Leonhard Euler

Quick Info

Born: 15 April 1707, Basel, Switzerland.

Died:  18 September 1783, St Petersburg, Russia.

Introduction

Leonhard Euler (1707-1783) was a Swiss mathematician and physicist who made significant contributions to various branches of mathematics and physics. Euler is widely regarder as one of the greatest mathematicians of all time.

Here are some key aspects of Euler’s work:

• Inventing functions for sine, cosine and tangent.

• Introducing and conceptualizing functions for mathematical and complex analysis.

• Discovering the relationships  between trigonometric functions and exponential functions.

• Creating an entirely new branch of mathematics, analytic Number Theory.

• Establishing and proving a product (known as Euler's product) that is the Riemann Zeta Function.

• First formulating the law of quadratic reciprocity in number theory.

• And more...

Constants and Theorems named after Euler

Constants

• Euler’s Number: 

\begin{equation}  e=2.718281828459... \end{equation}

• Euler-Mascheroni Constant:

\begin{equation}  \gamma =0.577215664901... \end{equation}

• Euler-Gompertz constant:

 \begin{equation} \delta =0.596347362323... \end{equation}

Theorems

• Euler’s theorem : If a and n are coprime positive integers, then

\begin{equation} a^{\phi \left ( n \right )}\equiv 1 \left ( modn \right ) \end{equation}

• Euler's partition theorem, used in combinatorics and numbers theory, states that the amount of partitions of an integer n into odd parts is equal to the numbers  of partitions of n  into distinct parts.

• The Euclid-Euler theorem, which connects Mersenne primes to perfect numbers, states that an even number is perfect if can be written as:

\begin{equation} 2^{p-1}\left ( 2^{p}-1 \right ) \end{equation}

Formulas and identities

• Euler's Formula:

\begin{equation} e^{i\theta }=cos\left ( \theta  \right )+isin\left ( \theta  \right ) \end{equation}

With a special case known as Euler's Identity:

\begin{equation} e^{i\pi  }+1=0 \end{equation}

• Euler’s reflection formula:

 \begin{equation} \Gamma \left ( z \right )\Gamma \left ( 1-z \right )=\frac{\pi }{sin\left ( \pi z \right )} \end{equation}

• Euler’s polyhedron formula:

\begin{equation} V-E+F=2 \end{equation}

• Euler's Totient functions Phi is a method for determining the amount of relatively prime numbers.