The Madhava Trignometric series
The Kerala mathematician, Madhava of Sangramagrama, gave birth to the Trignometric series. They are the infinite series of the functions, sine, cosine and the arctangent. These infinite series are called by his name, Madhava sine series and Madhava Cosine series.
While the power series expansion of ArcTan is called Madhava-Gregory series, the power series are collectively called the Madhava Taylor series. The Pi series is known as Madhava Gregory series.
Sankara
Varier, one of the foremost disciples of Madhava, had translated his poetic
verses in
in his
Yuktideepika commentary on Tantrasamgraha-vyakhya, in verses 2.440 and 2.441
Multiply the arc by the square of the arc, and take the result of
repeating that (any number of times). Divide (each of the above numerators)
by the squares of the successive even numbers increased by that number and
multiplied by the square of the radius. Place the arc and the successive results so
obtained one below the other, and subtract each from the one above. These
together give the jiva, as collected together in the verse beginning with
"vidvan" etc.
These can be rendered in modern terms.
Let r denote the radius of the circle and s the arc-length.
The following numerators are formed first:
s.s^2,
s.s^2.s^2
s.s^2.s^2.s^2
These are then divided by quantities specified in the verse.
1)s.s^2/(2^2+2)r^2,
2)s. s^2/(2^2+2)r^2. s^2/4^2+4)r^2
3)s.s^2/(2^2+2)r^2.s^2/(4^2+4)r^2. s^2/(6^2+6)r^2
Place the arc and the successive results so obtained one below the other, and
subtract each from the one above to get jiva:
Jiva = s-(1-2-3)
We can change it to current terms
If x is the angle subtended by the arc s at the center of the Circle, then s =
rx and jiva = r sin x.
Sin x = x - x^3/3! + x^5/5! - x^7/7!...., which is the infinite power series
of the sine function.
By courtesy www.wikipedia.org and we thank Wikipedia for
publishing this on their site.
