Maths seminar i (1)

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125th Birth Anniversary Celebrations of Genius

Ramanujan

Topic : Ramanujan as a genius par Excellence

Venue : Sri Sarada Niketan College for Women, Amaravathipudur

Presenter : Dr.(Mrs).S.SelvaRani, Principal & H.O.D of Mathematics,

Sri Sarada Niketan College for Women.

22nd December 1887

Born in Erode

January 1898

Entering Town High School Kumbakonam. Won Prizes for proficiency in Mathematics & English.

1900 Began to work on his own Mathematics – summing Geometric & Arithmetic series. Inspiration : G.S Carr’s Book

1902 Solving Cubic Equations

1904 Involved in Deep research Investigated series

∑1/n & calculated Euler’s Constant to 15 decimal places began to study Bernoulle’s Nos. Entered Government College Kumbakonam.

Life History & his sojourn in Mathematics

1906 Entered Pachayappa’s College, Madras

1908 Studied continued fractions & Divergent series

14th July 1909 Married S.Janaki Ammal

1910 Pose & Solve problems in the Journal of the Indian Mathematical Society. Developed relations between elliptic modular equations.

1911 Published a brilliant Research paper on Bernoulli numbers in the Journal of the Indian Mathematical Society & became popular as mathematical genius in Madras area.

1912 Got the clerk post in the accounts section of the Madras Post Trust.

1913 Ramanujan”s letter to Prof G.H.Hardy sending his book ‘orders of infinity’.

April,1914 Hardy brought Ramanujan to Trinity College Cambridge to begin an extra ordinary collaboration. It led to important results .

1915 Health problems due to food and climate.

March 1916 Ramanujan graduated from cambridge with a Bachelor of Science by Research. (Which is Ph.D degree from 1920. Dissertation on “Highly Composite numbers”.

February 28, 1919

Elected as the Fellow of the Royal society – First Indian Mathematician.

March 27, 1919 Returned to India, as a celebrity, after devoting 5 years 1914 – 1919 at the camridge.

April 26, 1920 Due to severe illness, He passed away.

Some Memories in his Mathematics Sojourn1. Mother – Smt.Komalathammal

2. Goddess Namagirithayar

School Anecdotes:1)0/0 cannot be determined.

2)Solving Simultaneous Equations in seconds.

3) Timetable ->Magic squares.

Well Wishers of Ramanujan• GH Hardy Mentor of Ramanujan

• E.H.Nevellie Trinity College

• S.Narayana Iyer , Treasurer, port trust.

• V.Ramasami Iyer, Founder of IMS & JIMS

• Seshu Iyer Prof. of Maths, Kumbakonam. Dewan Bahadur collector of Nellore.

• Sir Francis Spring, Chairman Madras Post Trust.

Ramanujan’s Infinite Series – the basis to compute

When k = 0, the result is 3.14159273, k=1, it is 3.141592654.

The value of to 14 decimal places is

3.141592653589793.

The Value of

Ramanujan’s equation arises at value of Pi to large numbers of decimals plus more rapidly than just about any other known series.

Ramanjuan’s also gave 17 other series formulas for .

Even now, with the help of a mathematical tools such as computer software, find it hard to generate the kind of identities that Ramanujan already found.

Even computer find it hard to generate such series.

Glimpses of

Ramanujan’s Work

Redefining Euler’s Constant Ramanujan gave many beautiful formulas for π and . Euler’s constant γ =-Γ (1) =0.57721566…, which occurs in many well-known formulas involving The Gamma function, the Riemann zeta function, the divisor function d(n), etc.

Gamma FunctionIt is an infinite integral

T1 (n)

T1(n) Converges for

t1(n+1) = nT1(n), if

Euler’s identityIn analytical mathematics, Euler’s identity (also know as Euler’s equation), named for the Swiss-German mathematician Leonhard Euler, is the equality

Where

e is Euler’s number, the base of natural algorithms,

i is the imaginary unit, which satisfies , and

is pi, the ratio of the circumference of a circle to its diameter.

Euler’s formula from complexanalysis

Ramanujan’s Formula For , Euler’s Constant

At the top of page 276 in[13],Ramanujan writes

the last term of the nth group being

Let 0 be the centre & PR any diameter bisect op at H & trisect OR at TDraw PK =PM &PL =MNDraw CD parellel KL. Then, = Circle PQR

Ramanujan’s NotebookSquare the circle

To Construct a square equal to a given circle

The Hardy-Ramanujan Number : 1729

“The sum of two positive cubes in two different ways”.

Ramanujan’s Work in Continued FractionThe General Form of a Continued Fraction

(P and Q are whole, positive numbers) expressing it in the form of a continued fractions as follows :

=a+1/(b+1)/(c+1/(d+…….)))

Where a, b, c, d, e, etc are all whole numbers. If P/Q is less than 1, then the first number ,a, will be 0.

Ramanujan developed a number of interesting closed-form expressions for non-simple continued fractions. These include the almost integers

1)

2)

3)

4)  

Applications of Ramanujan’s work

•It has found applications in Polymer Chemistry, Computer Science, Physics and even in Cancer Research.

•Ramanujan’s graphs have found applications in Communication Engineering, characterizing certain efficient networks.

•Ramanujan graphs for some problems in Coding Theory.

Thank You