Ramanujan's continued fractions

Ramanujan's continued fractions

Ramanujan's continued fractions are a series of interesting closed-form expressions for non-simple continued fractions developed by Indian mathematician Srinivasa Ramanujan.

Examples

Among the expressions developed by Ramanujan are two which are nearly equal to one:

Nearly one

:{1over 1+{e^{-2pi}over 1+{e^{-4pi}over 1+dots} = left({sqrt{5+sqrt{5}over 2}-{sqrt{5}+1over 2 ight)e^{2pi/5} = e^{2pi/5}left({sqrt{phisqrt{5-phi} ight) = 0.9981360dotswhere phi is the golden ratio (Approximately 1.618)

The multiplicative inverse of this expression is:

:1 + {e^{-2pi}over 1+{e^{-4pi}over 1+{e^{-6pi}over 1+dots}=frac{1}{2}left [1+sqrt{5}+sqrt{2(5+sqrt{5})} ight] ,e^{-2pi/5}:= frac{e^{-2pi/5{sqrt{phisqrt{5-{phi=1.0018674...

Even closer to one

:{1over 1+{e^{-2pisqrt{5over 1+{e^{-4pisqrt{5over 1+dots}:=left(frac{sqrt{5{1+ [5^{3/4}(phi-1)^{5/2}-1] ^{1/5-{phi} ight),e^{2pi/sqrt{5=0.99999920...

The multiplicative inverse of this expression is:

:1 + {e^{-2pisqrt{5over 1+{e^{-4pisqrt{5over 1+dots:=frac{e^{-2pi/sqrt{5}{frac{sqrt{5{1+left [5^{3/4}(phi-1)^{5/2}-1 ight] ^{1/5-{phi=1.000000791267...

References

[http://www.torinoscienza.it/img/orig/it/s00/00/0005/000005cf.jpg]


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Ramanujan, Srinivasa — ▪ Indian mathematician born Dec. 22, 1887, Erode, India died April 26, 1920, Kumbakonam       Indian mathematician whose contributions to the theory of numbers include pioneering discoveries of the properties of the partition function.       When …   Universalium

  • Ramanujan, Srinivasa (Aaiyangar) — born Dec. 22, 1887, Erode, India died April 26, 1920, Kumbakonam Indian mathematician. Extremely poor, he was largely self taught from age 15. In 1913 he began a correspondence with Godfrey H. Hardy (1877–1947) that took him to England, where he… …   Universalium

  • Srinivasa Ramanujan — Infobox Scientist name=Srinivasa Ramanujan thumb|Srinivasa Ramanujan birth date = birth date|1887|12|22|df=y birth place = Erode, Tamil Nadu, India death date = death date and age|1920|4|26|1887|12|22|df=y death place = Chetput, (Madras), Tamil… …   Wikipedia

  • List of mathematics articles (R) — NOTOC R R. A. Fisher Lectureship Rabdology Rabin automaton Rabin signature algorithm Rabinovich Fabrikant equations Rabinowitsch trick Racah polynomials Racah W coefficient Racetrack (game) Racks and quandles Radar chart Rademacher complexity… …   Wikipedia

  • Approximations of π — Timeline of approximations for pi …   Wikipedia

  • Pi — This article is about the number. For the Greek letter, see Pi (letter). For other uses, see Pi (disambiguation). The circumference of a ci …   Wikipedia

  • Square root of 5 — The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. This number appears in the formula for the golden ratio. It can be denoted in surd form as::sqrt{5}.It is an irrational algebraic number.… …   Wikipedia

  • Mathematical coincidence — This article is about numerical curiosities. For the technical mathematical concept of coincidence, see coincidence point. A mathematical coincidence can be said to occur when two expressions show a near equality that lacks direct theoretical… …   Wikipedia

  • Bruce C. Berndt — Bruce Carl Berndt (born March 13, 1939, in St. Joseph, Michigan) is an American mathematician. He attended college at Albion College, graduating in 1961,and received his master s and doctoral degrees from the University of Wisconsin Madison.He… …   Wikipedia

  • List of important publications in mathematics — One of the oldest surviving fragments of Euclid s Elements, found at Oxyrhynchus and dated to circa AD 100. The diagram accompanies Book II, Proposition 5.[1] This is a list of important publications in mathematics, organized by field. Some… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”