Talk:Madhava of Sangamagrama
From Wikipedia, the free encyclopedia
Contents |
[edit] Citation Needed
Please provide a citation for Madhava being "considered the father of mathematical analysis" . . . I know of no printed sources that make a reasonable claim of this . . . the father of analysis as we know them today are more likely to be consider Karl Weierstraß, as he was interested in the basis of the Calculus . . . before his time, there were no clear definitions regarding the fundamentals of calculus, and thus theorems could not be properly proven. Madhava in contrast, like many before him, merely introduced the idea of infinity to certain classes of functions, like many after him would do before analysis was put on a firm footing. Arundhati bakshi 18:45, 24 March 2006 (UTC)
- See Mathematical analysis and it's talk page for more detail (and a printed source for the claim). Also see [1] highlighting all pioneers in analysis - it considers Madhava the founder of mathematical analysis in contrast with Weierstrass as father of modern analysis. Weierstrass' page also says the same. --Pranathi 00:29, 6 April 2006 (UTC)
[edit] Put on Kerala School
Hi, I put these questions on the Kerala school.
Could we get some cleanup on the maths stuff?
- Infinite series expansions of functions.
- Ok. How? By Taylor series? Fourier series?
- The power series.
- Power series of what?
- The Taylor series.
- Taylor series of what? There are 4 or 5 trigonmetric functions listed below.
- Trigonometric series.
- ...
- Rational approximations of infinite series.
- ...
- Taylor series of the sine and cosine functions (Madhava-Newton power series).
- Ok. Wouldn't that fall above?
- Taylor series of the tangent function.
- Likewise
- Taylor series of the arctangent function (Madhava-Gregory series).
- Likewise?
- Second-order Taylor series approximations of the sine and cosine functions.
- What's the 2nd order Taylor series? If the "1st order" is just sin(x) ~ x, then the second order would be sin(x) ~ x + x^3/3!.
- Third-order Taylor series approximation of the sine function.
- So that would be sin(x) ~ x + x^3/3! + x^5/5! ?
- Which one? π = 4/1 - 4/3 + 4/5 - ...
- Power series of π/4 (Euler's series).
- Which one? Does this mean Leibniz above?
- Power series of the radius.
- What radius?
- Power series of the diameter.
- What diameter?
- Power series of the circumference.
- What circumference?
- Power series of angle θ (equivalent to the Gregory series).
- ??? What does this mean?
- Infinite continued fractions.
- Ok. As solutions to quadratics? Cubics?
- The solution of transcendental equations by iteration.
- As solutions to ...
- Approximation of transcendental numbers by continued fractions.
- Which transcendental number?
- Tests of convergence of infinite series.
- Ok...
- Correctly computed the value of π to 11 decimal places, the most accurate value of π after almost a thousand years.
- Ok...
- Sine tables to 12 decimal places of accuracy and cosine tables to 9 decimal places of accuracy, which would remain the most accurate upto the 17th century.
- Ok...
- A procedure to determine the positions of the Moon every 36 minutes.
- Ok...
- Methods to estimate the motions of the planets.
- Ok
- Including the fundamental theorem of calculus? Which rules? Integration of polynomials?
- Term by term integration.
- Ok
- Laying the foundations for the development of calculus, which was then further developed by his successors at the Kerala School.
- ...
Thanks! --M a s 01:57, 12 May 2006 (UTC)
[edit] Obvious Thing
Okay in the article, the expansion for pi/4 is a direct result of the Madhava-Gregory series(theta=pi, n=1), but the article points to some expansion of the arctangent as the source. Is this a mistake?
[edit] Disputed
None of the questions under Put on Kerala School have been answered.
To give an example, the article claims that Madhava invented the fundamental ideas of:
- Infinite series expansions of functions.
- Power series.
- Taylor series.
- Maclaurin series.
- Trigonometric series
First question, were these expansions really infinite as claimed? Further, for some functions he gave what looks like the first terms of a Taylor or Maclaurin series of that function. That is a nice accomplishment. But the fundamental idea of a Maclaurin or Taylor series consists of giving the rule how to develop a function into a series. No evidence that he knew this is given. -- Zz 17:56, 29 March 2007 (UTC)
[edit] Tag removed
Just added a number of facts, such as the location of Sangramagrama, the uncertainty regarding which of the results are specifically attributable to Madhava (including a sketch of some of the arguments), ensuring that the reader is aware of the degree of doubt on this matter.
Removed many of the indirect references (e.g. several instances of the Mactutor pages) with journal articles. Merged the "bibliography" section into references (though some people may disagree, I feel this is cleaner - you get a sense of what the text does; if we need to repeat the citations in a bibliography, it should have annotations).
Also added a citation for integration along with the original malayalam text.
As for the disputed facts, I think among the examples given, with citations to various texts, we have many examples of these three :
- Infinite series expansions of functions (mostly trig functions)
- Power series : all are power series expansions
- Trigonometric series : many involve the trig functions in the expansion
The following two however appear to be unlikely:
- Taylor series.
- Maclaurin series.
Since these express functions as powers of the derivative for a general function, it is unclear that the understanding of higher-order derivatives was sufficiently understood. While derivatives are being computed, it is not clear whether power series of derivatives were used. Unless other facts come to light, I am removing reference to these two. mukerjee (talk) 17:42, 6 September 2007 (UTC)