Madhava series

In mathematics, Madhava series is any one of the collection of infinite series expressions believed to have been discovered by Sangamagrama Madhava (c. 1350 – c. 1425) the founder the Kerala school of astronomy and mathematics. These expressions are the infinite power series expansions of the trigonometric sine, cosine and arctangent functions, and the special case of the power series expansion of the arctangent function yielding a formula for computing π. The power series expansions of sine and cosine functions are respectively called Madhava's sine series and Madhava's cosine series. The power series expansion of the arctangent function is somtimes called Madhava–Gregory series or Gregory–Madhava series. These power series are also collectively called Taylor–Madhava series. The formula for π is referred to as Madhava–Newton series or Madhava–Leibnitz series or Leibniz formula for pi or Leibnitz–Gregory–Madhava series. These further names for the various series are reflective of the names of the western discoverers or popularizers of the respective series.

No surviving works of Madhava contain explicit statements regarding the expressions which are now referred to as Madhava series. However, in the writing of later members of the Kerala school of astronomy and mathematics like Nilakantha Somayaji and Jyeshthadeva one can find unambiguous attributions of these series to Madhava. It is also in the works of these later astronomers and mathematicians one can trace the Indian proofs of these series expansions. These proofs provide enough indications about the approach Madhava had adopted to arrive at his series expansions.

Madhava's series in modern notations

In the writings of the mathematicians and astronomers of the Kerala school, Madhava's series are described couched in the terminology and concepts fashionable at that time. When we translate these ideas into the notations and concepts of modern day mathematics, we obtain the current equivalents of Madhava's series. These present-day counterparts of the infinite series expressions discovered by Madhava are the following:

No.	Series	Name	Western discoverers of the series and approximate dates of discovery
1	sin x   =   x − x3 / 3! + x5 / 5! − x7 / 7! + ...	Madhava's sine series	Issac Newton (1670) and Wilhelm Leibnitz (1676)
2	cos x   =   1 − x2 / 2! + x4 / 4! − x6 / 6! + ...	Madhava's cosine series	Issac Newton (1670) and Wilhelm Leibnitz (1676)
3	tan−1x   =   x − x3 / 3 + x5 / 5 − x7 / 7 + ...	Madhava's series for arctangent	James Gregory (1671) and Wilhelm Leibnitz (1676)
4	π / 4   =   1 − 1 / 3 + 1 / 5 − 1 / 7 + ...	Madhava's formula for π	James Gregory (1671) and Wilhelm Leibnitz (1676)

Madhava series in "Madhava's own words"

None of Madhava's works containing any of the series expressions attributed to him has survived. These series expressions are found in the writings of the followers of Madhava in the Kerala school. At many places these authors have clearly stated that these are "as told by Madhava". Thus the enunciations of the various series found in Tantrasamgraha and its commentaries can be safely assumed to be in "Madhava's own words". The translations of the relevant verses as given in the Yuktidipika commentary by Sankara Variar (also known as Tantrasamgraha-vyakhya) of Tantrasamgraha by Nilakantha Somayaji are reproduced below. These are then rendered in current mathematical notations.

Madhava's sine series

In Madhava's own words

Madhava's sine series is stated in verses 2.440 and 2.441 in Yukti-dipika commentary (Tantrasamgraha-vyakhya) by Sankara Variar. A translation of the verses follows.

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.

Rendering in modern notations

Let r denote the radius of the circle, s denote the arc-length.

• The following numerators are formed first:

$s\cdot s^{2},\qquad s\cdot s^{2}\cdot s^{2},\qquad s\cdot s^{2}\cdot s^{2}\cdot s^{2},\qquad ...$

• These are then divided by quantities as specified in the verse.

$s\cdot {\frac {s^{2}}{(2^{2}+2)r^{2}}},\qquad s\cdot {\frac {s^{2}}{(2^{2}+2)r^{2}}}\cdot {\frac {s^{2}}{(4^{2}+4)r^{2}}},\qquad s\cdot {\frac {s^{2}}{(2^{2}+2)r^{2}}}\cdot {\frac {s^{2}}{(4^{2}+4)r^{2}}}\cdot {\frac {s^{2}}{(6^{2}+6)r^{2}}},\qquad \cdots$

• Place the arc and the successive results so obtained one below the other, and subtract each from the one above to get jiva:

${\text{jiva}}=s-{\Big [}s\cdot {\frac {s^{2}}{(2^{2}+2)r^{2}}}-{\Big [}s\cdot {\frac {s^{2}}{(2^{2}+2)r^{2}}}\cdot {\frac {s^{2}}{(4^{2}+4)r^{2}}}-{\Big [}s\cdot {\frac {s^{2}}{(2^{2}+2)r^{2}}}\cdot {\frac {s^{2}}{(4^{2}+4)r^{2}}}\cdot {\frac {s^{2}}{(6^{2}+6)r^{2}}}-\quad \cdots {\Big ]}{\Big ]}{\Big ]}$

Transformation to current notation

Let θ be the angle subtended by the arc s at the centre of the circle. Then s = rθ and jiva = r sin θ. Substituting these in the last expression and simplifying we get

• $\sin \theta =\theta -{\frac {\theta ^{3}}{3!}}+{\frac {\theta ^{5}}{5!}}-{\frac {\theta ^{7}}{7!}}+\quad \cdots$

which is the iinfinte power series expansion of the sine function.

Madhava's reformulation for numerical computation

The last line in the verse ' as collected together in the verse beginning with "vidvan" etc. ' is a reference to a reformulation of the series expression effected by Madhava himself so as to make it amenable for easy computations for specifies values of the arc and radius. For such a reformulation, Madhava considers a circle one quarter of which measures 5400 minutes (say C minutes). He then develops a scheme for the easy computations of the jiva 's of the various arcs of such a circle. Let R be the radius of a circle one quarter of which measures C. Madhava had already computed the value of π using his series formula for π. Using this value of π = 3.1415926535922, the radius R can be computed as follows: Then

R = 2 × 5400 / π = 3437.74677078493925 = 3437 arcminutes 44 arcseconds 48 sixtieths of an arcsecond = 3437′ 44′′ 48′′′

Madhava's expression for jiva corresponding to any arc s of a circle of radius R is equivalent to the following:

${\begin{aligned}{\text{jiva }}&=s-{\frac {s^{3}}{R^{2}(2^{2}+2)}}+{\frac {s^{5}}{R^{4}(2^{2}+2)(4^{2}+4)}}-\cdots \\&=s-\left({\frac {s}{C}}\right)^{3}{\Big [}{\frac {R\left({\frac {\pi }{2}}\right)^{3}}{3!}}-\left({\frac {s}{C}}\right)^{2}{\Big [}{\frac {R\left({\frac {\pi }{2}}\right)^{5}}{5!}}-\left({\frac {s}{C}}\right)^{2}{\Big [}{\frac {R\left({\frac {\pi }{2}}\right)^{7}}{7!}}-\quad \cdots \quad {\Big ]}{\Big ]}{\Big ]}\end{aligned}}$

Madhava then computes the following values:

No.	Expression	Value	Value in Katapayadi system
1	R × (π / 2)3 / 3!	2220′   39′′   40′′′	ni-rvi-ddhā-nga-na-rē-ndra-rung
2	R × (π / 2)5 / 5!	273′   57′′   47′′′	sa-rvā-rtha-śī-la-sthi-ro
3	R × (π / 2)7 / 7!	16′   05′′   41′′′	ka-vī-śa-ni-ca-ya
4	R × (π / 2)9 / 9!	33′′   06′′′	tu-nna-ba-la
5	R × (π / 2)11 / 11!	44′′′	vi-dvān

The jiva can now be be computed using the following scheme:

jiva = s − (s / C)³ [ (2220′   39′′   40′′′) − (s / C)² [ (273′   57′′   47′′′) − (s / C)² [ (33′′   06′′′) − (s / C)² [ (44′′′ ) − ... ] ] ] ]

Further reading

1. G G Joseph, The crest of the peacock (London, 1991).
2. K. V. Sarma, A History of the Kerala School of Hindu Astronomy (Hoshiarpur, 1972).
3. A. K. Bag, Madhava's sine and cosine series, Indian J. History Sci. 11 (1) (1976), 54–57.
4. D. Gold and D Pingree, A hitherto unknown Sanskrit work concerning Madhava's derivation of the power series for sine and cosine, Historia Sci. No. 42 (1991), 49–65.
5. R. C. Gupta, Madhava's and other medieval Indian values of pi, Math. Education 9 (3) (1975), B45–B48.
6. R. C. Gupta, Madhava's power series computation of the sine, Ganita 27 (1–2) (1976), 19–24.
7. R. C. Gupta, On the remainder term in the Madhava–Leibniz's series, Ganita Bharati 14 (1–4) (1992), 68–71.
8. R. C. Gupta, The Madhava–Gregory series, Math. Education 7 (1973), B67–B70.
9. T. Hayashi, T. Kusuba and M. Yano, The correction of the Madhava series for the circumference of a circle, Centaurus 33 (2–3) (1990), 149–174.
10. R.C. gupta, The Madhava–Gregory series for tan⁻¹x, Indian Journal of Mathematics Education, 11(3), 107–110, 1991.