Lecture 30. Euler, Our Master in Everything

Figure 30.1 Leonhard Euler and Riehen

Leonhard Euler Leonhard Euler(1707-1783) was a Swiss mathematician who madeenormous contributions to a wide range of mathematics and physics including analytic ge-ometry, trigonometry, geometry, calculus and number theory. Euler was the most prolificmathematical writer of all times finding time to publish over 800 papers in his lifetime.

Leonhard Eulers father Paul Euler was a friend of Bernoulli family (see Lecture 29).In fact Paul Euler had studied theology at the University of Basel and had attended JacobBernoullis lectures there.

Leonhard Euler was born in Basel and was brought up in Riehen, not far from Basel.

Living with his grandmother, Leonhard went to a local school in Basel. This school wasso poor that Euler learned no mathematics at all from the school. Paul Euler had somemathematical training and he was able to teach his son elementary mathematics along withother subjects. As a result, Leonhards interest in mathematics was sparked by his fathersteaching, and he read mathematics texts on his own and took some private lessons.

Eulers father wanted his son to follow him into the church and sent him to the Universityof Basel to prepare for the ministry. As he age of 14, Euler entered the University in 1720.Johann Bernoulli soon discovered Eulers great potential for mathematics. Euler wrote asfollows:

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... I soon found an opportunity to be introduced to a famous professorJohann Bernoulli. ... True, he was very busy and so refused flatly to give meprivate lessons; but he gave me much more valuable advice to start reading moredifficult mathematical books on my own and to study them as diligently as Icould; if I came across some obstacle or difficulty, I was given permission to visithim freely every Sunday afternoon and he kindly explained to me everything Icould not understand ...

In 1723 Euler completed his Masters degree in philosophy. As his fathers wish, he beganhis study of theology in the same year, but he could not find the enthusiasm for the studyof theology. After persuaded by Johann Bernoulli, his father agreed to let him change tomathematics.

Euler completed his studies at the University of Basel in 1726 where he had studiedmathematics. By 1726 Euler had already a paper in print. In 1727 he published anotherarticle and submitted an entry for the 1727 Grand Prize of the Paris Academy. Eulers essaywon him second place which was a very good achievement for the young graduate. SinceNicolaus(II) Bernoulli died in St Petersburg in July 1726, it created a vacancy at ImperialRussian Academy of Sciences in St Petersburg. Euler was offered the post, and he acceptedthe post.

Figure 30.2 St Petersburg and Imperial Russian Academy of Sciences there

He had a phenomenal memory, and once did a calculation in his head to settle anargument between students whose computations differed in the fiftieth decimal place. Eulerlost sight in his right eye in 1735, and in his left eye in 1766. Nevertheless, aided by hisphenomenal memory (and having practiced writing on a large slate when his sight was failing

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him), he continued to publish his results by dictating them. In his life time, Euler publishedover 800 papers. He won the Paris Academy Prize 12 times. When asked for an explanationwhy his memoirs flowed so easily in such huge quantities, Euler is reported to have repliedthat his pencil seemed to surpass him in intelligence. Francois Arago 1 said of him hecalculated just as men breathe, as eagles sustain themselves in the air.

Euler was the director of the Berlin Academys mathematics section. With numerousmathematica papers, he became recognized as the primier mathematician of Europe.

In 1766 Euler returned to the St. Petersburg Academy and spent the rest of his life inRussia. Euler passed away in St. Petersburg on Stetember 18, 1983.

Figure 30.3 Eulers grave at the Alexander Nevsky Lavra

Eulars mathematical contribution Euler worked in every field of mathematics whichexisted in his day. Many of his results are of fundamental interest. Eulers name is associatedwith a large number of topics. Here are some of his works.

Prestigious textbooks Euler not only published his results in articles of variedlength, but also in impressive number of large textbooks.

In several fields Eulers presentation has been almost final. An example is our presenttrigonometry with it conception of trigonometric values as ratios and its useful nota-

1Arago (1786 - 1853) was an important French mathematician.

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tion, which dates from Eulers Introductio in analysin infinitorum (1748). The tremen-dous prestige of his textbooks settled forever many notations. Lagrange, Laplace andGauss followed Euler in all their works.

Our notation is almost Eulers Euler introduced and popularized several nota-tional conventions through his numerous and widely circulated textbooks, notations.He sharpened the concept of a function and was the first to denote by () the function of a variable . He used to denote the base of the natural logarithm (now alsoknown as Eulers number). He used

to denote summations. He used to denote the

imaginary unit1. He used to denote the ratio of a circles circumference to its di-

ameter. He also introduced the modern notation for the trigonometric functions. MITProfessor Struk said2: Since Eulers Latin is very simple and his notation is almostmodern or perhaps we should better say that our notation is almost Eulers.

Basel problem Thanks to the influence of Bernoullis family, studying calculusbecame the major focus of Eulers work. His ideas led to many great advances. Euleris well-known for his frequent use and development of power series, the expression offunctions as sums of infinitely many terms, such as the power series expansions for and for the inverse tangent function.

Figure 30.4 Euler

The Basel problem is a famous problem in number theory, first posed by Pietro Mengoliin 1644, which asks for the precise summation of the series

1

2

2D. Struk, A Concise History of Mathematics, fourth edt., Dover Publications, Inc., 1987, p.124.

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The series is approximately equal to 1.644934. Euler found the exact sum to be 2

6

and announced this discovery in 1735. Since the problem had attracted the leadingmathematicians of the day, Eulers solution brought him immediate fame when he wastwenty-eight.

The most remarkable formula In 1697, Jacob Bernoulli studied lim(1+ 1),which had been implicit in earlier work on natural logarithms. In 1748, Euler definedthe two functions:3

= lim

(1 +

), = lim

(

1). (1)

and also proved his famous formula:

= + (2)

Euler proved that the infinite series (2) of both sides being equal. 50 years later,the view of complex numbers as points in the complex plane arose. Until Euler thetrigonometrical quantities sine, cosine, tangent, etc., were regarded as lines connectedwith the circle rather than functions. Even the derivation of the series expansion forthe sine in dependence of the arc by Newton and Leibniz did not change this view. Itwas Euler who introduced the functional point of view. 4 A special case of the aboveformula is known as Eulers identity:

+ 1 = 0.

called the most remarkable formula in mathematics by Richard Feynman because ituses of the notions of

addition, multiplication, exponentiation, 0, 1, e, i, , sine, and cosine.

In 1988, readers of the Mathematical Intelligence voted it the Most Beautiful Math-ematical Formula Ever. In total, Euler was responsible for three of the top fiveformulas in that poll.5

3Morris Kline, Mathematical Thought from Ancient to Modern Times, volume 2, New York Oxford,Oxford University Press, 1972, p.404.

4Hans Niels Jahnke (editor), A History of Analysis, AMS, 2003, p.115-116.5Here is the list of the top five: 1. Eulers formula; 2. Eulers formula for a polyhedron: + = + 2.

3. The number of primes is infinite. 4. There are 5 regular polyhedrons. 5. 1 + 122 +132 +

142 + ... =

2

6(Euler). cf., David Wells, Are these the most beautiful? Mathematical Intelligencer 12(3)(1990), 37-41.

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Figure 30.5 Eulers formulas

How to define log a? It was a difficulty problem how to define when is anegative number. Between 1712 and 1713, Bernoulli held the view that = ()because () =

, while Leibniz believed that () must be imaginary. Between

1727 and 1731, the question was taken up by Bernoulli and Euler without at a solution.It was only in the period between 1749 and 1751, Euler developed ideas far enough tostudy logarithm so that it leads to a satisfying solution. Eulers approach is to dealwith logarithm of general complex numbers. Eulers argument is as follows: by thedefinition (1), = lim (

1). For each positive integer , ( 1) has

different roots (Euler realized it!) so that the limit indeed has infinitely many differentvalues. When = 1, he got 1 = 2, = 1, 2, 3, ....6In 1735, Euler define a constant:

:= lim

( =1

1

)which is the well-known Euler constant. Its numerical value to 50 decimal places is

0.57721566490153286060651209008240243104215933593992

6Hans Niels Jahnke (editor), A History of Analysis, AMS, 2003, p.117-118.

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Gamma functions Euler introduced the gamma function:

() :=

0

1

which has the property: () = ( 1)! so that it is a natural generalization of thefactorial.

Complex analysis He also found a way to calculate integrals with complex limits,foreshadowing the development of modern complex analysis.

Calculus of variations He invented the calculus of variations including its best-known result, the Euler - Lagrange equation.

Number theory Euler proved that that the divergence of the harmonic seriesimplied an infinite number of Primes. He factored the fifth Fermat number (thusdisproving Fermats conjecture), proving Fermats lesser theorem, and showing that was irrational.

Three-body problem In 1772, he introduced a synoptic coordinates (rotating) co-ordinate system to the study of the three-body problem (especially the Moon ).

Topology The Euler characteristic was classically defined for the surfaces of poly-hedral, according to the formula

= +

where , , and are respectively the numbers of vertices (corners), edges and facesin the given polyhedron.

Any convex polyhedrons surface has Euler characteristic 2. This result is known asEulers formula.

Graph theory Konigsberg was a city in Prussia situated on the Pregel River,which served as the residence of the dukes of Prussia in the 16th century (Today,the city is named Kaliningrad, and is a major industrial and commercial center ofwestern Russia). The river Pregel flowed through the town, creating an island, as inthe following picture.

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Figure 30.6 Seven Bridges of Konigsberg

A famous problem concerning Konigsberg was whether it was possible to take a walkthrough the town in such a way as to cross over every bridge once, and only once.All who tried ended up in failure, including Euler. However, Euler did succeed inexplaining why such a journey was impossible, not only for the Konigsberg bridges,but whether such a journey was possible or not for any network of bridges anywhere.

This is the earliest work on graph theory by Leonhard Euler in 1736.

Ordinary differential equations Euler invented the Euler method in ODE. Itis a first order numerical procedure for solving ordinary differential equations (ODEs)with a given initial value. It is the most basic kind of explicit method for numericalintegration for ordinary differential equations. In Eulers textbook Institutiones calculiintegralis, the section of linear, exact and homogeneous equations is still themodel of our elementary texts on the this subject.

Others He also made major contributions in optics, mechanics, electricity, and mag-netism, astronomy, hydraulics, ship construction, artillery ......

In a testament to Eulers proficiency in all branches of mathematics, the great Frenchmathematician and celestial mechanic Laplace told his students, Read Euler, read Euler,he is our master in everything.

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