Good,Bad and Ugly: India's contribution to mathematics world!

Pandit Raghunath Murmu - Indian, created Ol Chiki script in 1925.
Mangei Gomango - Indian, invented Sorang Sompeng script in 1936.
Saint Shahjalal - Bengali, ascribed invention of Syloti Nagri alphabet c. 1300, according to tradition.
Dhawan Turi - possibly mythical, ascribed invention of Varang Kshitri alphabet c. 1250 (?)

$\sqrt{1+2\sqrt{1+3 \sqrt{1+\cdots}}}.$

$x+n+a = \sqrt{ax+(n+a)^2 +x\sqrt{a(x+n)+(n+a)^2+(x+n) \sqrt\mathrm{\cdots}}}$

He also devised a method of calculating B_n based on previous Bernoulli numbers. One of these methods went as follows:

It will be observed that if n is even but not equal to zero,
(i) B_n is a fraction and the numerator of ${B_n \over n}$ in its lowest terms is a prime number,
(ii) the denominator of B_n contains each of the factors 2 and 3 once and only once,
(iii) $2^n(2^n-1){b_n \over n}$ is an integer and $2(2^n-1)B_n\,$ consequently is an odd integer.

One of the theorems Hardy found hard to believe was found on the bottom of page three (valid for 0

$\int_0^\infty \cfrac{1+{x}^2/({b+1})^2}{1+{x}^2/({a})^2} \times\cfrac{1+{x}^2/({b+2})^2}{1+{x}^2/({a+1})^2}\times\cdots\;\;dx = \frac{\sqrt \pi}{2} \times\frac{\Gamma(a+\frac{1}{2})\Gamma(b+1)\Gamma(b-a+\frac{1}{2})}{\Gamma(a)\Gamma(b+\frac{1}{2})\Gamma(b-a+1)}.$

Hardy was also impressed by some of Ramanujan's other work relating to infinite series:

$1 - 5\left(\frac{1}{2}\right)^3 + 9\left(\frac{1\times3}{2\times4}\right)^3 - 13\left(\frac{1\times3\times5}{2\times4\times6}\right)^3 + \cdots = \frac{2}{\pi}$

$1 + 9\left(\frac{1}{4}\right)^4 + 17\left(\frac{1\times5}{4\times8}\right)^4 + 25\left(\frac{1\times5\times9}{4\times8\times12}\right)^4 + \cdots = \frac{2^\frac{3}{2}}{\pi^\frac{1}{2}\Gamma^2\left(\frac{3}{4}\right)}.$

Good,Bad and Ugly

Tuesday, April 28, 2009

India's contribution to mathematics world!

0 comments:

Post a Comment

Trial

Labels

Blog Archive