the arithmetico-geometric mean of gausssury/agmreso.pdf · the sum pop1 + pip2 + "'"...

FEATURE I ARTICLE

The Arithmetico-geometric Mean of Gauss How to find the Perimeter of an Ellipse

72

B Sury

The author enjoys

writing about such great

masters as:

A person who does arouse

much admiration and a

million wows!

Such a mathematicians'

prince

has not been born since.

I talk of Carl FHedrich

Gauss!

Elementary Calculus

As all of us learn very early in school, the length of a circle is 7r • diameter. Somewhat later, one learns tha t e lementary calculus gives the length of an arc of the unit circle to be the value of the integral s = s ( x ) = f~ ~ = sin-l(x).

Before going further, it is worthwhile to recall how length of (part of) a curve is defined. Suppose first tha t the curve is given on the plane and is described by the pair of equations x = x ( t ) , y = y ( t ) in terms of a pa ramete r t. Intuitively, one would think of the length of any par t of the curve to be tha t of parts of a polygon which in- creasingly approximates tha t part of the curve. Thus, one divides the interval [a, b] where t varies into finitely many parts a = to < tl < " " < tn = b. The points P~ = ( x ( t , ) , y ( t i ) ) can be joined by line segments and

the sum PoP1 + PIP2 + "'" + P n - l P n of the lengths of these segments is an approximation to the length of the part of the curve which corresponds to the pa ramete r t varying from a to b. This approximat ion might be a crude one but it is at once clear (see Figure 1) tha t one can take more points to obtain a be t te r approximation. At this point, calculus comes to the rescue. It is heuris- tically clear tha t the best possible notion of length is obtained as the l imi t ing case as one increases the number n of points Pi indefinitely while s imultaneously let t ing the lengths of all the line segments PiPi+l diminish indefinitely. In other words, if the limit exists, it can be defined as the length of the curve. A curve for which this limit exists is called rectifiable for evident reasons. Clearly, circles, ellipses and hyperbolae are rectifiable

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curves.

E x e r c i s e S h o w tha t the curve def ined by

y = x 2 s i n l / x , O < x < 1 , y(O) = O

is rectif iable while tha t def ined by

y = x s i n l / x , O < x < 1 , y(O) = O

is not . W h a t is the length o f the f i r s t curve?

Is there a nice condit ion which is sufficient to ensure tha t a given curve is rectifiable? Let us look at a curve wi th a parametr i sa t ion x = x ( t ) , y -- y ( t ) for a < t < b where these functions are assumed to have continuous first derivatives. This condition is indeed enough to en- force rectifiability as we show now. Let a = to < tl < �9 . . < tn = b be a subdivision; then the approximating

per imeter n--1

S n = P i P i + l = i=0

n -1 - x(t,))2 + {y(t,+,) - y(t,)}2].

i=0

By the mean-value theorem, there are hi, bi E (ti, t i+l)

such tha t x ( t i + l ) - x ( t i ) = dz and y(t ,+l)- y(ti) = ~ ( b i ) ( t , + l - ti). When we let the number of intervals (ti, t i+l) tend to infinity while simultaneously le t t ing their lengths tend to 0, the limit is, by defini-

tion, just the integral fb ~ / (~)2 + (~)2dt" We note tha t

the above limit exists by uniform continui ty of d~ ~ _ dt ~ d t

consequence of the assumption tha t the parametr is ing functions x = x ( t ) , y = y ( t ) for a < t < b are functions which have continuous first derivatives. Actually, one can even allow the functions d~ and ~ to have finitely many discontinuities and still the integral makes sense.

It is clear from the change of variable formula for integrals tha t the above notion of length is independent of

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Figure 1.

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which parametr isa t ion we use. For this reason, if one is given the curve in nonparametr ic form as y = f (x ) , then one can write its length from x = a to x = b to be the in-

tegral ~b 9/1 + [dy~2dXkdz/ . If the curve is given in polar co- ordinates as x = r(0)cosO, y = r(O)sinO, the length can,

therefore, be wri t ten as the integral fo~ 1 ~ / r 2 + (~)2d0 where r(Oo)COSOo = a, r(O1)cos01 = b.

Let us look at the circle of radius to; its polar equation is r = ro (constant); the integral gives the length (01 - Oo)ro.

Already, it might be clear tha t even if a sequence of functions fn defined on some interval (a, b) converges in the interval to a function f , the corresponding integrals f : fn(x)dx need not converge to f : f (x )dx . The same problem persists wi th the integrals of their derivatives etc.

This, for instance, provides the correct explanat ion of one of the 'Think-it-over ' problems: 'Where is the miss- ing string?' posed in Resonance, Vol. 3, No. 1, 1998. The area under the string in its various positions can get arbitrari ly close to that under the diagonal s t r ing without the arc lengths tending to the diagonal length.

Elliptic Integrals

After the brief digression to recall the fundamental con- cept of length of a curve, let us come back a whole circle viz., look at x 2 + y2 = I. The length s = s(xo) from (x,y) = (0, 1) to (xo, Yo)is given as f~o : ~ = s in- lxo.

We express this as x = sin(s) and, viewed this way, the addition formula for the sine function amounts to the formula

foX dt foU dt fo z + ~ =

where z = x ~ + y ~ .

W h a t happens when we t ry to calculate the length of an

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arc in an ellipse X2 2 ~+~=i? Such a length is given by an integral of the form

s(xo) = ~0=~ l+(d'~y)2dxax

a 2 _ e2x 2 . . . . d x -_ f o

where R is a rational function (i.e. a quotient of two polynomials) and p is a polynomial of degree 4 in x.

Here e = ~ is the eccentricity of the ellipse.

Similarly, let us look at a lemniscate which can be described as the locus of points, the product of whose distances from two fixed points is constant. It is given in polar co-ordinates by the equation r 2 = cos(20) , the arc length is given by the integral

s(00) = J~0O~ (r2"t-(dr)2)d0d0

= fO O~ cos(20)-l/2dO

= - - 1 = o ( 1 _ x4)-l/2dx. Jo

The analogy to the circular length f~~ - x2)-l/2dx is evident. For the circle, one evaluates the integral by rationalising the integrand i.e., by making a change of variables which makes the integrand into a rational function i.e., into a quotient of two polynomials. One can rationalise the integrand by the substitution

2t (How?). Guided by this analogy in 1718, x = 1--$~ 2t2 for the lem- Fagnano tried the substitution x =

niscate. Although the integrand does not rationalise, he noticed that f~~ - x 4 ) - l / 2 d x = V~f~~ + t 4 ) - l / 2 d t


Lemniscate can be

described as the

locus of points, the

product of whose

distances from two fixed points is constant.

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76

Using a ruler and

compass, one can

double the arc

length of a

lemniscate.

if 0 _< x _< 1. Following this up with the subst i tut ion t - 2=2 - 1-=4, one obtains f0x~ - x4)-l/2dx = 2j'~~ - u4)-l/2du. Let us note the interesting consequence of this tha t using a ruler and compass, one can double the arc length of a lemniscate - see [1] for a more general discussion of this aspect.

A fundamenta l idea in the general problem of evaluat-

ing integrals of the form f~o R(x, ~ ) d x is due to Abel and Jacobi. This is simply to invert the problem! So, one considers x0 as a function of s in the formula

s(xo) = f~o R(x, ~ - ~ ) d x . Note tha t the integrand is a multi-valued function while however their inverse functions arise as single-valued functions as Abel and Jacobi realised in the 1820's. This yields addi t ion formulae for these functions akin to those for the t r igonometr ic functions. These integrals and, in general, the integrals of the form s(xo) = f~o R(x, x/~)dx are called elliptic integrals where R is a rat ional function and p is a polynomial of degree 3 or 4 in x (why did we leave out degrees 1 and 2?). The inverse functions x = x(s) are called elliptic functions. They have properties analogous to those for t r igonometric functions; in place of the usual periodicity, they are doubly 1 periodic!

Although we can ' t prove it here, the fundamenta l fact involved here is tha t any cubic curve over the complex numbers which is smooth i.e., has no singularities, is biholomorphic to a complex torus. We digress now to discuss a notion first introduced by Gauss and s tudied in relation with elliptic integrals to get some beaut iful results.

T h e a g M

Let a > b > 0 be two real numbers. Const ruct the following sequence of ar i thmetic and geometric means:

al---- a+b2 ' b l = V f ~ ' a2 = al+b12 , b2 ---- av/-~lbl, " ' " , a n + l =

a=+b. , b n + l = ~ / a n b n , �9 �9 �9 etc. 2


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For example, look at a - 2, b -- 1. We have the following sequence of values correct to eight decimal places:

n an 0 2 1 1.5 2 1.45710678 3 1.45679105 4 1.45679103

bn 1

1.41421356 1.45647531 1.45679101 1.45679103

It is natural to guess tha t the two sequences approach a common number. This is indeed t rue in general and is seen quite easily as follows. From the familiar inequality which asserts tha t the ari thmetic mean is at least as big as the geometric mean, one has an _> bn. In fact,

a >_ al >_ a2 >_ "" >_ an >_ bn >_ bn-1 >_ "'" >_ bl >_ b. Can one say more?

Indeed, it is easy to see tha t any of the a's is at least as big as any of the b's (why?) i.e.,

a >__al >_a2>_"" >_an>_"" >_ bm >_ bin-1 >_""

>_bl >_b.

Moreover, by mathemat ica l induction, it follows tha t an - bn < ~ for all n > 1. What do these two observa- tions give us? The first implies tha t the limits limn--.~an and limn--.o~bn exist. The second implies tha t these two limits are actually equal! This common limit, denoted by M (a, b) is called the arithmetico-geometric mean and denoted by agM, a nota t ion introduced by Gauss him- self. Some of the evident properties of the agM are:

M ( a , a ) = a

M(a ,b) --- M(an, bn) V n

M( ta , t b ) = t M ( a , b )

The fundamental

fact involved here

is that any cubic

curve over the

complex numbers

which is smooth is

biholomorphic to a

complex torus.

RESONANCE j August 2000 77

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78

a g M a n d t h e P e r i m e t e r

Gauss proved the following beautiful result on the perimeter of an ellipse:

T h e o r e m

If a > b > 0, then

fo'~ /2( a 2 cos2 0 ~r + b2sin20)-t/2dO = 2M ~a, b).

Gauss's proof is ingenious and goes as follows. If I(a, b) denotes the integral above, the key step is to show tha t I(a, b) = I(al , bl) for then, repeat ing this, and going to the limit, one will get I(a,b) = I ( M ( a , b ) , M ( a , b ) ) = M(a, b)- l~ , which is the assertion. The proof of the key step is achieved in the following manner by making the subst i tut ion

2asinr sin0 =

a + b + (a -- b)sin2r

This is now called Gauss's t ransformation.

The change of variables gives easily tha t

a2cos2 0 + b2sin2 0 = a2 (a + b - (a - b)sin2r 2 (a + b + (a - b)sin r 2

Let us differentiate both sides of

a + b - (a - b)sin2r (a2cos20 -]- b2sin20) 1/2 ~- a . . . ( A)

a + b + (a - b)sin2r

~:o compute the integral in terms of the new variable r Derivative of the left hand side of (A) is

(a2cos20 -I" b2sin20 ) - l / 2 ( b 2 - a2)sinOcosOdO.

Differentiating the right hand side of (A), one gets the expression

4a(b 2 - a2)sinr162 (a + b + (a - b)sin2r 2dr


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Subst i tut ing for the denominator, this further simplifies to 4a(b 2 - a 2 ) s i n r 1 6 2 1 6 2 i.e., to the expression

(b 2 - a2)cosCsin28sinr162

asinr

On the other hand, using a + b = 2al and ab = b21, one has

(al2cos2r + bl2sin2r162 = a 2 _ (2b 2 - a~)sin2r

(al2-b2)sin4r = 4

(a + b + (a - b)sin2r 2 4a2sin2r

Thus,

(a + b) 2 - 2(a 2 + b2)sin2r + (a - b)2sin4r

a2cos28sin2r

4 - sin28

(a2cos2r + b2sin2r _ sinScosr . acos~sinr

Mutiplying this with (b 2 - a2)sinScos6dr yields just the derivative of the right hand side of (A). In other words, we have verified tha t

(a2cos28 + b2s in28) - l /2d~ =

(al2cos2r + b21sin2r162 From this, the key step follows immediately and so does Gauss's theorem.

The agM M (a, b) can be expressed in terms of one pa- rameter , viz., the eccentricity e = ~ / a of the ellipse. Indeed, M(1 + e, 1 - e) = M(1, x/T-L-'~) = M(1, b) = M ( a , b ) / a . Therefore, the theorem can be restated as

7F

M ( l + e , l - e ) - 2K(e)

where K ( e ) = fo~/2(1 - e2sin2•)-l/2d•.

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80

There is a transformation similar to the one used by Gauss in the proof above. It is due to a little-known English mathematician J Landen and has turned out to be very fruitful in evaluating elliptic integrals. Landen's transformation is the substitution esint9 = sin(2r -/9).

Exercise. Verify that this transformation can also be used to prove the above theorem of Gauss.

Yet another result that can be proved by a repeated usage of Landeri's transformation is the following. For a > b > 0, denote by J(a, b), the integral f~/2(a2cos20 + b2sin2O)W2dS. Note that J(a, b) = aE(e) where E(e) = f~/2(1 - e2sin20)l/2dO. Then, we have:

Proposition o o

J(a,b) = (a 2 - Z 2n-l(a2 - b2))I(a'b)" n=O

We shall return to this proposition in the next section. For some other applications of Landen's transformation, the interested reader may see [2].

In the case of a lemniscate, the total perimeter is

f01r cos(2O)-l/2d/9 = 4 f0~r/2(2COS2r + sin2~)-l/2d~

from the change of variables

cos(2O) = cos2a.

So, in this case, the constants a and b in Gauss's theorem are V~ and 1, respectively. Therefore, one has

- fo M(v~, 1 ) = 2w where w = (1-z4)- l /2dz . (0)

Gauss's 92nd entry made in 1798 in his famous math- ematical diary reads "I have obtained most elegant results concerning the lemniscate, which surpasses all ex- pectation - indeed, by methods which open an entirely


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new field to us". The 98th entry made in May 1799 says that"demonstration of this fact (the identity ~ ) will open an entirely new field of analysis". What is this 'new field of analysis' that is being talked about? W h e n a and b are complex numbers, the square-root causes trouble i.e., at each stage bn has two possible choices. Thus, M(a, b) is a multi-valued function and the various different values are related. Determining the relations between them requires the so-called theory of modular forms of weight 1 for congruence subgroups of SL(2, Z). This is what Gauss refers to as ' the new field of analysis.'

W a y s t o A p p r o x i m a t e 7r

The fascination of calculating approximations to 7r is said to have existed among mathematic ians for as long as 4000 years - even laymen have had their fingers in the ~r! The study of zr has led to deep questions - an example is the ancient Greek problem of squaring the circle - see the discussion of this problem and other related ones in [3]. It was only at the end of the 19th century that L indemann (1852-1939) finally proved tha t 7r is tran- scendental i.e., tha t there is no nonconstant polynomial with rational coefficients that has ~r as a root.

Several great mathemat ic ians haste given'formulae' or approximations. Perhaps, the earliest was Archimedes (287-212 B.C.) who gave us the approximation ~13 < 7r < -~. The notat ion 7r is also due to h i m . Aryabhat ta

6 2 8 3 2 _ _ (476-550 A.D.) gave the approximation 7r ,,~ 2000o - 3.1416. Leibniz (1646-1716) and Euler (1707-1783) gave the respective formulae:

lr 1 1 1 ~ = 1 - ~ + g - 7 + " "

~2 1 1 1 - - = 1 + + + + . . . 8

Legendre gave the following formula for lr in terms of the elliptic integrals ([4] has a new and rather easy proof):


The fascination of

calculating

approximations to

is said to have

existed among mathematicians for as long as 4000

years - even

laymen have had

their fingers in the

m

FEATURE I ARTICLE

Legendre's Formula

Let O < e < 1 and e' = ~ . Then

7r K(e )E(e ' ) + K(e t )E(e ) - K ( e ) K ( e ' ) = 7"

Let us use this formula with the special value e = e ~ = 1/x/~ to give an approximation for 7r which involves the agM. This will be seen to be reminiscent of Archimedes's method. His method was to approximate the unit circle by inscribed and circumscribed regular polygons. Then, one has Pn < 21r < Qn where Pn, Qn are, respectively, the perimeters of the inscribed and the circumscribed regular polygons of n sides. In fact, the numbers an := 1/Q2, and b, = 1/P2~ satisfy the recursion an+l = (an + bn)/2, bn+l = x/a,+lbn. This is very similar to the defining recursion for M(1/2, 1/4). However, Archimedes's sequences converge much more slowly.

Now, Legendre's formula gives

2 K ( 1 / V ~ ) E ( 1 / v ~ ) - K(1/V~)2 = 7"

On the other hand, the proposition above implies (with a = 1, b - l /v/2) that

E(1/v/2) = (1 - E 2n-l( a2 - b2) )K(1 /V~) �9 n = 0

Combining these two with Gauss's theorem, we have

~r(1 - E 2i+1( a2 - b2)) = 4M(1, l/V/'2) 2" i=1

Thus, one has the approximations

2 n i+1 ^2 ~rn = 4an+ f f ~'~i-=1 2 (~i - b2) which converge quadrat- ically to ~r. In other words, the number of correct digits doubles at each step.


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Finally, we end with a formula due to Ramanujan (see [5]). Ramanujan used the theory of modular functions (this was alluded to at the end of the last section) to give some amazing formulae for ~r as well as for the perimeters of ellipses. One of these is:

1 oo 1103 + 26390n (4n)!

2~rv~ = ~ 9 9 4 ~ + 2 44~(n!) 4" n=0

Using just the first term, one gets the approximation 0.112539536... to ~ = 0.112539539..:

Suggested Reading

[1] B Sury, Resonance, Vol.4, No.12, p.48, 1999. [2] B Berndt, Ramanujan'$ Notebooks, Part III, Springer-Verlag, 1991. [3] Shashidhar Jagadeesan, Resonance, VoL 4, No. 3, 1999. [4] G Almkvist and B Berndt, Amer. Math. Monthly, Vol.94 585-608, 1987. IS] J H Borwein and P B Borwein, P/and theAGM, Wiley, Canada, 1987. [6] D Cox, L'Enseignonent Matlwmatique, Vol.30, 275-330,1984.

Address for Correspondence B Sury

Statistics & Mathematics Unit Indian Statistical Institute Bangalore 560050, India.

Email: [email protected]

The Square Root by Infinite Descent

One of the first instances of a proof by contradiction that one learns in school is that of the irrationality of x/~. Here is a proof due to Tannenbaum by 'infinite descent' - the method employed by Fermat

a 2 and by Brahmagupta even earlier in other contexts. Suppose 2 = ~- where a, b are natural numbers. In the figure, ABCD is a square of side a, PBQV and STRD are both squares of side b. As a 2 = 2b 2, one has Area(TUVW) = Area(PWSA) + Area(QCRU). This means

( 2 b - a ) 2 = 2 ( a - b) 2 i.e., 2 = ~ . On the other hand, it is clear from

the figure tha t 0 < 2b - a < a which produces a smaller numerator than the original one setting in a process of infinite descent. The resultant

contradiction forced on us proves the irrationality of v~.

A T

S W

D V

D R r Kanakku Puly

RESONANCE t August 2000 83

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