The Euler we never knew and his contributions in mathematics

 
 

Leornard Euler

He was known as the greatest mathematician of the eighteenth century, Leonard Euler(1707-1783), grew up in the city of Basel in Switzerland and he was a student of Johann Bernoulli, he followed his friend Daniel Bernoulli to St Petersburg in 1727. Daniel Bernoulli is a son of johann, migrated to Basel in 1733 as Professor of Botany and later, physics. his interests were primarily on differential equations and their applications.
Euler is regarded as the most prolific mathematician of all time, his collected works fills more than 70 large volumes.his interests ranged over all the fields of mathematics and their applications.
Even though he was blind during the last 17years of his life, his works continues undiminished until the very last breadth of his life.
Among his notable works in differential equations was giving the general solution of a differential equation.
he also proposed a numerical procedure in 1768-1769,he made important contribution in partial differential equations and gave the first symmetric treatment of the calculus of variation.
After a while Euler's formula was disputed by a mathematician named Carl Heun who suggested that Euler's formula was not accurate in solving numerical solutions of ordinary differential equations, he suggested that a better formula can be obtained and the formula was later called improved euler's formula or Heun's formula.
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 did not originate with him.
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

${\displaystyle e^x=\sum _{n=0}^{\mathrm{\infty }}\frac{x^n}{n!}=\underset{n\to \mathrm{\infty }}{lim}(\frac{1}{0!}+\frac{x}{1!}+\frac{x^2}{2!}+\cdots +\frac{x^n}{n!}).}$

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) 

${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}=\underset{n\to \mathrm{\infty }}{lim}(\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots +\frac{1}{n^2})=\frac{{\pi }^2}{6}.}$

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.

Euler's success in mathematics is so large and wide that it can not be put into written with just a few pages of note but rather would require volumes of books.

Earlier mathematicians like Euler would never be forgotten in history as men of their courage would be difficult to find in this 21st century.