}\)

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 integrate music theory as part of mathematics.

**Number Theory**

"Euler proved Newton's identities, Fermat's little theorem, Fermat's theorem on sums of two squares, and made distinct contributions to the Lagrange's four-square theorem. He also invented the totient function φ(n) which assigns to a positive integer n the number of positive integers less than n and coprime to n. Using properties of this function he was able to generalize Fermat's little theorem to what would become known as Euler's theorem." - Wikipedia

**Analysis **

"The development of 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. Understanding the infinite was the major focus of Euler's research. While some of Euler's proofs may not have been acceptable under modern standards of rigor, his ideas were responsible for many great advances. First of all, Euler introduced the concept of a function, and introduced the use of the exponential function and logarithms in analytic proofs.

Euler frequently used the logarithmic functions as a tool in analysis problems, and discovered new ways by which they could be used. He discovered ways to express various logarithmic functions in terms of power series, and successfully defined logarithms for complex and negative numbers, thus greatly expanding the scope where logarithms could be applied in mathematics. Most researchers in the field long held the view that \({\displaystyle \log(x)=\log(-x)}\) for any positive real\( {\displaystyle x} \)since by using the additivity property of logarithms \({\displaystyle 2\log(-x)=\log((-x)^{2})=\log(x^{2})=2\log(x)}.\) In a 1747 letter to Jean Le Rond d'Alembert, Euler defined the natural logarithm of −1 as \( {\displaystyle x} \) a pure imaginary.

Euler is well known in analysis for his frequent use and development of power series: that is, the expression of functions as sums of infinitely many terms, such as

\({\displaystyle e=\sum _{n=0}^{\infty }{1 \over n!}=\lim _{n\to \infty }\left({\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+\cdots +{\frac {1}{n!}}\right).}\)

Notably, Euler discovered the power series expansions for e and the inverse tangent function

\({\displaystyle \arctan z=\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n+1}}{2n+1}}.}\)

His use of power series enabled him to solve the famous Basel problem in 1735:[5]

\({\displaystyle \lim _{n\to \infty }\left({\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots +{\frac {1}{n^{2}}}\right)={\frac {\pi ^{2}}{6}}.}\)

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 complex analysis. Euler invented the calculus of variations including its most well-known result, the Euler–Lagrange equation.

**Geometry**

In geometry, Euler's theorem states that the distance, d, between the circumcentre and incentre of a triangle is given by

\({\displaystyle d^{2}=R(R-2r)}\)

Euler showed in 1765 that in any triangle, the orthocenter, circumcenter and centroid are collinear.

Source: Wikipedia

I hope this helped,

Gavin

GYanggg Aug 14, 2018