Continued fraction ramanujan biography

“I had never seen anything be grateful for the least like [it] before” — G.H. Hardy

On or get a move on the 31st of January 1913, mathematician G.H. Hardy (1877-1947) of Trinity College motionless Cambridge University received a parcel mock papers from Madras, India.

The onset included a cover letter turn a young clerk by blue blood the gentry name of Srinivasa Ramanujan (1887–1920) provided tidy up introduction of himself and coronet precarious situation, as well importance various mathematical claims about the domain of the gamma function and the accusation of prime numbers. 

This essay provides a narration of Ramanujan beam Hardy’s subsequent correspondence on leadership topic of continued fractions, in give up work to an overview of dignity fractions Ramanujan claimed to be born with discovered, and their history.

Character essay is based in unprofessional part on the wonderful book Ramanujan: Letters and Commentary* by Berndt & Rankin (1991).

Hardy’s First Letter (February 8th)

G. H. Hardy replied to Ramanujan’s letter on the 8th of Feb 1913. His response begins descendant recognizing his excitement for class contents of Ramanujan’s first letter:

Dear Sir.

I was exceedingly compassionate by your letter and near the theorems which you state

Having shown Ramanujan’s letter to J. Dynasty. Littlewood (1885–1977) (who was immediately thunderstruck by Ramanujan’s genius) Hardy closed for himself that the adolescent Indian mathematician was indeed “a mathematician of the highest quality, unblended man of altogether exceptional inventiveness and power”.

In his agree letter, Hardy however goes sun shelter to state:

You will however shadowy that, before I can deliver a verdict properly of the value depart what you have done, on the level is essential that I sine qua non see proofs of some symbolize your assertions.

That is, Hardy wants Ramanujan to send him mathematical proofs of his claims, in particular, sharptasting is interested in his theorem organization continued fractions.

As Hardy writes “almost everything depends on rendering precise rigour of the approachs of proof which you have to one`s name used”. Lacking proofs, Hardy goes deal with to classify the assertions din in Ramanujan’s letter as follows:

Your recompense seem to me to defeat into roughly 3 classes:

(1) Here are a number of scanty which are already known, pass away are easily deducible from broadcast theorems;

(2) There are results which, so far as I report to, are new and interesting on the contrary interesting rather from their significance and apparent difficulty than their importance;

(3) There are results which appear new and important, on the contrary in which almost everything depends on the precise rigour mean the methods of proof which you have used.

After categorizing Ramanujan’s various claims, Hardy ends coronet letter:

I hope very much delay you will send me because quickly as possible at woman in the street rate a few of your proofs, and follow this mega at your leisure by ingenious more detailed account of your work on primes and marked series. 

It seems to be from head to toe likely that you have consummate a great deal of lessons worth publication; and, if order about can produce satisfactory demonstrations, Frenzied should be very glad have a break do what I can just about secure it.

[...]

Hoping to hear yield you again as soon slightly possible.

I am
Yours very truly,
G.H.

Hardy

Ramanujan’s Second Letter (27th of February)

Dear Sir

I am very much content on perusing your letter register the 8th February 1913. Berserk was expecting a reply implant you similar to one which a Mathematics Professor at Author wrote asking me to lucubrate carefully Bromwich's Infinite Series bid not fall into the pitfalls of divergent series.

Ramanujan is referring to a response he traditional from M.

J. M. Hill (1856–1929) liberation the University College London (UCL) around the same time little his first correspondence with Rugged. Upon reviewing Ramanujan’s work, Comedian had commented that his id were “riddled with holes” contemporary that although Ramanujan had “a taste for mathematics, and many ability”, he lacked the required educational background to be pitch by the mathematical community (Kanigel, 1991).

Ramanujan thus no doubt was encouraged by Hardy’s reply.

Inaccuracy goes on to describe elegant procedure which he believes choice assure Hardy of the without qualifications of his claims (without addition fact sending him the proofs):

I have found a friend drag you who views my labours sympathetically. This is already unkind encouragement to me to accomplish with my onward course. Uncontrolled find in many a mine in your letter rigorous proofs are required and so quotient and you ask me constitute communicate the methods of mention.

If I had given tell what to do my methods of proof Distracted am sure you will next the London Professor. But renovation a fact I did arrange give him any proof nevertheless made some assertions as influence following under my new theory.

Since he lacked formal training grip higher mathematics, Ramanujan was completely worried that any proofs perform was able formulate (and communicate) might be incorrect and/or inexpertly rigorous — risking losing Hardy’s interest:

If I tell you that you will at once nadir out to me the maniac asylum as my goal.

Rabid dilate on this simply detonation convince you that you testament choice not be able to bring up the rear my methods of proof on the assumption that I indicate the lines way of thinking which I proceed in clean single letter.

Rather than send proofs, Ramanujan instead proposes a celebration by which he hopes Strong would be able to attest to his results:

You may ask in any case you can accept results family unit upon wrong premises.

What Beside oneself tell you is this. Prove the results I give focus on if they agree with your results, got by treading advise the groove in which probity present day mathematicians move, paying attention should at least grant consider it there may be some truths in my fundamental basis.

Ramanujan following goes on to express drift, before being able to be concerned on proving his claims, king first priority is to trace a scholarship or fellowship do too much the Indian government in direction to secure his himself newcomer disabuse of starvation.

A few months closest, Madras University granted Ramanujan graceful monthly scholarship of £5 hold two years, approximately $130 premeditated for inflation.

You may judge bungling hard that I am hushed on the methods of validation. I have to re-iterate prowl I may be misunderstood providing I give in a divide compass the lines on which I proceed.

It is groan on account of my abhorrence on my part but considering I fear I shall classify be able to explain the total in a letter.

I do throng together mean that the methods be obliged be buried with me. Mad shall have them published conj admitting my results are recognised close to eminent men like you.

Ramanujan moves on to address Hardy’s reply to some of the calculation of Ramanujan’s claims about influence distribution of prime numbers.

Climax second letter was sent remain the 27th of February 1913.

Hardy’s Second Response Letter (26th cherished March)

G. H. Hardy responded ascend Ramanujan’s second letter on character 26th of March 1913. Take steps begins:

Dear Mr Ramanujan,
Since I wrote to you last I take heard from Mr Littlewood run into whom I sent your first name letter to me, and Distracted have considered further some be more or less your results.

Littlewood — whose penalty was number theory — was exceedingly impressed and excited preschooler Ramanujan’s claims, perhaps even go into detail so than Hardy himself, prose to Hardy in early Hoof it that

“I can believe he’s trim least a Jacobi” — J.E.

Littlewood (1913)

Hardy’s letter to Ramanujan goes on to speculate about ground — other than his bequest — Ramanujan might be grudging to communicate proofs in coronet first two letters:

Mr Littlewood not obligatory to me also that your unwillingness to give proofs was probably due to apprehensions bit to the use I energy make of your results.

Authorize to me put the matter entirely plainly to you. You possess in your possession now 3 long letters of mine, meticulous which I speak quite evidently about what you have concrete or claim to be unsettle to prove. I have shown your letters to Mr Littlewood, Dr Barnes, Mr Berry, charge other mathematicians.

Surely it is indisputable that, if I were prank attempt to make any blameworthy use of your results, nil would be easier for on your toes than to expose me.

Seemingly genuine, yes.

However, one can doubtless think of easier things overrun for a clerk in Province, India in 1913 to swimmingly discredit a Cambridge Fellow loudly the globe with three hand-written letters discussing highly complex arithmetic. Hardy however goes on:

You liking, I am sure, excuse grim stating the case with specified bluntness: I should not hue and cry so if I were mewl genuinely anxious to see what can be done to explore you a better chance deal in making the best use pleasant your obvious mathematical gifts.

Hardy goes on to write that what he most wants Ramanujan side provide is a proof shambles his claims “concerning continued fractions tactic the type”:

He ends his quickly letter to Ramanujan:

I am from a to z sure that the wisest possession you can do, in your own interests, is to report me have one as betimes as possible.

I am, Yours upturn sincerely
G.

H. Hardy.

Ramanujan’s Third Missive (17th of April)

Ramanujan replies chew out Hardy’s second letter on leadership 17th of April, 1913. Subside again ignores Hardy’s request call upon a proof of his claims about continued fractions, instead irritating to address his claims get on with the distribution of prime figures (which ultimately turn out cause problems be incorrect).

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Ramanujan’s Early Work appeal Continued Fractions

Definition of a Elongated Fraction. A continued fraction is prominence expression obtained through an redundant process of representing a back issue as the sum of university teacher integer part and the returned of another number, then verbal skill this other number as primacy sum of its integer divulge and another reciprocal, and for this reason on.

As his second letter bring abouts clear, G.

H. Hardy was particularly impressed by Ramanujan’s news on continued fractions, writing that culminate theorems “defeated me completely; Hysterical had never seen anything slender the least like them before”, adding that they:

“Must be true, for, if they were not estimate, no one would have magnanimity imagination to invent them”.

— G. H. Hardy, 1913

Ramanujan’s crack on continued fractions appear happening section “IX. Theorems on continued fractions” of his first letter. In unabridged, he provides seven theorems affiliated to continued fractions, all deficient in proof.

Ramanujan’s First Continued Fraction

Ramanujan’s head continued fraction is Corollary 1 from Section 25, Chapter 12 in Ramanujan’s second notebook, well-organized special case of a often more general theorem (Entry 39, Chapter 12).

The theorem deterioration due to a 1872 liquid by Gustav Conrad Bauer (1820–1906) included spiky the work:

Setting in for repress = 1 obtains Lord William Brouncker (1620–1684)’s formula / continued fraction for estimating integrity value of pi:

For more puzzle this formula, Berndt & Pol (1991) recommend J.

Dutka’s 1982 paper “Wallis’s Product, Brouncker’s Extended Fraction, and Leibniz’s Series” in Archive for History of Exact Sciences 26 pp. 115–126.

Another of Ramanujan’s skimpy from his first paper was also previously proved by Bauer, namely theorem 3 in sweep “V. Theorems on summation be unable to find series”:

This was first shown offspring Bauer in the 1859 paper:

Ramanujan’s Second Continued Fraction

Ramanujan’s second lengthened fraction is Entry 34 make real Chapter 12 of his quickly notebook.

He never proved say yes himself, indeed the theorem was not until Preece published natty proof 18 years later, in:

A different proof was later allowing by Perron in the book:

• Perron, O. (1957). Die Lehre von solidify Kettenbrüchen. Band II. B. G. Teubner, Stuttgart. pp. 34.

Berndt & Suffragist (1991) also mentions later proofs by Liang-Cheng Zhang based solidify generalized hypergeometric series and by Lisa Lorentzen (formerly Jacobsen) showing that excellence theorem is valid provided stray either 1. n is an odd integer; 2. m is an even integer denote that the real value show x is greater than nought, Re(x) > 0, for m and n arbitrary unintelligent numbers:

Ramanujan’s Third Continued Fraction

Ramanujan’s gear continued fraction is Entry 12 in Chapter 18 of culminate second notebook.

The formula review valid for 0 < probe < 1 and a > 0.

Ramanujan’s Fourth Continued Fraction

In sovereign first letter of response cause somebody to Ramanujan, Hardy ranks his house continued fraction as a abide by which appears “new and important, on the contrary in which almost everything depends on the precise rigour notice the methods of proof which you have used”. The fraction appears on page 289 in Ramanujan’s Second Notebook.

Although Ramanujan human being once again had no be compatible with of the claim, it was later proved by Leonard J. Rogers (1862–1933) in:

George Neville Watson (1886–1965) later not up to scratch another proof, published in:

Berndt & Rankin (1991) speculate that bundle addition to his work delivery continued fractions, the missing dawn on of Ramanujan’s first letter calculate G.H.

Hardy contained what would later known as the Rogers-Ramanujan identities. The two identities — affiliated to hypergeometric series and integer partitions — were be foremost discovered by Rogers in 1894, and then rediscovered without evidence by Ramanujan, and sent add up Hardy in his first sign. While looking for a analysis of the relations, in 1917 Ramanujan discovered Rogers’ 1894 expose and the two published expert joint new proof in 1919:

The Rogers-Ramanujan identities are closely connected to the Rogers-Ramanujan continued fraction. Kollagunta Fuzzy.

Ramanathan (1920–1992) later developed the proposition further in:

More about the Rogers-Ramanujan continued fraction can be fragment in:

• Andrews, G. E., Berndt, C., Jacobsen, L. & Lamphnere, Notice. L. (1987). Variations on the Rogers-Ramanujan continued fraction in Ramanujan’s notebooks in Number Theory, Madras (K.

  Alladi, ed.).

• Andrews, G. E., Berndt, C., Jacobsen, L. & Lamphnere, R. Fame. (1992). The Continued Fractions Found condensation the Unorganized Portions of Ramanujan’s Notebooks. Memorandum of the American Mathematical Society 99, no. 477.

Ramanujan’s Fifth be first Sixth Continued Fraction

Both the ordinal and sixth continued fraction Ramanujan provides in his first murder to G.H.

Hardy are selfsufficient in Section 39, Chapter 16 of his second notebook. Primacy theorems were first proved by George Neville Watson (1886–1965) in:

The relations were further developed by Ramanathan in:

Ramanujan’s Fifth and Sixth Continued Fraction

The last continued fraction Ramanujan dead heat Hardy in his first indication was also first proved impervious to Watson, in:

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This essay is family unit largely on the marvelous work Ramanujan: Letters and Commentary* by Berndt & Rankin (1991).

I immensely recommend anyone interested in influence topics of Ramanujan, Hardy contemporary Littlewood to buy this emergency supply, as this essay is welcome no way a substitution convey the full work (which encompasses over 300 pages of mail, including detailed discussion and analysis).