gauss, landen, ramanujan, the arithmetic-geometric mean, ellipses, pi, and the ladies diary

7/26/2019 Gauss, Landen, Ramanujan, the Arithmetic-Geometric Mean, Ellipses, Pi, and the Ladies Diary

1/25

Gauss, Landen, Ramanujan, the Arithmetic-Geometric Mean, Ellipses, , and the Ladies DiaryAuthor(s): Gert Almkvist and Bruce BerndtReviewed work(s):Source: The American Mathematical Monthly, Vol. 95, No. 7 (Aug. - Sep., 1988), pp. 585-608Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2323302.

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2/25

Gauss,Landen,

Ramanujan,

heArithmetic-Geometric

ean,

Ellipses, rr,nd the

LadiesDiary

GERTALMVIST, UniversityfLund

BRUCE

BERNDT*,Universityf llinois

GERT

ALMKVISTeceived

is Ph.D. at theUniversityf Californian

1966

and has been t

Lund ince

967.His main nterestsre lgebraic

-theory,

invariant

heory,

nd

elliptic unctions.

BRUCEBERNDT

eceived

is

A.B. degree romAlbion

College,Albion,

Michigan

n

1961 nd his

Ph.D. fromhe

UniversityfWisconsin,adison,

in196.

Since

977,

he

hasdevotedll

ofhisresearchffortso

proving

he

hitherto

nproven

esultsn

Ramanujan's

otebooks.is

book,

Ramanujan's

Notebooks,

art (Springer-Verlag,985), s thefirstfeitherhree rfour

i

volumes to be

publshed

on

this

project.

Virtuendsense,

with emale-softness

oin'd

(All that ubdues nd

captivates ankind )

In

Britain's atchlessair

esplendent

hine;

They ule ove's mpire y rightivine:

Justly

heir harmshe stonished orld

dmires,

Whom

oyal

Charlotte's

rightxample

ires.

1. Introduction.he

rithmetic-geometric

eanwas first iscovered

y

Lagrange

and

rediscovered

y

Gauss a few

years

aterwhilehe was a

teenager.

owever,

Gauss's

major

contributions,ncluding

n

elegant ntegral epresentation,

ere

madeabout7-9 yearsater. he firsturposef this rticles, then,oexplain he

arithmetic-geometric

ean nd to describe ome

of ts

major

properties, any

f

which re due to

Gauss.

*Research

artially

upportedy

the

Vaughnoundation.

585

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3/25

586

GAUSS, ANDEN,

RAMANUJAN...

[August-September

Because f

tsrapid

onvergence,he

rithmetic-geometric

eanhas

been ignifi-

cantly

mployed

n

the

astdecaden

fast

machine

omputation.

second urpose

of

this rticle s

thus o

delineatets

role n

the omputationf

7T.

Weemphasizethat thearithmetic-geometriceanhas muchbroader

pplications,

.g., to the

calculation

f

elementary

unctions

uch s logx,

ex, sinx,

and

cosx. The

inter-

ested

reader

houldfurther

onsult he

several

eferences

itedhere,

specially

Brent's aper

14]

and the

Borweins'

ook 13].

The

determinationf

the

rithmetic-geometricean

s

intimately

elated o the

calculation

fthe

erimeter

f n

ellipse.

ince he ays

f

Kepler nd

Euler, everal

approximate

ormulas ave

been

devised o

calculate

he

perimeter.

he

primary

motivation

n

derivinguch

pproximations

as

evidently

hedesire o

accurately

calculate he

elliptical

rbits f

planets. third

urpose f

this

rticles

thus o

describehe onnectionsetweenhe rithmetic-geometricean nd theperimeter

of

an

ellipse, nd to

survey

any

f

the

pproximate

ormulashat

avebeen

given

in

the

iterature.he

most

ccurate f these

s due to

Ramanujan, ho

lso

found

some

extraordinarilynusual nd

exotic

pproximations

o

elliptical

erimeters.

The

atter

esultsrefoundn

hisnotebooks

ndhave

never

een

published,nd so

we shall

pay particularttention

o these

pproximations.

Also

contributing

o

this

circle f

ideas is

the

English

mathematician

ohn

Landen.

n

the

tudy fboth

he

rithmetic-geometric

ean

nd the

determination

of

elliptical

erimeters,

here rises

is most

mportant

athematical

ontribution,

whichs nowcalledLanden's ransformation.anyverymportantndseemingly

unrelated

uises f

Landen's

ransformationxist

n

the

iterature.hus,

fourth

purpose

f

this

rticles to

delineate

everal

ormulationsf

Landen's

ransforma-

tion

s well as

to

provide short

iography

f

this

ndeservedly,ather

bscure,

mathematician.

For

several

ears, anden

ublished

lmost

xclusively

n

the

Ladies

Diary. his

is,

historically,hefirst

egularly

ublished

eriodicalo

contain

section

evoted

to the

posing

f

mathematical

roblemsnd their

olutions.

ecause n

important

feature f

the

MONTHLY

has

its roots

n

the

Ladies

Diary,

it

seems

hen

dually

appropriatenthis aper oprovide brief escriptionf theLadiesDiary.

2.

Gauss and the


ean.As we

previously

lluded,

he


ean

was first et forth

n

a

memoir f

Lagrange30]

pub-

lished

n

1784-85.

However,

n

a

letter,

ated

April

6,

1816,

o a

friend,

. C.

Schumacher,

auss

confidedhathe

independently

iscovered

he

rithmetic-geo-

metricmean

n

1791

t the

ge

of

14. At about he

ge

of

22

or

23,

Gausswrote

longpaper

23]

describing

is

many

iscoveriesn

the

rithmetic-geometric

ean.

However,

his

work,

ike

many

thers

y

Gauss,

was

not

published

ntil fter

is

death.Gauss's

fundamental

aper

hus

id

not

ppear

ntil 866

when .

Schering,

theeditor f Gauss'scompleteworks, ublishedhepaperas partof Gauss's

Nachlass.

Gauss

obviously

ttached

onsiderable

mportance

o his

findings

n the


ean,

or everal fthe

ntries

n

his

diary,

n

particular,

rom

the

years

799 to

1800,pertain

o the


ean.Some

of these

entries

re

quite

vague,

ndwe

may

till

otknow

verything

hat

Gauss

discovered

aboutthe

rithmetic-geometric

ean.

For

an

English

ranslationf

Gauss's

diary

together

ith

ommentary,

ee a

paper y

J.J.

Gray 24].)

By

now,

hereaders anxious o earn bout

he

rithmetic-geometric

ean

nd

what

he

young

Gaussdiscovered.



4/25

1988]

GERT ALMKVIST

AND BRUCE BERNDT

587

Let

a and b denotepositive

umbers

ith

a

>

b. Construct

sequence

f

arithmetic

eans nd

a

sequence

f

geometric

eans s

follows:

1

a,l=

-(a

+b),

b=

4,b

1

a2

=

-(a,

+

bl)9 b2= Ha1,1,

1

a

=

+

-(an

+

bn),

bnl

=

b

Gauss

[23]

gives

our

umerical

xamples,

f

whichwe

reproduce

ne. Let

a

=

1

and

b

=

0.8. Then

al

=

0.9,

b,

=

0.894427190999915878564,

a2

=

0.8972135954999579392829 b2

=

0.8972092687327349

a3

=

0.897211432116346,

b3

= 0.897211432113738,

a4

=

0.897211432115042, b4

=

0.897211432115042.

(Obviously,

auss

did

not hirk

rom umerical

alculations.)

t

appears

rom his

example

hat

an)

and {

bn)

converge

o the

ame

imit,

nd that urthermorehis

convergence

s

very apid.

his

we nowdemonstrate.

Observe hat

b

1,

we see that

11)

reduces

o

a

Wenow et

n

tend o

x.

Since

n

tends o

M(a,

b)

and

xn

tends o

0,

we

conclude

that

a

a7T

K(x)-=M(a,

b)

K(O)

2M(a,

b)

Landen's

ransformation

10)

was

ntroduced

y

him n

a

paper

31]

published

n

1771

and

n

more

eveloped

orm

n

his

most amous

aper 32]

published

n

1775.

There xist everal

ersionsf Landen's

ransformation.

ften

anden's ransfor-

mation

s

expressed

s an

equality

etweenwodifferentials

n

the

heory

f

elliptic

functions17], 37].The importancef Landen's ransformations conveyedy

Mittag-Leffler

ho,

nhis

very erceptive

urvey

37, .

291]

n the

heory

f

lliptic

functions,

emarks,

Euler's ddition heoremnd the

transformationheoremf

Landen and

Lagrange

were

the two fundamentaldeas of which

he

theory

f

elliptic

unctions as n

possession

hen

his

heory

as

brought

p

for enewed

consideration

y Legendre

n

1786."

In

Section

,

we

shall

rove

he

ollowing

heorem,

hich

s often alled anden's

transformationor

omplete

lliptic

ntegrals

f

the

first ind.

THEOREM

2.

If

0

2

VJb

He [28, p.

368]

furthermore

emarks

hat 1/2)(a +

b) >

V,

and

so

concludes

that

L

T

h-(a

+

b).

2

Kepler ppears o be

using he dubious

rinciple

hat uantities

arger

han he

same

number

must

e

about

qual.

Approximationsf

several

ypes,

epending

pontherelativeizesof a and

b,

exist

n

the

iterature.n

this

ection,

e

concentraten

estimateshat rebest or

close

to

b.

Thus,

we

shall write ll of

our

approximations

n

terms f

X

(a - b)/(a + b) andcomparehemwithhe xpansion

25).

Forexample, epler's

second

pproximation

an be

written

n

theform

L

7T(a

+

b)(1

-

A

)

We now how

how

heformula

/

1

00

C\

L(a, b)

=

4J(a,

b)

=

M( bIa2-

2E

2ncn),

(27)

arising romTheorems ' and 4, can be used to findapproximationso the

perimeter

f an

ellipse.

Replacing

M(a, b) by a2

and

neglecting

he termswith

n

>

2,

we

find

hat

L(a,

b)

-

(a2

-

- 12 = T

This

formula

was first btained

y

Ekwall

19]

n

1973 as a

consequence

f a

formula

y Sipos

from 792

54].



17/25

600

GAUSS, LANDEN,

RAMANUJAN...

[August-September

If

we

replaceM(a, b)

by a3

in

27)

andneglect ll terms

ithn

>

3, we

find,

after ome

alculation,hat

2(a

+

b)2 (

a _

b)4

L(a,b) 27Tr

2

(Va

+

V)

+

2V4/a

b

Vab>

This formula

s complicated

nough

o dissuade s from

alculating urther

p-

proximations

y thismethod.

We nowprovide

table

f approximations

or (a, b) that ave

beengiven

n

the

iterature.t the

eft,

e ist he

discovereror source)

ndyear fdiscovery

if

known).

The approximation

(X) forL(a,

b)/7T(a

b) is given n the

second

column

n

two

forms.

n

the

astcolumn,he

first onzero erm

n thepower

eries

for

AA-7T(a

+

b)

=A )-

-2 '2;1

)

is offered

o that

he ccuracy f the

pproximating

ormulaan

be discerned.or

convenience,

e

note

hat

F(

-

-

'

-

2

;

1;

A?)

=

1

+

-

A?

+

43 A

+

4

+

47 A

+

48

A1+

Kepler 28], 1609

-

+=

(1

-

X2)1/2

3

Euler

21],

1773

2 1 +

b2)1/

2

+

a

+b

4

Sipos [54],

1792

2(a

+

b)

2

7

Ekwall

[19],

1973 (f

+

)2

1

+

1

64

Peano

[42],

1889 --

=---(1

-

K')1/2

34

2 a +b

2 2

6

2

(a3/2_+ b3/2

2/3

Muir

38],

1883

a+ b

2

1

1

1

64

=

/3

{(1

+

X)3/2

+

(1

-X)3/22/3

(

1

(a-b\22

Lindner

35,

p.

439],

1?-j

I

1

1904-1920 ( 8(a+b)

j

-1x6

Nyvoll 41],

1978

( lX2)2

28

Selmer

49,

975 1 +

4(a

-

b)

Selmer

49], 1975

1 +

(5a

+ 3b)(3a

+ 5b)

3

-1

+

x2

1-

-X2

-

4

1

X2



18/25

1988]

GERT

ALMKVIST

AND

BRUCE

BERNDT 601

Ramanujan

[44],

[45],

1914

-

a3)(ab

1

a

+

b

-A

Fergestad 49], 19513- 4- 2

Almkvist

1],

1978

2

2(+

-(V-

'i)

(a +b){(

V)+ 2V

a

+

bv%b

15

-

X8

(1+

+1

X22?X i-x2-

21

-2

~~~~~~~~~~2

(1

+

1 +2)114 2

Bronshtein nd

1

64(a

+

b

4-

3(a

-

b)

Semendyayev 16

(a

+ b

2

(3a +

b)(a

+

3b)

219

[15],

1964

64

-

3X4f

Selmer

49],

1975=64-1X

6{

2

166a+b)

(

a- b\2

2V2(a

+6+b2

Selmer

49],

1975

a

2

+

b

1

3-?- 2--1-X

2 8

2 2

Jacobsen nd 256

-

48X2 21X

33x1

Waadeland

[26],

1985

256

-

112X2

3X

218

Ramanujan

1+

3X2

3

[44],

[45],

1914

10+4-3f27

The

two

approximationsy

India's

great

mathematician,

.

Ramanujan,

ere

first tated

y

him

n

his notebooks

46,

p.

217],

nd then ater t theend of

his

paper

[44],

45, p. 39],wherehe saysthattheywere discoveredmpirically.

Ramanujan

44],45]

also

provides

rror

pproximations,

ut

they

re

in

a

form

different

rom

hat

iven

ere.

ince

a -b 1

-

11-

e2

e2

a +b

1?

1j-

e2

T4

we

find

hat,

or hefirst

pproximation,

(e2/4)

6

__2

__

~~(a?b)j~~a(1?

1-e2)

~29

2. Proceeding y nduction, e deduce that

an+1

2

af

for n > 2, and theproof f 34) is complete.

From

32)

and

(34),

it follows

hat

3

-

r4

-?2

7T/6.

Second,

we calculate

a when

=

1.

Thus,

X

=

1

and

0

=

a.

Therefore,

rom

25)

and

(32),

1 /11\

4

1 + 4sin2-a

=F--,--;1;1I

=

-. (35)

2 \2 2J

This evaluation

follows

from

general

theorem

f Gauss

on the evaluation

of

hypergeometric

eries t

the

argument [4, p.

2].

Moreover,

his

particular

eries s

found

n

Gauss's

diary

under

he date

June,

798

[24].

Thus,

1 1 1

sin2

-a

=-

-

-

=

0.0683098861.

2

l

4

It follows

hat

a

=

30'18'6"~.



21/25

604

GAUSS, ANDEN,

RAMANUJAN...

[August-September

Third,

we calculate

when

=

0. From 30) and

32),

sin2o

4

sin2-0

lim in2a lim

lim

2

00

21

=

M

im

2

adin

F=

ai~

A ?

n 1

4

Thus,

tends

o

ir/6

s e tends o 0.

Ramanujan

46, p. 224] offers nother

heorem,

hichwe do not state,

ike

Theorem

but which ppears o be

motivatedy

his second pproximation

or

L(a, b).

Ramanujan46,p. 224] states wo dditional ormulasachof which ombines

two

pproximations,

ne for near and the ther

or close o1. Again,we

give

just

one of

the

pair.

A completeroof

fTheorem belowwould

e too engthyor

this

aper,

nd

so

we

shall

ust

ketch

he

main deas

ofthe

proof.

ompleteetails

may

be found

n

[7].

THEOREM . Set

tan

L(a, b)-= r(a

+ b)

,

0

gauss, landen, ramanujan, the arithmetic-geometric mean, ellipses, pi, and the ladies diary

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