Transcendental Numbers. (AM-16) 9781400882359

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ANNALS OF MATHEMATICS STUDIES 0 NUMBER 16

ANNALS OF MATHEMATICS STUDIES Edited by Emil Artin and Marston Morse 1. Algebraic Theory of Numbers, by H e r m a n n W e y l 3. Consistency of the Continuum Hypothesis, by K u r t G o d e l 6. The Calculi of Lambda-Conversion, by A lo nzo C hurch

7. Finite Dimensional Vector Spaces, by P a u l R. H a lm o s 10. Topics in Topology, by S o lo m o n L e f s c h e t z

11. Introduction to Nonlinear Mechanics, by N. K r y l o f f and N. B o g o l i u b o f f 14. Lectures on Differential Equations, by S o lo m o n L e f s c h e t z 15. Topological Methods in the Theory of Functions of a Complex Variable, by M arston M o rse 16. Transcendental Numbers, by C a r l L udwig S ie g e l 17. Probleme General de la Stabilite du Mouvement, by M. A. L ia p o u n o ff 19. Fourier Transforms, by S. B o ch n er and K. C ha n d ra sek h a ra n

20.

Contributions to the Theory of Nonlinear Oscillations, Vol. I, edited by S. L e f s c h e t z

21.

Functional Operators, Vol. I, by J ohn

von

22. Functional Operators, Vol. II, by J ohn

Neum ann von

N eu m a n n

23. Existence Theorems in Partial Differential Equations, by D orothy L B e r n st e in

24. Contributions to the Theory of Games, Vol. I, edited by H. W. A. W. T u c k e r

Kuhn

and

25. Contributions to Fourier Analysis, by A. Zyg m un d , W. T r a n su e , M. M o r se , A. P. C a ld e r o n , and S. B o c h n e r 26. A Theory of Cross-Spaces, by R o b e r t S c h a t t en

27. Isoperimetric Inequalities in Mathematical Physics, by G. G. S z e g o 28. Contributions to the Theory of Games, Vol. II, edited by H. A. W. T u c k er

P olya

and

Kuhn

and

29. Contributions to the Theory of Nonlinear Oscillations, Vol. II, edited by S. L e f s c h e t z 30.

Contributions to the Theory of Riemann Surfaces, edited by L. A h l fo r s et al.

TRANSCENDENTAL NUMBERS B y Carl Ludw ig Siegel

PRINCETON PRINCETON UNIVERSITY PRESS 1949

C O P Y R IG H T

1949,

B Y P R IN C E T O N U N IV E R S IT Y PR ESS

L O N D O N : G E O F F R E Y C U M B E R L E G E , O X FO R D U N IV E R S IT Y PRESS

P R IN T E D IN T H E U N IT E D S T A T E S O F A M E R I C A

PREFACE T h is b o o k le t r e p ro d u c e s w it h s l i g h t ch an ges a co u rse o f le c tu r e s d e liv e r e d S p r in g term 19^6.

i n P r in c e t o n d u r in g th e

I t w ould be m is le a d in g t o c a l l i t a

t h e o r y o f t r a n s c e n d e n t a l num bers, o u r kn ow led ge c o n ­ c e r n in g t r a n s c e n d e n t a l numbers b e in g n a r r o w ly r e s t r i c t e d . The t e x t d e a ls w it h a few s p e c i a l tr a n s c e n d e n c y p roblem s o f some i n t e r e s t , b u t i t

i s more th a n a mere c o l l e c t i o n

o f s c a t t e r e d e x a m p le s , s in c e i t

i n v o lv e s a method w h ich

m igh t be u s e f u l i n th e s e a r c h o f more g e n e r a l r e s u l t s . C a r l Ludwig S i e g e l .

A p r il,

19^9

P r in c e t o n , New J e r s e y .

CONTENTS

Preface

v

CHAPTER I .

THE EXPONENTIAL FUNCTION............................

1

§1 . The I r r a t i o n a l i t y o f e ......................................................... 1 §2. The o p e r a t o r fTD)

..........................................k

§ 3 . A p p r o x im a tio n t o ex b y r a t i o n a l f u n c t i o n s

.

. 6

§4. The i r r a t i o n a l i t y o f e a f o r r a t i o n a l a ^ o .

. . 8

§5. The i r r a t i o n a l i t y o f t t .........................................................9 §6. The i r r a t i o n a l i t y o f t g a f o r r a t i o n a l a^O. 10 § 7 . The f u n c t i o n P e

P1X

+ .

. . + Pme

Pmx

. . . .

12

§8. E s t im a t io n o f R( 1 ) ....................................................

.

16

§9 . E s t im a t io n o f R ( 1 ) and i t s d en o m in a to r . a §10. The tr a n s c e n d e n c y o f e f o r r e a l

.

16

a l g e b r a i c a ^ 0 ..................................................................... 1 7 §11 . The d e te rm in a n t o f m a p p r o x im a tio n fo rm s .

.

18

.

2k

§ 1 2. A lg e b r a i c in d e p e n d e n c e ..................................... § 1 3 . A n o th e r e x p r e s s i o n f o r th e re m a in d e r t e r m §lt.

20

The i n t e r p o l a t i o n f o r m u l a ............................................27

§ 1 5 . C o n c lu d in g r e m a r k s ...............................................................30 CHAPTER I I .

SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS........................................................................ 51

§1 . F u n c tio n s o f ty p e E .......................................................... 33 § 2 . A r i t h m e t i c a l le m m a s .......................................................... 35 §3 . A p p r o x im a tio n f o r m s .................................

38

§ 4 . Normal s y s t e m s ........................................................................ ^0 § 5 . The c o e f f i c i e n t m a tr ix o f th e a p p r o x im a tio n f o r m s .......................................................... ^

v ii

viii

CONTENTS

§6. E s tim a tio n o f R^. and P ^ ..............................................^7 §7 . The rank o f E1 (

. . . ,Em( a ) ............................... §8. A lg e b r a ic in depen den ce.................................................. 52 a ),

§9. H ypergeom etric E - f u n c t io n s ......................................... 5 k §10. The B e s s e l d i f f e r e n t i a l e q u a tio n .............................. 59 §1 1 . D e term in atio n o f th e e x c e p tio n a l ca se . . .

63

§12. A lg e b r a ic r e la t i o n s in v o lv in g d i f f e r e n t B e s s e l f u n c t i o n s ............................................................... 65 § 13 - The n o rm a lity c o n d itio n f o r B e s s e l fu n c tio n s 68 § l t . A d d itio n a l rem arks...........................................................72 CHAPTER I I I .

THE TRANSCENDENCY OF ab FOR IRRATIONAL

ALGEBRAIC b AND ALGEBRAIC a ^0, 1 .......................... 75 § 1 . S c h n e id e r 's p r o o f .........................................................'7 6 §2. G e lfo n d 's p r o o f ............................................................... 80 §3. A d d itio n a l rem arks........................................................... 83 CHAPTER §1 . §2. §3.

IV . ELLIPTIC FU N C T IO N S..............................................83 A b e lia n d i f f e r e n t i a l s .................................................. 85 E l l i p t i c i n t e g r a l s ...........................................................87 The a p p roxim ation form .................................................. 90

§ij-. C o n clu sio n o f th e p r o o f ..............................................92 §5. Some o th e r r e s u l t s ........................................................... 95 B ib lio g r a p h y ........................................................................................1 01 1

CHAPTER I THE EXPONENTIAL FUNCTION The most w id e ly known r e s u l t on t r a n s c e n d e n t a l numbers i s th e tr a n s c e n d e n c y o f x p ro ved b y Lindemann i n 1882.

H is method i s b ased on H e rm ite f s p r e v io u s w ork

who d is c o v e r e d th e tr a n s c e n d e n c y o f e i n 1 873.

B oth

r e s u l t s a r e c o n ta in e d i n th e g e n e r a l Lindem annW e ie r s t r a s s th eo rem w h ich w i l l be p ro ved i n §1 2.

We

s h a l l s t a r t w it h some s im p le r p r o b le m s , n am ely th e i r r a t i o n a l i t y o f e and * and r e l a t e d q u e s t i o n s . § 1 . The i r r a t i o n a l i t y o f e The u s u a l p r o o f o f th e i r r a t i o n a l i t y o f e run s a s fo llo w s .

From th e s e r i e s

we g e t th e d e c o m p o s itio n e

k! (n = i, 2 , .

S in c e

1

2 we f in d t h a t

I . THE EXPONENTIAL FUNCTION

th e r e fo r e

0 < r n < ( n + 1 )! Put n! Rn = a n , th e n th e number a n i s

n! r n = bQ,

i n t e g r a l and

n fo r n = 1 , 2 , . . . . a fo r t io r i,n e is

T h is p r o v e s t h a t n! e = a n + bn a n d , i s n e v e r an in t e g e r .

ir r a tio n a l. The p r o o f i s

still

f o r e ” 1 in s te a d o f e . -i

„

" n * »n'

In o t h e r w o rd s , e

s im p le r , i f we u se th e s e r i e s

Then

lzl Lk

k!

.

-

' ' n ' fesT

k!

(n=l, 2 , . . . ) and ( 2 |a| + | c | .

On th e o t h e r hand we h ave th e

r e c u r s io n fo rm u la “V i

" Rn = a (n b n - i

-

V

+

CV 0 n -1 '

3 n>

= a + (-1 )n c . I t f o ll o w s t h a t a t l e a s t one o f th e t h r e e numbers ^ 1 ,

Rn+1 i s d i f f e r e n t fro m 0 , s in c e o th e r w is e

a+ c= o, a - c =0 and a= 0 , c= 0 .

T h is .shows th e e x i s t e n c e o f

a p o s i t i v e i n t e g e r v su ch t h a t E^ i s n o t i n t e g r a l , and t h e r e f o r e th e number

I.

THE EXPONENTIAL FUNCTION b + tj ! = ae + b + ce 1

i s d i f f e r e n t from 0, f o r a l l i n t e g r a l b .

T h is means

th a t ae

2

+ be + c + 0

f o r a r b i t r a r y i n t e g e r s a , b , c , n o t a l l 0.

In o t h e r

w o rd s , e i s n o t a q u a d r a t ic i r r a t i o n a l i t y . §2.

The o p e r a t o r f ( D )

We d e n o te b y D th e d i f f e r e n t i a t i o n w it h r e s p e c t t o th e v a r i a b l e x .

If

f(t) = aQ + a t

+ a 2t

2

+ ...

i s a pow er s e r i e s w it h r e a l o r com plex c o e f f i c i e n t s a Q, a 1 ,

...

and