These include both Ramanujan's own discoveries, and those developed or proven in collaboration with Hardy.
- Properties of highly composite numbers
- The partition function and its asymptotics
- Mock theta functions He also made major breakthroughs and discoveries in the areas of:
- Gamma functions
- Modular forms
- Ramanujan's continued fractions
- Divergent series
- Hypergeometric series
- Prime number theory
It is said his discoveries were unusually rich; that is, in many of them there was far more than initially met the eye.
source:http://en.wikipedia.org/wiki/Srinivasa_Ramanujan
Sankalp Unit
Comments
Small Correction..
In the last step, there is a small correction.
When you take square root of any number; it can take either negative or positive. Considering only one is not completely correct.
(3-5/2)(3-5/2) = (2-5/2)(2-5/2)
After taking square root,
-+(3-5/2) = -+(2-5/2)
So,
either 3-5/2 = (2-5/2) Holds false
or -(3-5/2) = (2-5/2) Holds True
or -(3-5/2) = -(2-5/2) Holds False
or (3-5/2) = -(2-5/2) Holds True
It's similar to,
4 = 4
Talking square root both side,
2 = -2
Hope this clear the illusion...;)
Can U Prove 3=2??
This seems to be an anomaly or whatever u call in mathematics.
It seems, Ramanujam found it but never disclosed it during his life time
and that it has been found from his dairy.
See this illustration:
-6 = -6
9-15 = 4-10
adding 25/4 to both sides:
9-15+(25/4) = 4-10+(25/4 )
Changing the order
9+(25/4)-15 = 4+(25/4)-10
(this is just like : a square + b square - two a b = (a-b)square. )
Here a = 3, b=5/2 for L.H.S and a =2, b=5/2 for R.H.S.
So it can be expressed as follows:
(3-5/2)(3-5/ 2) = (2-5/2)(2-5/ 2)
Taking positive square root on both sides:
3 - 5/2 = 2 - 5/2
3 = 2