Leonhard Euler Contribution

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Abstract— This paper is on the Swiss astronomer, physicist, engineer and mathematician Leonhard Euler. Euler made fundamental contributions to the world of mathematics and science. His mathematical studies range from analytic geometry, infinitesimal calculus, graph theory and topology. He introduced many of the notations one uses in today’s modern mathematics. This paper will focus on Leonhard Euler’s life and some of his scientific and mathematical works.

Index Terms—Calculus, Geometry, Leonhard Euler, Number Theory.

Introduction
Leonhard Euler was a Swiss physicist, astronomer and mathematician. Euler is one of the greatest mathematicians of the 18th century. He made great contributions to many areas of mathematics such as geometry, calculus,
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1) was born in Basel, Switzerland on April 15, 1707. He is the son of a minister. Euler’s father wanted his son to follow him into the church [6]. In 1720, Euler enrolled at the University of Basel to study theology, Greek and Hebrew, as his father wanted him to. However, with much persuasion, Johann Bernoulli convinced Euler’s father to let Euler pursue mathematics. Johann Bernoulli being a prominent mathematician at that time became Euler’s teacher.
In 1727, Euler joined St. Petersburg Academy of Science as a physics professor. Within the next two years, Euler married and had 13 children; unfortunately, only 5 survived their infancy. He claimed that he made some of his greatest discoveries while holding a baby on his arm [6]. In 1738, Euler became blind from one eye and he was in danger of losing the other.
Euler left St. Petersburg in 1741 to take up a post at the Berlin Academy of Science, where he remained for 25 years [3]. Within his stay in Berlin, Euler wrote over 200 articles. Two of his most famous works are Introductio in analysin infinitorum, a text on functions, and Institutiones calculi differentialis on differential calculus
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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 [3].

Infinitesimal Calculus
Since infinitesimal calculus had just been developed in the 17th century, the majority of Euler’s work focused on the study of calculus. Euler is well known in calculus for his frequent use and development of power series, the expression of functions as the sum of infinitely many terms, such as e^x=∑_(n=0)^∞▒〖x^n/n!=lim┬(n→∞)⁡〖(1/0!+x/1!+x^2/2!+⋯+x^n/n!)〗 〗 [3]. He then proved this expansion for e and the inverse tangent function. The series expansion also led him to solve the Basel problem in his number theory studies.
Euler stated that mathematical analysis was the study of functions in is Introductio in analysin infinitorum. This work bases the calculus on the theory of elementary functions and it gives Euler’s formula e^ix=cos⁡〖x+i sin⁡x 〗 [5].
The analysis infinitorum was followed by the Institutiones calculi differentialis. This work was published in 1755 and it focused on finite differences. The work makes a thorough investigation of how differentiation behaves under substitution

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