AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL 6

Fourier Series and Orthogonal

Polynomials

Dunham Jackson

The C a m s M a t h e m a t i c a l M o n o g r a p h s

NUMBER SIX

FOURIER SERIES AND

ORTHOGONAL POLYNOMIALS

By

DUNHAM JACKSON Professor of Mathematics, The University of Minnesota

Published iy

T H E M A T H E M A T I C A L A S S O C I A T I O N O F A M E R I C A

10.1090/car/006

Copyright 1941 byThe Mathematical Association of America

Library of Congress Catalog Card Number: 41024829

Paperback ISBN 978-0-88385-135-7

eISBN 978-1-61444-006-2

Hardcover (out of print) ISBN 978-0-88385-006-0

Hardcover history:

Fourth Impression, April, 1957

Fifth Impression. January, 1961

Sixth Impression April, 1963

T H E

C A R U S M A T H E M A T I C A L M O N O G R A P H S

Published by

T H E M A T H E M A T I C A L A S S O C I A T I O N O F A M E R I C A

Publication Committee

G I L B E R T A M E S B L I S S

D A V I D R A Y M O N D C U R T I S S

D U N H A M J A C K S O N

A U B R E Y J O H N K E M P N E R

S A U N D E R S M A C L A N E

J O S E P H M I L L E R T H O M A S

THE CARUS MATHEMATICAL MONOGRAPHS are an expression of the desire of Mrs. Mary Hegeler Carus, and of her son, Dr. Edward H. Carus, to contribute to the dissemina-

tion of mathematical knowledge by making accessible at nominal cost a series of expository presentations of the best thoughts and keenest researches in pure and applied mathematics. The publication of the first four of these monographs was made possible by a notable gift to the Mathematical Association of America by Mrs. Carus as sole trustee of the Edward C. Hegeler Trust Fund. The sales from these have resulted in the Carus Monograph Fund, and the Mathematical Association has used this as a revolving book fund to publish the fifth and sixth monographs.

The expositions of mathematical subjects which the monographs contain are set forth in a manner comprehensible not only to teachers and students specializing in mathematics, but also to scientific workers in other fields, and especially to the wide circle of thoughtful people who, having a moderate acquaintance with elementary mathe-matics, wish to extend their knowledge without prolonged and critical study of the mathematical journals and treatises. The scope of this series includes also historical and biographical monographs.

The following books in this series have been published to date: No. 1. Calculus of Variations, by G . A . BLISS No. 2. Analytic Functions of a Complex Variable, by D . R . CURTISS No. 3. Mathematical Statistics, by H. L. RIETZ No. 4. Projective Geometry, by J . W . YOUNG No. 5. A History of Mathematics in America before 1900, by D . E.

SMITH and JEKUTHIEL GINSBURG No. 6 . Fourier Series and Orthogonal Polynomials, by DUNHAM

JACKSON

P R E F A C E

T h e under lying t heme of th is monograph is t h a t the fundamenta l s implici ty of the propert ies of or thogonal functions and the deve lopments in series associated wi th t hem no t only commends t h e m to the a t t en t ion of the s t u d e n t of pure ma thema t i c s , b u t also renders t hem in-evi tably i m p o r t a n t in the analysis of na tu ra l phenomena which lend themselves to ma themat i ca l descript ion.

I t is the essence of ma thema t i c s t h a t i t concerns it-self wi th those relat ions which lie so deep in the na tu re of things t h a t they recur in the mos t varied s i tuat ions . Th i s is par t icu lar ly t rue , of course, of the rud imen ta ry not ions of a r i thmet i c and geomet ry which have forced themselves on the a t t en t ion of mank ind since the earli-es t beginnings of though t . B u t wi th the advance of sci-ence and the accompanying extension of the range of phenomena subjected to q u a n t i t a t i v e discussion, more highly organized groups of concepts , gradual ly simpli-fied b y reduct ion to their essentials, have come to mani-fest themselves wi th similar persistence.

Among these a re the formulat ions relat ing to the general analy t ica l concept of or thogonal i ty , and , in a more restr ic ted field, the par t i cu la r types of or thogonal sys tems discussed in the following chapte rs .

T h e choice of mate r ia l is guided b y two main lines of deve lopment , which are in t imate ly associated from the beginning, b u t u l t imate ly diverge in accordance wi th a l t e rna t ive principles of general izat ion.

For the s t a n d a r d me thod of solution of the "par t ia l differential equa t ions of ma thema t i ca l physics" b y means of or thogonal functions the Fourier series m a y

ν

vi PREFACE

be regarded as t h e p ro to type , corresponding to the use of rec tangula r coordinates for purposes of geometr ic representa t ion . Different choices of coordinate sys tem lead to t h e series of Legendre , Laplace , and Bessel, as well as o thers no t considered here a t all. T h e words Fourier series in the t i t le of the book are in tended to typify, if n o t adequa te ly to describe, th is phase of the discussion.

The reader whose primary interest is in the physical applications may find it most satisfactory to begin with Chapters IV and V, referring b a c k to the earlier chap-ters as the need for such reference becomes appa ren t , and pos tponing the s t u d y of convergence for la ter con-siderat ion.

As orthogonal polynomials, under the second half of the t i t le , the Legendre polynomials are in a sense the simplest , hav ing the cons t an t un i t y as weight function. O the r weight functions correspond to the Fourier series of sines and cosines separa te ly (in connection wi th an auxi l iary change of var iable) , t he der ivat ives of the Legendre polynomials , the more general polynomials of Jacobi , and those of He rmi t e and Laguerre . F rom com-parison of these var ious types it is na tu r a l to proceed to t h e considerat ion of o r thogona l polynomials wi th an a rb i t r a ry weight function.

Wi th in the compass of the m a t t e r to be presented, t h e order of topics has n o t been governed b y str ict adherence to e i ther of t h e courses of deve lopmen t thus out l ined. Various depa r tu re s from w h a t might be re-garded after the event as the mos t d i rec t logical sequence have appeared expedient in the in teres t of compactness of expression or facility of comprehension. In case of conflict be tween these des idera ta the la t te r has been re-garded as of p a r a m o u n t impor tance .

PREFACE vii

For the reading of mos t of the book no specific prepara t ion is required beyond a first course in the cal-culus. In Chap te r s V I I I , I X , and X , and in some of the exercises on C h a p t e r I I I , an acqua in tance with funda-men ta l facts a b o u t the G a m m a and Be ta functions is assumed. A certain a m o u n t of "ma themat i ca l m a t u r i t y " is presupposed, or should be acquired in the course of the reading. T h e reader m u s t be able to apprecia te the general not ion of function, a p a r t from representat ion b y a preconceived t ype of formula; he m u s t be ready to th ink of the value of a definite integral as a numerical magni tude , subject to relat ions of greater and less, and only incidental ly as the result of a par t icular process of calculat ion. Especially, he m u s t be prepared to accept the word orthogonal as a t e rm of mathemat ica l analysis, on the basis of i ts definition, with a geometr ic associa-tion only in the remote background or temporar i ly in abeyance a l together , as he has already learned to speak of a linear equat ion or the square of a number wi thou t feeling obliged to visualize a geometr ic figure in connec-tion wi th every occurrence of the words.

Under the c i rcumstances , "rigor" in the sense of literal completeness of s t a t e m e n t has been ou t of the quest ion. I t is hoped however t h a t the reader who is familiar with the me thods of rigorous analysis will be able w i thou t a n y difficulty to read between the lines the requisi te supp lemen ta ry specifications, and will find t h a t w h a t has ac tual ly been said is ent irely accura te in the l ight of such in terpre ta t ion .

A set of exercises relat ing to the var ious chapters , and in tended for the mos t p a r t to i l lustrate and extend the tex t r a the r than to serve for purposes of "dril l ," has been grouped together a t the end of the book.

A bibl iography of suggestions for supp lementa ry

viii PREFACE

reading h a s also been placed a t the end, and c i ta t ions in the text , when n o t otherwise given in deta i l , refer to the i tems of this b ibl iography. A shor t list of references to the b ib l iography is a t t a c h e d a t the end of each chapte r . A few more specific references h a v e been inserted in footnotes.

T h i s book is based on a course which I have given a t the Univers i ty of Minneso ta over a long period of years . I t would be impossible now to m a k e acknowledg-m e n t to all t he individuals in successive generat ions of s t uden t s whose c o m m e n t s I have consciously or uncon-sciously t aken in to account . I a m grateful to the Carus M o n o g r a p h C o m m i t t e e , and in par t i cu la r t o i ts Chair-m a n , Professor Saunders M a c Lane , for the most cordial cooperat ion and for numerous i l luminat ing suggestions.

DUNHAM JACKSON

T H E UNIVERSITY OF MINNESOTA

Apri l , 1941

T A B L E O F C O N T E N T S

CHAPTER PAGE

I. FOURIER SERIES

1. Definition of Fourier series 1 2. Orthogonality of sines and cosines 2 3. Determination of the coefficients 3 4. Series of cosines and series of sines 6 5. Examples 8 6. Magnitude of coefficients under special hypotheses . 11 7. Riemann's theorem on limit of general coefficient . 14 8. Evaluation of a sum of cosines 17 9. Integral formula for partial sum of Fourier series . 17

10. Convergence at a point of continuity . . . . 18 11. Uniform convergence under special hypotheses . 21 12. Convergence at a point of discontinuity . . . . 22 13. Sufficiency of conditions relating to a restricted

neighborhood 24 14. Weierstrass's theorem on trigonometric approxima-

tion 25 15. Least-square property 27 16. Parseval's theorem 29 17. Summation of series 31 18. Fejfir's theorem for a continuous function . . . 32 19. Proof of Weierstrass's theorem by means of de la

Vallee Poussin's integral 35 20. The Lebesgue constants 40 21. Proof of uniform convergence by the method of

Lebesgue 42

II. LEGENDRE POLYNOMIALS

1. Preliminary orientation 45 2. Definition of the Legendre polynomials by means of

the generating function 45 3. Recurrence formula 46 4. Differential equation and related formulas . . . 48 5. Orthogonality 50

ix

χ CONTENTS

CHAPTER PAGE

6. Normalizing factor SI 7. Expansion of an arbitrary function in series 53 8. Christoffel's identity 54 9. Solution of the differential equation 55

10. Rodrigues's formula 57 11. Integral representation 58 12. Bounds of i>„(*) 61 13. Convergence at a point of continuity interior to the

interval 63 14. Convergence at a point of discontinuity interior to

the interval 65

III. BESSEL FUNCTIONS

1. Preliminary orientation 69 2. Definition of /„(*) 69 3. Orthogonality 71 4. Integral representation of Jo{x) 74 5. Zeros of 7 0(x) and related functions 76 6. Expansion of an arbitrary function in series 79 7. Definition of /„(*) 80 8. Orthogonality: developments in series . . . . 82 9. Integral representation of /»(*) 84

10. Recurrence formulas 85 11. Zeros 87 12. Asymptotic formula 87 13. Orthogonal functions arising from linear boundary

value problems 88

IV. BOUNDARY VALUE PROBLEMS

1. Fourier series: Laplace's equation in an infinite strip 91 2. Fourier series: Laplace's equation in a rectangle 95 3. Fourier series: vibrating string 96 4. Fourier series: damped vibrating string . . . . 100 5. Polar coordinates in the plane 101 6. Fourier series: Laplace's equation in a circle; Pois-

son's integral 103 7. Transformation of Laplace's equation in three di-

mensions 105 8. Legendre series: Laplace's equation in a sphere 107

CONTENTS xi

CHAPTER PAGE

9. Bessel series: Laplace's equation in a cylinder . 109 10. Bessel series: circular drumhead 112

V . DOUBLE SERIES ; LAPLACE SERIES

1. Boundary value problem in a cube; double Fourier series 115

2. General spherical harmonics 118 3. Laplace series 121 4. Harmonic polynomials 126 5. Rotation of axes 129 6. Integral representation for group of terms in the

Laplace series 132 7. Completeness of the Laplace series 137 8. Boundary value problem in a cylinder; series involv-

ing Bessel functions of positive order . . 1 3 8

V I . T H E PEARSON FREQUENCY FUNCTIONS

1. The Pearson differential equation 142 2. Quadratic denominator, real roots 142 3. Quadratic denominator, complex roots . . . . 145 4. Linear or constant denominator 146 5. Finiteness of moments 147

V I I . ORTHOGONAL POLYNOMIALS

1. Weight function . 149 2. Schmidt's process 151 3. Orthogonal polynomials corresponding to an arbi-

trary weight function 153 4. Development of an arbitrary function in series . . 155 5. Formula of recurrence 156 6. Christoffel-Darboux identity 157 7. Symmetry 158 8. Zeros 159 9. Least-square property 160

10. Differential equation 161

V I I I . JACOBI POLYNOMIALS

1. Derivative definition 166 2. Orthogonality 167 3. Leading coefficients 169

xii CONTENTS

CHAPTER PAGE

4 . Normalizing factor; series of Jacobi polynomials . 171 5 . Recurrence formula 172 6. Differential equation 1 7 3

I X . HERMITE POLYNOMIALS

1. Derivative definition 1 7 6 2 . Orthogonality and normalizing factor . . . . 177 3 . Hermite and Gram-Charlier series 1 7 8 4 . Recurrence formulas; differential equation . 1 7 9 5 . Generating function 1 8 1 6 . Wave equation of the linear oscillator . 1 8 1

X . LAGUERRE POLYNOMIALS

1. Derivative definition 1 8 4 2 . Orthogonality; normalizing factor; Laguerre series . 1 8 4 3 . Differential equation and recurrence formulas 1 8 6 4 . Generating function 1 8 7 5 . Wave equation of the hydrogen atom 1 8 8

X I . CONVERGENCE

1. Scope of the discussion 191 2 . Magnitude of the coefficients; first hypothesis . 192 3 . Convergence; first hypothesis 1 9 4 4 . Magnitude of the coefficients; second hypothesis 197 5 . Convergence; second hypothesis 1 9 9 6. Special Jacobi polynomials 2 0 0 7. Multiplication or division of the weight function by

a polynomial 2 0 1 8. Korous's theorem on bounds of orthonormal poly-

nomials 2 0 5

EXERCISES 2 0 9

BIBLIOGRAPHY 2 2 9

INDEX 2 3 1

F O U R I E R S E R I E S A N D

O R T H O G O N A L P O L Y N O M I A L S

E X E R C I S E S

CHAPTER I

1. Calcula te t h e coefficients in the Fourier series for a function f(x) of period 2π which is equal to — 1 for — π<χ<0 and equal to 1 for 0 < x < i r .

4 Γ sin 3x sin 5x Ί Ans. f(x) = — sin χ -) 1 h · · · .

π L 3 5 J

2. Calculate the coefficients in the Fourier series for an even function f(x) of period 27r which is equal to χ for Ο^χ^^π and equal to 57Γ for %π^χ^π.

3ΤΓ " 2 / \ Ans. f(x)= \- 2-, \ c o s 1 ) c o s

kx 8 k - i irk2 \ 2 /

3π 2 Γ cos3x cos5x Ί = cos x-\ 1 h

8 7 Γ L 3 2 5 s J 4 rcos2a; cos6x Ί

3. Calculate the coefficients in the Fourier series for an odd function f(x) of period 2ττ which is equal to \π — %x for 0<χ<π, a) b y in tegra t ion, b) b y subst i tu-tion of ir—χ for χ in the series inside the b racke t s in (12).*

s in2x s in3x Ans. f{x) = sin χ -\ 1 } - • • · .

* In connection with the exercises on any particular chapter, numbers of formulas or sections refer to the corresponding chapter, unless otherwise indicated.

209

EXERCISES

F I G . 2

1 Graphs of y = sin χ sin 3x for O S s S r

and of y = —* for 0 i ϊ ί —, y = — ( τ — χ) for — ί= * S τ . 4 2 4 2

(See Exs. 3, 4, 5 of Chapter I.)

EXERCISES 211

4. Calcula te the coefficients in the Fourier series for an odd function f{x) of period 2-r which is equal to χ for 0 = χ ^ j 7 r and equal to π — χ for %π ^χ ^ π, a) by integra-t ion, b) b y subs t i tu t ion of x-\-%ir for χ in (11) and sub-t ract ion of a cons tan t .

Ans. f(x) =

5. Cons t ruc t g raphs of the first few par t ia l sums of the series in Exs. 3 and 4, proceeding from one par t ia l sum to the next by drawing a curve for the single te rm to be added and combining ord ina tes graphical ly. (See Figs. 1 and 2.*)

6. Using the iden t i ty

cos (k + l)x = 2 cos kx cos χ — cos (k — l)x,

show b y induct ion t h a t cos nx can be expressed as a polynomial of the n t h degree in cos x, for a rb i t r a ry posi-t ive integral n .

No te the following corollaries: a) Any cosine sum of the n t h order (i.e. t r igonometr ic

sum involving only cosines) is a polynomial of the n t h degree in cos x.

* In a diagram of this size, the inclusion of one more term would make the graph of the trigonometric sum in Fig. 2 almost indistin-guishable from that of the function to which the Fourier series con-verges, except near the peak.

In each case the complete graphs of the trigonometric sum and of the function represented by the series, for unrestricted values of x, would be symmetric with respect to the origin, and of period 2ir.

212 EXERCISES

b) T h e function cos" χ is expressible as a cosine sum of the » t h order .

c) Any polynomial of the n th degree in cos χ is ex-pressible as a cosine sum of the n t h order.

7. Using Ex. 6 and the iden t i ty

sin (k + l)x = 2 cos kx sin χ + sin (k — l)x,

show by induct ion t h a t sin nx can be expressed as the p roduc t of sin χ and a polynomial of degree η — 1 in cos x, for a rb i t r a ry posit ive integral n.

8. Show by adap ta t i on of the proof of §14 t h a t a n y even cont inuous function of period 2x can be uniformly approx imated b y a cosine sum wi th a n y assigned degree of accuracy . (The function g(x) can be taken as an even function.)

9. If f(x) is a n y function which is cont inuous for — l _ x ^ l , show t h a t / ( x ) can be uniformly approxi-ma ted b y a polynomial in χ with a n y assigned degree of accuracy on the in terval . (Let χ = cos Θ, and use Exs. 8 and 6 to show t h a t / ( cos Θ) as an even function of θ can be uniformly approx imated by a polynomial in cos Θ.)

Hence show by a linear change of independent varia-ble t h a t a function f(x) continuous on an arbitrary closed intervala^x^b can be uniformly approximated by a poly-nomial on the interval with any assigned degree of ac-curacy. Th i s is Weierstrass's theorem for polynomial approximation. *

10. Show t h a t if p and q are a n y two dis t inct non-negat ive integers the functions sin (/>+4)x and sin (q+%)x are or thogonal to each o ther on the inter-val (0, TT).

* K. Weierstrass, Uber die analytische Darstellbarkeit sogenannter iMlkiirlicher Functionen einer reeUen Verdnderlichen, erste Mitthei-lung, Berliner Sitzungsberichte, 1885, pp. 633-639.

EXERCISES 213

11. A func t ion / (χ ) being given on the in terval (0, ir), de te rmine the coefficients in a formal representa t ion of f(x) on the in terval in a series of the form 2o°c* s m + ϊ)χ·

12. Prove for the series of Ex. 11a least-square prop-e r t y corresponding to t h a t of §15.

13. P rove for the series of Ex. 1 1 a theorem analo-gous to t h a t obta ined in the first two pa rag raphs of §7.

14. Obta in for the series of Ex . 11 inequali t ies corre-sponding as far as possible t o those of §6, assuming t h a t f(x) vanishes for χ = 0.

15. Obta in for the par t ia l sum of the series of Ex. 11 a formula corresponding to (19).

1 r T Γ sin (« + 1)(/ - x) Ans. sn(x) = - I f(t)\ \ " '

π J ο L 2 sin $(t — x)

sin (n + l)(t + x)l dt.

2 sin %{t + x) J

CHAPTER II

1. T h e Legendre polynomials Po(x), • · · , Pt(x) be-ing known from §2, calculate Pt(x) and Pe(x) b y means of (5).

Ans. Pf,(x) = | ( 6 3 * 6 - 70x 3 + 15*),

-Pe(*) = lhrC231ze - 315** + 105* 2 - 5).

2. Calculate the roots of the equa t ions Pk(x) = 0 and P * ' ( x ) = 0 for £ = 5.

3. Using the results of Ex. 2, plot g raphs of the poly-nomials Po(x), • • • , Pi(x).

4. Calculate the value of -P*(0) for a rb i t r a ry k by means of the recurrence formula.

214 E X E R C I S E S

P*'(0) = ( - !)(*-«/*-2-4-6 • • • ( * - 1)

= *P»_i(0).

Verify this result by mathematical induction based on the recurrence formula.

6. Find the coefficients in the Legendre series for I x | . (Integrate by parts, and use (14), as in §14, for the integration of the Legendre polynomials.)

Ans. α* = 0 for odd k; a 0 = £; fl2 = f; for even k>2:

(2*+l ) ak= — P * ( 0 )

( * - l ) ( * + 2 )

1-3-5 · · · (* -3 ) = (- l ) (*/ i )+i(2*+i) . 2-4-6 · · • k(k+2)

7. Construct graphs for the first few partial sums of the series in Ex. 6, proceeding from one partial sum to the next by drawing a curve for the single term to be added and combining ordinates graphically.

8. Construct graphs similarly for the series of the first two paragraphs of §14, taking c = 0.

1-3-5 · · · (k-l) Ans. P*(0) = ( - 1 ) * / 2 for even *:>0,

2-4-6 · · · k

P*(0)=0 for odd

Check this result by finding directly the coefficient of r* in the power series expansion of H(0, r), when H(x, r) is the generating function of §2.

5. Find the value of P£ (0) as coefficient of r* in the power series expansion of dH/dx for x = 0.

Ans. Pi (0) = 0 for even k; for odd k:

3 - 5 - 7 - · · *

E X E R C I S E S 215

9. Evaluate Jo cos 2 'φ άφ for j = 0, 1,2, • • - bysetting_ χ = 0 in (31) and applying Ex. 4.

f 1-3-5 • • • (2j-l) Ans. I cos 2 ' φ do = i r for/>0.

Jo 2-4-6 • · • (2j) J

10. Prove that / L i | Pn(x) \ dx does not exceed a con-stant multiple of rrw, a) by reference to (36), b) by means of (18) and Schwarz's inequality (see §19 of Chapter I).

11. Obtain theorems for Legendre series correspond-ing to those of §6 in Chapter I for Fourier series, using (14) for the evaluation of fPt(x)dx.

It is found that if f(x) has a continuous derivative, I α* I does not exceed a constant multiple of k~112; if f(x) has a continuous second derivative, or is a broken-line function, |a*| does not exceed a constant multiple of k~*12. For a fixed value of χ in the interior of the interval it follows from (36) that | a^Pkix) | has an upper bound of the order of 1/k in one case and 1/k2 in the other, in closer analogy with the results obtained for Fourier series.

Note further that when the upper bound obtained for [α*I is of the order of k~il2 the series is uniformly convergent throughout the interval, inclusive of the end points. According to §13 the sum of the series is/(*) in the interior of the interval. By the theorem of the theory of functions which states that the sum of a uniformly convergent series of continuous functions is continuous, it follows that the sum of the series is equal to the value of f(x) a t the ends of the interval also.

12. Prove for Legendre series the least-square prop-erty corresponding to that of §15 in Chapter I for Fourier series: If f(x) and [f(x)]2 are integrable over ( — 1, 1), and if sn(x) is the partial sum of the Legendre series for f(x), the integral f-ι[f(x) — sn(x)]2dx has a

2 1 6 EXERCISES

smaller value t han t h a t which is ob ta ined if sn(x) is re-placed b y a n y o the r polynomial of the same (or lower) degree.

13. Using Ex. 12 and Weiers t rass ' s theorem for poly-nomial approximat ion (see Ex. 9 of Chap te r I ) , prove for Legendre series the analogue of Parseval ' s theorem in §16 of C h a p t e r I : lif(x) is cont inuous for — 1 _ x _ 1, and if ak is the general coefficient in i ts Legendre series,

1. E v a l u a t e xJo(x)dx. (Obta in an a l te rna t ive ex-pression for xJB(x) from the differential equat ion (1).)

2. F ind the coefficients in the expansion of the func-t i o n f(x) = \ on the in terval (0, 1) in a series of the form discussed in §6, the X's being the roots of the equat ion / „ ( * ) = 0.

Ans. ak = - 2/Mo'(λ*)].

W h a t is the form of the series if the X's are the roots of Λ ' ( χ ) = 0 ?

3. Verify (10) by subs t i tu t ing in (12) the value of / O c o s 2 ' φ άφ given b y Ex. 9 of C h a p t e r I I .

(The evaluat ion of / O c o s 2 n $ άφ as needed for the pur-poses of §9 can be t aken direct ly from the same source, ins tead of being m a d e to depend on the proof of (10) by the me thod of §4.)

4. Show t h a t the series inside the bracke ts in (17)

J -l k-o Ik •+• 1

CHAPTER III

Ans. ο

EXERCISES 217

represents (sin x)/x if « = f, and represents cos a; if

5. Show b y direct subs t i tu t ion t h a t the differential equa t ion

to which (14) reduces for η = + \, is satisfied b y the func-t ions (sin x ) / x 1 / 2 and (cos x)/xl/i.

For a rb i t r a ry n, except for the need of a supplemen-t a r y in te rpre ta t ion in the case of the negat ive integral values — 1 , — 2, · • · , Jn(x) is defined by giving to the cons t an t c 0 in the last sentence of §7 the value

2-4(2» + 2)(2« + 4) J

In par t icu lar , as Γ ( | ) = ^ τ τ 1 ' 2 and Γ ( ^ ) = π 1 ' 2 ,

7 1 / 2 ( x ) = ^ / ( T T X ) ] 1 ' 2 sin x, / _ 1 / 2 ( x ) = [ 2 / ( « ) ] " ! c o 5 * .

I t is obvious t h a t each of these functions vanishes for infinitely m a n y posit ive values of χ (as well as o thers) .

In the following exercises the value of /0*sin2n<pi/<p is assumed as known for a rb i t r a ry η > — 5 : with the sub-s t i tu t ion / = s in 2 φ,

1 / [ 2 T ( « + 1 ) ] :

2 I sin 2"(M4>= I i n - ( 1 / 2 > ( l -tyvHt / • ( τ / 2 ) f l

= B{n + hi)= T(n + ^ )Γ(^) /Γ(« + 1)

= irl'*T(n + ^ ) /Γ (« + 1).

2 1 8 EXERCISES

I t will be unders tood th roughou t t h a t η is real, though some of the s t a t e m e n t s would be t rue also for complex values of n.

6. Not ing t h a t the der ivat ion of (23) is independent of the assumpt ion t h a t η is integral , provided t h a t η > — and t h a t the same is t rue of the discussion of the solution of (15) by means of power series in §7, show t h a t

χη / . τ

Jn(x) = I s in 2 " φ cos (x cos φ)άφ

2»7Γ1' 2Γ(» + ! ) Λ

for all n> — 7. Verify the formula of Ex . 6 for n = \ by explicit

in tegra t ion . 8. Show b y means of Ex. 6 and (23) t ha t (24) is

valid for all η > — §.

9. Show b y means of Ex. 6 and the method of §5 t h a t Jn{x) vanishes for infinitely m a n y posit ive values of χ if - i < » < i .

10. T h e integrat ion by pa r t s in the second para-g raph of §10 being independent of the assumpt ion t h a t η is integral , provided t h a t n>%, show t h a t (25) is valid for all η >\.

11. Using Exs. 8 and 10 and the facts noted previ-ously wi th regard to J±w{x), show t h a t the equat ion Jn(x) = 0 has infinitely m a n y posit ive roots for any value of n ^ - i

12. Show by subs t i tu t ion of the power series expres-sions for the var ious t e rms and comparison of coeffi-cients t h a t (24) and (25) hold wi thou t a n y restriction on η except t h a t (for direct appl icabi l i ty of the definition of Jn(x) as given above) i t is no t a negat ive integer, and in the case of (25) is no t zero.

EXERCISES 219

13. Show t h a t the equat ion Jn(x)=0 has infinitely m a n y posit ive roots for a n y value of η which is no t a negat ive integer. (Even this restr ict ion is unnecessary when Jn(x) is defined for negat ive integral n, since the definition is such t h a t J-n{x) = { — l)nJn(x) when η is in-tegral .)

When η is no t an integer, the functions Jn{x) and J-n{x) are linearly independent , since one vanishes for x = 0 and the o ther becomes infinite the re ; and the gen-eral solution of (14) is AJn(x)+BJ-„(x). A second solu-tion of the differential equat ion for η = 0 is given by the following exercise. T h e form of a second solution for o ther integral values of η will no t be discussed here.

14. If y is a solution of (1) and if

ζ = y — Jo(x) log x,

show t h a t ζ satisfies the differential equat ion

dH 1 dz 2 — + - — + «= - - / „ ' ( * ) , dx2 χ dx χ

and find a solution of this equat ion in the form of a power series. When ζ is such a solution, Jo(x) log x + z is a solution of (1) independen t of J0{x).

Ans. « = f - - ( i + l ) - ^ - + ( l + J + J)- X

22 2*4* 2 2 4 s 6 2

(or the sum of this function and an a rb i t r a ry cons tan t mult iple of /ο (*) ) ·

220 EXERCISES

CHAPTER IV

1. Wr i t e explicitly (i.e. wi th the appropr ia te numeri-cal coefficients) the series represent ing the solution of the b o u n d a r y va lue problems of §§1 and 2, for the differ-ential equat ion (2) wi th the auxil iary condit ions (3), (4), (5) and (6) in one case and (3), (4), (5) and (9) in the o ther , w h e n / ( x ) = l on the in terval (0, i r ) . (See Ex. 1 of C h a p t e r I.)

2. Show b y subs t i tu t ion in the differential equat ion and auxil iary condit ions t h a t the function

2 sin χ u = — arc tan --

χ sinh y

is a solution of the b o u n d a r y value problem of §1 for the s t r ip of wid th ir wi th (except a t the corners, which are points of d iscont inui ty) .

T h e relation of this expression to the solution in se-ries m a y be recognized th rough the in t roduct ion of func-t ions of complex variables b y regarding u as the real p a r t of the function

2 1 + f — log 1 f = e~"(cos χ + t sin x), iri 1 — f

and expanding th is in series of powers of ζ. 3. Wri te explicitly the series represent ing the solu-

tion of the problem of the plucked s t r ing in §3 when f(x) is equal to hx for O^x^^ir and equal to h(ir — x) for %ir = xuir, h being a cons tan t . (See Ex. 4 of Chap te r I.)

4. Show t h a t the function

y(x, t) = *[/(* + at) + f{x - at)]

is a solution of the problem of the plucked s t r ing in §3, if t he definition of f(x) is extended from the interval

EXERCISES 221

(Ο, π) to the whole range of real values of χ in such a way t h a t it becomes an odd function of period 2ir.

Show how this form of y can be obta ined from the series in (17).

5. Solve the problem of the s t ruck s t r ing in §3 when φ(χ) is equal to 0 for 0^χ<%π — δ and for 4π+ό*<χ = ιτ and equal to h for §π — δ < χ < % π + δ , h and δ being posi-t ive cons tan t s .

6. F ind explicitly the solution of the problem of §8 when / ( 0 ) = c o s 4 0 for O ^ 0 ^ i r ; when / (0) is equal to 0 for Ο = 0<£ττ and equal to 1 for %π<θ^π (see §14 of C h a p t e r 11); when / (0 ) = | cos 0 | for 0 g 0 g χ (see Ex. 6 of C h a p t e r I I ) .

7. Wr i t e explicitly the series for the solution of the problem defined b y (30) and (31) in §9 w h e n / ( r ) ^ 1 for 0 = > < 1 . (See Ex. 2 of Chap te r I I I . )

8. Discuss the v ibra t ion of a circular d rumhead with d a m p i n g propor t ional to the velocity, the d rumhead be-ing initially a t res t wi th a distort ion which is a function only of d is tance from the center ; ζ as a function of r and / satisfies a differential equa t ion of the form

in which k and a 2 are positive cons tan ts . Discuss the corresponding problem in which the

d r u m h e a d s t a r t s from its position of equil ibrium with initial velocities depending only on dis tance from the center .

Ans.

sin (2k + 1)5 sin (2k + l ) x s i n (2k + l)at

222 EXERCISES

·/ ο xPm(x)dx = —

{m - l)(m + 2)

(see Ex. 6 of Chapter I I ) .

4. Wr i t e the series for the solution of the bounda ry value problem of §3 when f(9, φ) is the function of the preceding exercise.

5. Discuss the v ibra t ion of a square membrane fas-tened a t the edges, ζ being a solution of the equat ion

CHAPTER V

1. Wr i t e the spherical harmonics umn{Q, φ), νΜη(θ, φ) explicitly for m = 3.

2. Express s in 2 θ cos 2 φ as a linear combinat ion of spherical ha rmonics ; the representat ion can be regarded as a Laplace series which reduces to a finite number of t e rms .

Ans. sin 2 θ cos 2 φ = f«oo — $«20 + £«22-

3. Expand t h e function

f(fl, φ) = (1 - I cos 9\)(1 + cos 2φ)

in a Laplace series.

00 00

AnS. / ( 0 , φ) = Σ, ?amO«mO + Σ «<»2«m2, m—0 τη—2

a m o - a m 2 = 0 for odd m;

αοο = 1; for even m = 2:

flrnO = — 2(2W + l)Cm,

( w - 2 ) ! a m 2 = -, - (2m + 1) [6cm - P . ( 0 ) ] ,

(w + 2)!

Λ. (0)

EXERCISES 223

6. Discuss the vibrat ion of a circular membrane with initial condi t ions which are no t independen t of φ.

CHAPTER VI

1. D r a w graphs of the function (1 — x)"(l + * ) " on the in terval ( - 1 , 1) : a) for α=β = 2, b ) for α = β = \, c) for α = 0 = 1, d) for α=β= - J , e) for α = - | , 0 = 2, f) for β = 1

2. D r a w graphs of the function x"e~x for x>0: a) for α = 2, b) for a = 1, c) for α = 5, d ) for α = 0, e) for a = — 5.

3. Plot wi th the same axes and the same scale the curves

1 2 1 y - e ~ x / 2 , y = •

Ο ) " 2 τ ( l + .r2)2

(Note t h a t the second function is of the type discussed in §3, wi th a = 0. Stat is t ical ly each curve represents a dis t r ibut ion wi th to ta l frequency 1, mean 0, and s tand-ard deviat ion 1.)

CHAPTER VII

1. Find successively the first few polynomials in the o r thonormal sys tem corresponding to \x\ as weight function on the in terval ( — 1 , 1).

2. If pn(x) is the general polynomial in the o r tho-normal sys tem for weight p(x), and if m is an a rb i t r a ry positive integer, show t h a t xmpn(x) is connected by a recurrence formula wi th the 2 m + 1 polynomials pn-m(x), • • • , pn+m(x).

3. Show b y adap ta t i on of the method of §6 t h a t Kn(x, t) has a representa t ion of the form

« 2 - Χ 1 ) " 1 Σ - pr(x)P.(t)],

the coefficients c r , being cons tan t s , and the summat ion

2 2 4 EXERCISES

being extended over the three pairs of values (« + 2, «) , (» + l , « ) , (w + 1, η— 1) for the subscr ipts (r, s).

4. If g0(x), gi(x), • • · form an or thonormal sequence on (a, b) and if

η η

«(*) = Σ »(*) = Σ bkgk{x),

show t h a t

* - 0

5. If n = l in Ex. 4, if Ο, P, Q are the points (0, 0), («ο, O i ) , (bo, h) wi th respect to a plane rec tangular co-ord ina te sys tem, and if θ is the angle POQ, show t h a t

1 / 2

In par t icular , t h e functions M, D are orthogonal if the lines O P , OQ are perpendicular t o each other .

No te t h a t this formulation is applicable to any two functions (u, v) which are integrable with their squares and linearly independen t (in the sense of the concluding sentence of the first pa rag raph of §2), since u, ν can themselves be taken as φο, φι in apply ing the Schmidt process, and expressed linearly in te rms of the corre-sponding go, gl.

F u r t h e r m o r e , since | cos θ\ ^ 1, the relation (i) consti-tu t e s an a l t e rna t ive proof or in terpre ta t ion of Schwarz's inequality (see C h a p t e r I, §19).

6. If η = 2 in Ex. 4, if Ο, P, Q are the points (0, 0, 0) , (ao, a\, O i ) , (bo, δι, fa) wi th respect t o a rectangular co-ord ina te sys tem in space, and if θ is t he angle POQ, show t h a t cos θ is again represented by the formula (i) of Ex. 5.

EXERCISES 225

7. Develop a theory of t r igonometr ic sums or thogo-nal and normal ized wi th respect to a posit ive weight function p(x) of period 2π, the o r thonormal sys tem be-ing cons t ruc ted b y appl icat ion of Schmid t ' s process to the functions p 1 / 2 , p 1 / 2 cos x, p 1 / 2 sin x, p I / 2 cos 2x, pl/i sin 2x, • • • .

CHAPTER VIII

1. Show t h a t when α=β=— \, so t h a t pn{x) = ( 2 / 7 r ) 1 / 2 cos ηθ for w > 0 if x = cos 9, the recurrence

formula (10) of C h a p t e r V I I reduces to the ident i ty connect ing the cosines of successive integral multiples of 9.

2. In the differential equa t ion a t the end of the first pa rag raph of §6 wi th α=β= — | set a: = cos 9, and show t h a t cos ηθ is a solution of the t ransformed equat ion .

3. M a k e the same change of independent variable in the differential equat ion with a = β = | , and show t h a t the t ransformed equat ion has sin (» + l )0 / s in 9 as a solu-t ion.

4. Solve the differential equa t ion for general α, β by t h e me thod of unde te rmined coefficients, t o the ex ten t of ob ta in ing the formula from which the coefficients can be calculated in succession, and wi th the hypothesis t h a t α > — l, β> — 1, show t h a t except for an a rb i t r a ry con-s t a n t factor the equat ion has jus t one polynomial solu-t ion, η being a non-negat ive integer.

5. By differentiation of the differential equat ion wi th η replaced by n +1 and reference to Ex. 4 show t h a t (d/dx)Pn

a+?(x) is a cons tan t mult iple of Ρ < α + 1 · " + 1 ) ( * ) . 6. Develop a theory of t h e "ul t raspherical polyno-

mials ," cons t an t mult iples of the Jacobi polynomials wi th /3 = a , b y defining t h e m as coefficients in the repre-senta t ion of the genera t ing function (1 — 2xr-\-r2)~a~an)

226 EXERCISES

by a power series in r, proceeding as far as possible along the lines of C h a p t e r I I . ( I t is to be supposed t h a t

CHAPTER IX

1. T a k i n g the iden t i ty which concludes §5 as an al-t e rna t ive definition of the Hermi t e polynomials , obtain the relat ion (6) direct ly from this definition.

2. Der ive the differential equat ion (7) by combina-tion of (5) and (6). In view of §5 and Ex. 1 the differ-ential equat ion is thus obta inable from the definition in te rms of the genera t ing function.

3. Der ive the p roper ty of or thogonal i ty of the H e r m -ite polynomials from the differential equat ion .

4. Solve the differential equat ion b y the method of unde te rmined coefficients, and show t h a t except for an a rb i t r a ry cons t an t factor there is jus t one polynomial solution for each non-negat ive integral n.

5. Obta in the relat ion (6) b y differentiation of the differential equa t ion and use of Ex. 4, together wi th the fact t h a t the leading coefficient in Hn(x) is 1. (Note t h a t there has been one der ivat ion of the differential equa-tion which does no t depend on (6), namely t h a t based on §10 of C h a p t e r V I I . )

6. Deno t ing b y C„ the integral which forms the first member of (3), obta in a relation of recurrence connect-ing successive C's, after the analogy of the method fol-lowed in §6 of Chap te r I I , and thus find the value of C„, on the assumpt ion t h a t C0 is known.

CHAPTER X

1. For a = 0, ob ta in the differential equat ion (3) from a definition of the Laguerre polynomials in te rms of t h e genera t ing function in §4. (Use the relations

EXERCISES 227

dH dH dH ( 1 - / ) +tH = 0, t(l-t) (x-l+t) = 0.)

dx dt dx

2. Solve the same problem for general a. 3. Obta in (3) for a r b i t r a r y a from the der ivat ive

definition of the Laguerre polynomials as given in §1. (Differentiate the iden t i ty

Χφή (x) + (x — a — η)φη(χ) = 0

n + l t imes, differentiate the relation

Φη [χ) = ( — 1) χ e Ln{x)

twice, and combine the results.) 4. Der ive the p rope r ty of or thogonal i ty of the La-

guerre polynomials from the differential equa t ion . 5. Denot ing the integral ^xtte~x[Ln{x) ]2dx by C„,

obta in a relat ion of recurrence connect ing successive C's, after the manne r of §6 in C h a p t e r I I , and thus find the value of C„.

6. Solve the differential equa t ion (3) by the method of unde te rmined coefficients, and show (for a > — 1) t h a t it has j u s t one polynomial solution for each non-nega-t ive integral n, except for a cons tan t factor.

7. By differentiation of (3) and applicat ion of Ex. 6 show t h a t (d/dx)L(£1(x) = (n + l)Ln

a+1)(x).

CHAPTER XI

In Exs. 1-3 it is unders tood t h a t pn(x) is the poly-nomial of the rath degree in the o r thonormal sys tem for weight p(x) on a finite in terval (a, b), and t h a t ck is the coefficient of pk in the expansion of f(x) in series of these polynomials .

1. Using the least-square p roper ty (Chap te r V I I , §9) and Weiers t rass ' s theorem for polynomial approxi-

228 EXERCISES

mat ion (Chap te r I, Ex . 9) show t h a t iff{x) is cont inuous on (a, b) f Ρ ( * ) [ / ( * ) ] 2 ^ = Σ < * ·

J a k-0

2. By means of Schwarz ' s inequal i ty show t h a t

(i) j p(x) I pn(x)\d**[f P(x)dx^ ' .

3. If a polynomial 7 r n _ i ( x ) of degree η — 1 a t most and a n u m b e r e„_i are such t h a t

I }(x) — 7r„_i(z) I = e„_i

t h roughou t (a, b), show t h a t | c„ | ^ 7 « η - ι , where y de-notes the r igh t -hand member of (i) in Ex. 2. (Note t h a t pn(x) is or thogonal to rn-i(x) for weight p.)

4. Show t h a t the o r thonormal t r igonometr ic sums corresponding to a weight function p(x) (see Chap te r V I I , E x . 7) are uniformly bounded if ρ (χ) is a t r igonomet-ric sum which is everywhere posit ive, or the reciprocal of such a sum.

B I B L I O G R A P H Y O F S U G G E S T I O N S F O R

S U P P L E M E N T A R Y R E A D I N G

G. D. Birkhoff, Boundary value and expansion problems of ordinary linear differential equations, Transactions of the American Mathe-matical Society, vol. 9 (1908), pp. 373-395.

M. Bocher, Introduction to the theory of Fourier's series, Annals of Mathematics, (2), vol. 7 (1906), pp. 81-152.

W. E. Byerly, Fourier's Series and Spherical Harmonics, Boston, 1895.

H. S. Carslaw, Fourier's Series and Integrals, London, 1930, and earlier editions.

R. V. Churchill, Fourier Series and Boundary Value Problems, New York, 1941.

R. Courant and D. Hubert, Methoden der mathematischen Physik, vol. I, Berlin, 1931.

G. Darboux, Mtmoire sur Vapproximation des functions de tres-grands nombres, et sur une classe ttendue de developpements en strie. Journal de Matl^matiques pures et appliquees, (3), vol. 4 (1878), pp. 5-56, 377-416.

W. Palin Elderton, Frequency-Curves and Correlation, London, 1906.

L. Fejer, Untersuchungen iiber Fouriersche Reihen, Mathematische Annalen, vol. 58 (1904), pp. 51-69.

A. Gray, G. B. Mathews, and Τ. M. MacRobert, Bessel Functions, London, 1922.

E. W. Hobson, The Theory of Functions of a Real Variable and the Theory of Fourier's Series, Cambridge, 1907, and later editions. (Cited as Hobson(l).)

E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, Cambridge, 1931. (Cited as Hobson (2).)

E. L. Ince, Ordinary Differential Equations, London, 1927; Chap-ters IX, X, XI.

S. Kaczmarz and H. Steinhaus, Theorie der Orthogonalreihen, War-saw, 1935.

O. D. Kellogg, Foundations of Potential Theory, Berlin, 1929. H. Lebesgue, Lemons sur les stries trigonomttriques, Paris, 1906.

229

230 B I B L I O G R A P H Y

Karl Pearson (Editor) , Tables for Statisticians and Biometricians,

Cambridge, 1914; pp. lx- lxx . G. Polya and G. Szego, Aufgaben und Lehrsatze aus der Analysis,

Berlin, 1925; vol . II , Section 6, Polynome und trigonometrische Poly-

nome.

"Riemann-Weber," Differentialgleichungen der Physik, seventh edit ion, b y P. Frank and R. von Mises , Braunschweig, 1925 (vol. I) , 1927 (vol. I I ) .

H. L. Rietz , Mathematical Statistics, Carus Mathemat ica l M o n o -graphs, N o . 3, Chicago, 1927.

J. Shohat , Thiorie ginerale des polynomes orthogonaux de Tchebichef

(Memoriale des Sciences Mathemat iques , N o . 66) , Paris, 1934.

Μ . H. Stone, Developments in Legendre polynomials, Annals of Mathemat ics , (2), vol . 27 (1926), pp. 315-329 .

G. Szego, Orthogonal Polynomials, American Mathematical Soci-e t y Colloquium Publications, vol . 23, N e w York, 1939.

E . C. Ti tchmarsh, The Theory of Functions, Oxford, 1932; Chap-ter X I I I .

L. Tonell i , Serie Trigonometriche, Bologna, 1928. J. V. Uspensky, On the development of arbitrary functions in series

of Hermite's and Laguerre's polynomials, Annals of Mathemat ics , (2), vol . 28 (1927) , pp. 593-619 .

B. L. v a n der Waerden, Die gruppentheoretische Methode in der

Quantenmechanik, Berlin, 1932.

G. N . Watson , Theory of Bessel Functions, Cambridge, 1922.

H . Wey l , The Theory of Groups and Quantum Mechanics (trans-lated b y H. P. Robertson) , London, 1931.

Ε . T . Whittaker and G. N . Watson, A Course of Modern Analysis,

Cambridge, 1920, and other edit ions.

A. Zygmund, Trigonometrical Series, Warsaw, 1935.

See also J. Shohat , E . Hille, and J. L. Walsh, A Bibliography on

Orthogonal Polynomials, Bullet in of the Nat ional Research Council , N o . 103, Washington, 1940.

I N D E X O F N A M E S

This index does not contain references to the repeated occurrence of the names Fourier, Legendre, Bessel, Laplace, Jacobi, Hermite, and Laguerre, or to the bibliographical lists at the ends of chapters.

Birkhoff, G. D., 9C, 229 Bocher, M., 24, 229 Byerly, W. E., 112, 229

Carslaw, H. S., 229 Charlier, C., 178, 179 Christoffel, Ε. B., 54, 55, 157,

158, 206 Churchill, R. V., 229 Courant, R., 88, 189, 229

Darboux, G., 157, 158, 175, 206, 229

de Moivre, Α., 126

Elderton, W. Palin, 229

Fej6r, L., 32, 229 Frank, P., 230

Gauss, K. F., 55 Gibbs, J. Willard, 24 Gram, J. P., 178, 179 Gray, Α., 229 Green, G., 103, 107

Heine, E., 175 Hubert, D., 88, 189, 229 Hille, E., 230 Hobson, E. W., 229

Ince, E. L., 90, 229

Jackson, D., 44

Kaczmarz, S., 229 Kellogg, O. D., 229

Korous, J., 205

Lebesgue, H., 5, 40, 41, 42, 43, 44, 196, 197, 229

Leibniz, G. W., 166, 167, 169, 184, 186, 187

Liouville, J., 89 Lipschitz, R., 44

Mac Lane, S., viii Maclaurin, C , 49 MacRobert, Τ. M., 229 Marden, M., 162 Mathews, G. B., 229 Moore, C. N., 84

Parseval, Μ. Α., 29, 31, 216 Pearson, K., 142, 147, 162, 165,

230 Peebles, G., 203 Poisson, S. D., 103, 105, 136, 137 Polya, G., 230

Riemann, B., 14, 16, 21, 25 "Riemann-Weber," 230 Rietz, H. L., 230 Robertson, H. P., 230 Rodrigues, O., 57, 166, 167 Rolle, M., 78

Schmidt, E., 151, 152, 224, 225 Schrodinger, E., 182, 188 Schwarz, Η. Α., 38, 39, 195, 202,

204, 207, 215, 224, 228 Shohat, J., 230 Steinhaus, H., 229 Stone, Μ. H., 230

231

232 INDEX

Sturm, C , 89 Szego, G., 174, 205, 230

Taylor, B., 78 Tchebichef, P. L., 150, 230 Titchmarsh, E. C , 230 Tonelli, L., 230

Uspensky, J. V., 230

de la Vallie Poussin, C. J.,

van der Waerden, B. L., 189, 230 von Mises, R., 230

Walsh, J. L., 230 Watson, G. N., 230 Weierstrass, K., 25, 26, 32, 35,

40, 137, 212, 216, 227 Weyl, H., 183, 189, 230 Whittaker, Ε. T., 230

Zygmund, Α., 24, 230

TOPICAL INDEX

Asymptotic formula for Jn(x), 87-88

Bessel functions of order zero, 69-80, 216, 219; of order ± } , 217; of order n, 80-88, 217-19

Bessel series, 79-80, 83-84, 109-14, 138-41, 216

Boundary value problems, 88-90, 91-141, 220-23

Bounds of Legendre polynomials, 61-63; of normalized Jacobi polynomials, 191-92, 200-01; of other orthonormal poly-nomials, 201-08

Broken-line function, 13-14, 21

Christoffel's identity, 54-55 Christoffel - Darboux identity,

157-58 Convergence of Fourier series,

12-14, 18-25 , 42-44; of Le-gendre series, 63-68, 215; of more general series of orthog-onal polynomials, 191-208

Cosine series, 6-7, 8-9 Cosine sum, 36 Cylindrical coordinates, 106,

109-14, 138-41

Damped vibrations, 100-01, 221 Derivative formula for Legendre

polynomials, 57-58; for Jacobi

polynomials, 166-67; for Her-mite polynomials, 176-77; for Laguerre polynomials, 184

Differential equation for sines and cosines, 88; for Legendre polynomials, 48-49, 55-57, 88-89; for derivatives of Le-gendre polynomials, 119-20; for /„(*), 69-71, 74-76, 89, 219; for /„(*), 80-82; for Sturm-Liouville functions, 89-90; for Pearson frequency functions, 142-47; for a class of orthogonal polynomials, 161-65; for Jacobi polyno-mials, 173-74, 225; for Her-mite polynomials, 180, 226; for Laguerre polynomials, 186, 226-27

Discontinuities, 9-11, 22-24, 65-67

Double Fourier series, 115-17

Even function, 6-7

Fejir's theorem, 32-35 Finite jump, 19 Flow of heat, 92 Fourier series, 1-44,91-101,103-

05, 111-12, 115-17, 209-11 Frequency functions, 142-48,

223

INDEX 233

Function (general notion), 10

Generating function for Le-gendre polynomials, 45-46; for Jacobi polynomials, 174-75; for Hermite polynomials, 181; for Laguerre polynomials, 187-88; for ultraspherical poly-nomials, 225-26

Gibbs phenomenon, 24 Gram-Charlier series, 179

Harmonic polynomials, 126-32, 137-38

Hermite polynomials, 176-83, 226

Hermite series, 178-79

Integrability, 5; in the sense of Lebesgue, 5, 197-99

Integral representation of Pn(x), 58-60; of /„(*). 74-76; of /„(*), 84-85, 218

Jacobi polynomials, 166-75, 200-01, 225-26

Jacobi series, 172, 201

Laguerre polynomials, 184-90, 226-27

Laguerre series, 186 Laplace's equation, 91-96, 101-

12, 115-41, 220-22 Laplace series, 121-38, 222 Leading coefficients of Legendre

polynomials, 47-48; of Jacobi polynomials, 169-71, 172; of Hermite polynomials, 177; of Laguerre polynomials, 184

Least-square property of Fourier series, 27-29; of Legendre series, 215-16; of general or-thogonal polynomials, 160-61

Lebesgue constants, 40-42 Legendre polynomials, 45-68,

213-16

Legendre series, 53, 63-68, 107-09, 214-16

Moments, 144-48

Normalizing factor for Legendre polynomials, 51-52; for Bessel functions, 73-74, 83; for de-rivatives of Legendre poly-nomials, 124-25; for Jacobi polynomials, 171-72; for Her-mite polynomials, 177-78; for Laguerre polynomials, 185

Odd function, 7 Orthogonal polynomials, 45-68,

123-25, 149-208, 213-16,223-28

Orthogonality, 3, 224; of sines and cosines, 2-3; of Legendre polynomials, 50-51; of Bessel functions, 71-73, 82-84; of functions of two variables, 117, 122-24, 140; of Jacobi poly-nomials, 167-68; of Hermite polynomials, 177, 226; of La-guerre polynomials, 184-85, 227

Orthonormal functions, 152 Orthonormal polynomials, 154

Parseval's theorem, 29-31 Period, 1 Poisson's integral, 105, 136-37 Polar coordinates, 101-03 Probability integral, 178

Recurrence formulas for sines and cosines, 211-12, 225; for Legendre polynomials, 46-47, 58-60; for Bessel functions, 85-87, 218; for general orthog-onal polynomials, 156-57; for Jacobi polynomials, 172-73; for Hermite polynomials, 179-

234 INDEX

80; for Laguerre polynomials, 186-87

Riemann's theorem, 14-16 Rodrigues's formula, 57-58

Schmidt's process of orthogonali-zation, 151-53

Schrddinger equation, 181-83, 188-90

Schwarz's inequality, 38, 224 Sine series, 7, 8-10,94,95,98-99,

100, 101, 112 Singular points of differential

equations, 70, 88-89 Spherical coordinates, 106-09,

118-20, 189 Spherical harmonics, 118-38,189 Sturm-Liouville boundary value

problem, 89-90 Summation of series, 31-35

Transformation of Laplace's equation, 101-03,105-07,129-31

Trigonometric sum, 25-26

Ultraspherical polynomials, 225-26

Uniform convergence, 21-22,42-44, 215

Vibrating membrane, 112-14, 221, 222-23

Vibrating string, 96-101, 220-21

Wave equation of linear oscil-lator, 181-83; of hydrogen atom, 188-90

Weierstrass's theorem for trigo-nometric approximation, 25-27, 35-40; for polynomial ap-proximation, 212

Weight function, 72, 149-50, 153-54; for Jacobi polyno-mials, 166; for Hermite poly-nomials, 176; for Laguerre polynomials, 184

Zeros of Legendre polynomials, 52-53; of Bessel functions, 72-73, 76-79, 82, 87, 218, 219; of general orthogonal polyno-mials, 159-60

AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS

The underlying theme of this monograph is that the funda-

mental simplicity of the properties of orthogonal functions

and the developments in series associated with them makes

those functions important areas of study for students of both

pure and applied mathematics.

The book starts with Fourier series and goes on to Legendre

polynomials and Bessel functions. Jackson considers a variety

of boundary value problems using Fourier series and Laplace’s

equation. Chapter VI is an overview of Pearson frequency func-

tions. Chapters on orthogonal, Jacobi, Hermite, and Laguerre

functions follow. The fi nal chapter deals with convergence.

There is a set of exercises and a bibliography.

For the reading of most of the book no specifi c preparation is

required beyond a fi rst course in the calculus. A certain amount

of “mathematical maturity” is presupposed, or should be

acquired in the course of the reading.