December 22 was Srinivasan Ramanujan's 114th birth anniversary. KRISHNASWAMI ALLADI pays tribute by describing the life and contributions of the great 18th Century mathematician Euler, and discusses connections with Ramanujan's work.
LEONHARD EULER was one of the greatest mathematicians of all time, on par with Gauss, Newton and Archimedes. Euler's fundamental contributions have influenced almost every branch of mathematics. He was without doubt the most prolific mathematician in history. Indeed, even the blindness that physically handicapped him in his later years did not diminish his furious productivity. Euler was a master of infinite series and products. Eric Temple Bel in his famous book Men of Mathematics says that with regard to the ease and brilliance with which Euler manipulated infinite processes, only Ramanujan rivalled him. Similarly, G. H. Hardy of Cambridge University compared Ramanujan to Euler and Jacobi for sheer manipulative ability.
Euler was born in 1707 near Basel, Switzerland. He was a precocious youth blessed with a gift of languages and extraordinary memory. Dunham says "Euler... carried in his head an assortment of curious information, including orations, poems and lists of prime powers. He was a fabulous mental calculator, able to perform arithmetical computations without benefit of pencil or paper." Ramanujan too had a prodigious memory.
Upon joining the University of Basel at the age of 14, Euler came in contact with its most famous professor, Johann Bernoulli (1667-1748). Bernoulli was much impressed with Euler and permitted the young lad to have frequent discussions with him. Very soon, the role of mentor and pupil was reversed and as Euler matured it was he who was showing the way for Bernoulli. In the case of Ramanujan, Hardy says that a single professor like him was insufficient for so fertile a pupil.
Euler's university education was not restricted to mathematics. He studied the history of law and obtained a masters degree in philosophy. Then he entered divinity school to become a priest. This required him to study Greek and Hebrew, but the call of mathematics was too strong so he quit divinity school. In the case of Ramanujan, mathematics was an obsession to the extent that he neglected other subjects totally and so did not even earn a college degree.
In 1725, Daniel Bernoulli, son of Johann, was appointed to a position in mathematics at the newly formed St. Petersburg Academy in Russia, and he invited Euler to join. The only opening at that time was in physiology/ medicine, and since Euler did not have any other option, he accepted that position. Happily for Euler, by the time he arrived in St. Petersburg in 1727, he found that he had been reassigned to physics. Euler stayed at the home of Daniel Bernoulli, and the two engaged in elaborate discussions of physics and mathematics that eventually influenced the development of these subjects in Europe. In 1733 Daniel Bernoulli returned to Switzerland for an academic position. While this was a loss to Euler, it enabled him to occupy the position at St. Petersburg Academy that Daniel Bernoulli had vacated.
It was in St. Petersburg that he had his first great triumph by solving the notorious Busel Problem. The question was to determine the exact value of the convergent infinite series obtained by summing the reciprocals of the squares of the positive integers. The problem was posed by Pietro Mengoli in 1644, but it was Jakob Bernoulli, Johann's brother, who brought it to the attention of the broader mathematical community in 1699.
With the Basel Problem behind him, Euler began writing research papers at a furious pace. Nearly half the papers published by the St. Petersburg Academy turned out to be his work. But Russia got into a serious political turmoil with the death of Empress Catherine I. So Euler accepted an offer from Prussia's Fredrick the Great, and made a move in 1741 to take up a position at the newly revitalised Berlin Academy.
Euler was at the Berlin Academy for a quarter century. It was the middle phase of his mathematical career. It was during this period that he published his two greatest works Introductio in analysin infinitorum, a 1748 text on functions, and Institutiones calculi differentialis, a 1755 volume on differential calculus. It was also in Berlin that he discovered the famous Euler's Identity giving the value of the exponential function in terms of the trigonometric functions sine and cosine.
While in Berlin, Euler maintained a cordial relationship with the St. Petersburg Academy and published numerous papers in their journal. He continued to receive a stipend from St. Petersburg even during the period when Russian forces entered Berlin during the Seven Year War. It turned out that over a period of time, the relationship between Euler and Fredrick the Great worsened because Euler spoke German, and business in Prussia was conducted in French. Even though Euler had brought considerable fame to the Berlin Academy, he was forced to leave. The political climate in Russia had improved considerably under Catherine the Great, and the St. Peterburg Academy was ready in 1766 to welcome back Euler.
A few years after his return to Russia, two tragedies struck Euler. The vision problems that began in St. Petersburg prior to moving to Berlin had worsened considerably, and in 1771 Euler became totally blind. Also in 1773, his wife Katerina died. Undeterred by these setbacks, his mind continued to pour out mathematical theorems in rapid succession. In fact during this period of blindness, he wrote a 775-page treatise on Algebra. Three years after the death of his wife, Euler married her half sister, who was with him until his death on September 18, 1783. Euler was in full control of his mathematical powers until the final moments of his last day. That morning, after spending some time with his grandchildren, Euler took up some mathematical questions concerning the flight of balloons, spurred by the exciting news of the Montgolfer brothers' balloon flight over Paris. In the afternoon, he made some calculations on the behaviour of Uranus. The peculiar orbit of this planet attracted Euler. Later generation astronomers used Euler's preliminary calculations to seek and discover the next planet Neptune. If Euler had lived longer, he probably would have predicted mathematically the existence of a planet beyond Uranus. He suffered a massive haemorrhage that day and died immediately.
Euler's Constant: It is now a well known fact to any student of calculus, that the harmonic series, namely, the infinite series obtained by summing the reciprocals of the positive integers, diverges. In the days of Euler, there was no clear understanding of the notions of convergence and divergence. The rigorous approach to convergence and divergence came with Cauchy. In Euler's works there are statements on convergence and divergence which would either be taken as incorrect or incomplete statements by present day rigour. However, like Ramanujan, Euler knew what he was stating, and how to deal with such situations. Indeed, as in the case of identities in Ramanujan's notebooks, if Euler's statements are interpreted properly, then they are not only correct, but very significant. The situation with harmonic series highlights this point.
Euler asserts that the sum of the harmonic series equals the natural logarithm of infinity plus a quantity that is nearly a constant. What Euler is saying is that if the harmonic series are summed, then the value of this is approximated by the natural logarithm of x. More precisely, he says, that if the natural logarithm of x is subtracted from the sum of the first x terms, then the difference tends to a constant as x approaches infinity. This constant is known today as Euler's Constant. By the study of the harmonic series and related sums, Euler systematically developed an important method of summation of series which is now known as Euler-Maclaurin summation.
Ramanujan had his own theory of summing infinite series. To an infinite series convergent or not, he would associate a value he called the constant of the series.
It is amazing that Ramanujan who (according to Hardy) did not posses a clear understanding of complex function theory, came up with this value - 1/12 by his theory of constants for infinite series. Ramanujan's theory of constants has connections with the Euler-Maclaurin summation.
The taxicab equation: Ramanujan was mainly interested which had solutions and in providing formulas for the solutions. This could be a reason why he was not interested in Fermat's Last Theorem. The famous Ramanujan taxicab number 1729 is an example of an integer which is expressible as a sum of cubes in two different ways (1729 is the sum of the cubes of 12 and 1, as well as the sum of the cubes of 10 and 9).
Euler's Identity: While in Berlin, Euler made the fundamental discovery, that the exponential of an imaginary angle could be expressed as cos + isin. This is called Euler's Identity, and is pivotal to the understanding of complex numbers. Euler's identity provides a fundamental link between the Cartesian and polar representations via the exponential function.
Ramanujan studied the sum of the powers of the primitive n-th roots of unity and established several interesting properties for them. These sums are now called Ramanujan sums. For example, the sum of all the primitive n-th roots of unity takes only the values 1, 0, or - 1, and is equal to the famous Mocbius function of n.
Partitions: The theory of partitions founded by Euler has today become a central topic of research owing to its interactions with many fields within and outside of mathematics. A partition of a positive integer n is a representation of that integer a sum of positive integers, two such representations being considered the same if they differ only in the order of the parts. Euler was interested in partitions because by means of their generating function representations, he could prove beautiful results. These generating functions are infinite power series, and Euler was a master in manipulating them.
One fundamental result he proved using generating functions was that the number of partitions of any integer into distinct (non-repeating) parts equals the number of partitions of that integer into odd parts (that could repeat). Similarly, Ramanujan too stated his results as identities for power series and did not emphasise the partition theoretic or combinatorial significance.
In summary, Euler who had an unbelievably productive career spanning more than half a century, made fundamental contributions to several branches of mathematics. In the case of the theory of partitions, it is no exaggeration to say that while Euler was its illustrious founder, it was under Ramanujan's magic touch that the subject underwent a glorious transformation. E. T. Bell in his classic Men of Mathematics, begins his description of the life of Euler with a quote of Arago "Euler could calculate with no apparent effort as men breathe or as eagles sustain themselves in the wind." I think the same could be said of the Indian mathematical genius Srinivasa Ramanujan.
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