Реферат: Leonhard Euler
For example Euler credits Albrecht, Krafft and Lexell for their help with his 775 page work on the motion of the moon, published in 1772. Fuss helped Euler prepare over 250 articles for publication over a period on about seven years in which he acted as Euler's assistant, including an important work on insurance which was published in 1776.
Yushkevich describes the day of Euler's death in:-
On 18 September 1783 Euler spent the first half of the day as usual. He gave a mathematics lesson to one of his grandchildren, did some calculations with chalk on two boards on the motion of balloons; then discussed with Lexell and Fuss the recently discovered planet Uranus. About five o'clock in the afternoon he suffered a brain haemorrhage and uttered only "I am dying" before he lost consciousness. He died about eleven o'clock in the evening.
After his death in 1783 the St Petersburg Academy continued to publish Euler's unpublished work for nearly 50 more years.
Euler's work in mathematics is so vast that an article of this nature cannot but give a very superficial account of it. He was the most prolific writer of mathematics of all time. He made large bounds forward in the study of modern analytic geometry and trigonometry where he was the first to consider sin, cos etc. as functions rather than as chords as Ptolemy had done.
He made decisive and formative contributions to geometry, calculus and number theory. He integrated Leibniz's differential calculus and Newton's method of fluxions into mathematical analysis. He introduced beta and gamma functions, and integrating factors for differential equations. He studied continuum mechanics, lunar theory with Clairaut, the three body problem, elasticity, acoustics, the wave theory of light, hydraulics, and music. He laid the foundation of analytical mechanics, especially in his Theory of the Motions of Rigid Bodies (1765).
We owe to Euler the notation f(x) for a function (1734), e for the base of natural logs (1727), i for the square root of -1 (1777), for pi, for summation (1755), the notation for finite differences y and 2 y and many others.
Let us examine in a little more detail some of Euler's work. Firstly his work in number theory seems to have been stimulated by Goldbach but probably originally came from the interest that the Bernoullis had in that topic. Goldbach asked Euler, in 1729, if he knew of Fermat's conjecture that the numbers 2n + 1 were always prime if n is a power of 2. Euler verified this for n = 1, 2, 4, 8 and 16 and, by 1732 at the latest, showed that the next case 232 + 1 = 4294967297 is divisible by 641 and so is not prime. Euler also studied other unproved results of Fermat and in so doing introduced the Euler phi function (n), the number of integers k with 1 k n and k coprime to n. He proved another of Fermat's assertions, namely that if a and b are coprime then a2 + b2 has no divisor of the form 4n - 1, in 1749.
Perhaps the result that brought Euler the most fame in his young days was his solution of what had become known as the Basel problem. This was to find a closed form for the sum of the infinite series (2) = (1/n2 ), a problem which had defeated many of the top mathematicians including Jacob Bernoulli, Johann Bernoulli and Daniel Bernoulli. The problem had also been studied unsuccessfully by Leibniz, Stirling, de Moivre and others. Euler showed in 1735 that (2) = 2 /6 but he went on to prove much more, namely that (4) = 4 /90, (6) = 6 /945, (8) = 8 /9450, (10) = 10 /93555 and (12) = 69112 /638512875. In 1737 he proved the connection of the zeta function with the series of prime numbers giving the famous relation
(s) = (1/ns ) = (1 - p-s )-1
Here the sum is over all natural numbers n while the product is over all prime numbers.
By 1739 Euler had found the rational coefficients C in (2n) = C2n in terms of the Bernoulli numbers.
Other work done by Euler on infinite series included the introduction of his famous Euler's constant, in 1735, which he showed to be the limit of
1 /1 + 1 /2 + 1 /3 + ... + 1 /n - loge n
as n tends to infinity. He calculated the constant to 16 decimal places. Euler also studied Fourier series and in 1744 he was the first to express an algebraic function by such a series when he gave the result
/2 - x/2 = sin x + (sin 2x)/2 + (sin 3x)/3 + ...
in a letter to Goldbach. Like most of Euler's work there was a fair time delay before the results were published; this result was not published until 1755.
Euler wrote to James Stirling on 8 June 1736 telling him about his results on summing reciprocals of powers, the harmonic series and Euler's constant and other results on series. In particular he wrote [60]:-
Concerning the summation of very slowly converging series, in the past year I have lectured to our Academy on a special method of which I have given the sums of very many series sufficiently accurately and with very little effort.
He then goes on to describe what is now called the Euler- Maclaurin summation formula. Two years later Stirling replied telling Euler that Maclaurin:-
... will be publishing a book on fluxions. ... he has two theorems for summing series by means of derivatives of the terms, one of which is the self-same result that you sent me.
Euler replied:-
... I have very little desire for anything to be detracted from the fame of the celebrated Mr Maclaurin since he probably came upon the same theorem for summing series before me, and consequently deserves to be named as its first discoverer. For I found that theorem about four years ago, at which time I also described its proof and application in greater detail to our Academy.
Some of Euler's number theory results have been mentioned above. Further important results in number theory by Euler included his proof of Fermat's Last Theorem for the case of n = 3. Perhaps more significant than the result here was the fact that he introduced a proof involving numbers of the form a + b-3 for integers a and b. Although there were problems with his approach this eventually led to Kummer's major work on Fermats Last Theorem and to the introduction of the concept of a ring.
One could claim that mathematical analysis began with Euler. In 1748 in Introductio in analysin infinitorum Euler made ideas of Johann Bernoulli more precise in defining a function, and he stated that mathematical analysis was the study of functions. This work bases the calculus on the theory elementary functions rather than on geometric curves, as had been done previously. Also in this work Euler gave the formula
eix = cos x + i sin x.
In Introductio in analysin infinitorum Euler dealt with logarithms of a variable taking only positive values although he had discovered the formula
ln(-1) = i
in 1727. He published his full theory of logarithms of complex numbers in 1751.