CERN LIBRARIES, GENEVA

CM-P00100591

CHEBYSHEV POLYNOMIAL EXPANSIONS OF AIRY FUNCTIONS, THEIR ZEROS, DERIVATIVES, FIRST AND SECOND INTEGRALS

G. Németh

MAGYAR TUDOMÁNYOS AKADÉM. MATEMATIKAI ÉS FIZIKAI TUDOMÁNYOK OSZTÁLYAM KÖZLEMÉNYEI. VOLUME 20, PAGE 13-33, BUDAPEST (1971).

Translated at CERN by B. Szeless and H.-H. Umstaetter (Original: Hungarian)

Not revised by the Translation Service

(CERN Trans. Int. 81-01)

Geneva

March 1981

C H E B Y S H E V P O L Y N O M I A L EXPANSIONS O F AIRY FUNCTIONS, THEIR ZEROS, DERIVATIVES, FIRST A N D S E C O N D INTEGRALS

by G. NÉMETH

Summary

Chebyshev polynomial expansions are determined in this paper for Airy Functions and related functions (zeros, derivatives, first and second integrals). These asymptotic type expansions have convergent character. Their coefficients evaluated to 15 digit accuracy are listed in tabulated form.

1. INTRODUCTION

FOR THE NUMERICAL EVALUATION OF THE AIRY FUNCTIONS -BECAUSE OF THEIR PRACTICAL IMPORTANCE- DETAILED TABLES HAVE BEEN PREPARED,REF [1],[2]. THE COMPUTATIONS ARE DONE IN GENERAL BY TAYLOR SERIES EXPANSIONS, BY ASYMPTUTIC SERIES, NUMERICAL INTEGRATIONS ETC. IT IS KNOWN THAT AMONG THE METHODS SUITABLE FOR COMPUTER CALCULATIONS THESE POLYNOMIAL APPROXIMATIONS ARE ECONOMICALLY APPLICABLE. IN THE FOLLOWING WE DETERMINE EXPANSIONS SIMILAR TO CHEBYSHEV POLYNOMIAL SERIES EXPANSIONS FOR THE AIRY FUNCTIONS. WE GIVE THE SERIES EXPANSION COEFFICIENTS TO 15 DECIMAL DIGITS IN THE TABLES. WE USE THE FOLLOWING NOTATIONS FOR X>0:

Ai(x), Bi(x), Ai(-x), Bi(-x) Ai'(x), Bi'(x), Ai'(-x), Bi'(-x),

Ai(1)(x) = X

Αi(t)dt, Ai(1)(-x) = Χ

Ai(-t)dt, Ai(1)(x) = ∫ Αi(t)dt, Ai(1)(-x) = ∫ Ai(-t)dt, Ai(1)(x) =

0 Αi(t)dt, Ai(1)(-x) =

0 Ai(-t)dt,

Bi(1)(x) = Χ

Bi(t)dt, Bi(1)(-x) = χ

Bi(-t)dt, Bi(1)(x) = ∫ Bi(t)dt, Bi(1)(-x) = ∫ Bi(-t)dt, Bi

(1)(x) = 0 Bi(t)dt, Bi(1)(-x) =

0 Bi(-t)dt,

Ai(2)(x) = x s

Ai(t)dt ds, Ai(2)(-x) = x s Ai(-t)dt ds, Ai(2)(x) = ∫ ∫ Ai(t)dt ds, Ai(2)(-x) = ∫ ∫ Ai(-t)dt ds, Ai(2)(x) =

0 0 Ai(t)dt ds, Ai(2)(-x) =

0 0 Ai(-t)dt ds,

Bi(2)(x) = x s

Bi(t)dt ds, Bi(2)(-x) = X I

Bi(-t)dt ds. Bi(2)(x) = ∫ ∫ Bi(t)dt ds, Bi(2)(-x) = ∫ ∫ Bi(-t)dt ds. Bi(2)(x) = 0 0 Bi(t)dt ds, Bi(2)(-x) = 0 0 Bi(-t)dt ds. 2. THE ANALYTICAL PROPERTIES ΟF THE AIRY FUNCTIONS

WE RESUME BRIEFLY THE ANALYTICAL PROPERTIES OF AIRY FUNCTIONS, WHICH SERVE AS A BASIS FOR THE DERIVATION OF THE CHEBYSHEV SERIES EXPANSIONS. PART OF THESE FORMULAE CAN BE FOUND IN REFERENCE [1] OR IN LUKE'S BOOK [3]. THE AIRY FUNCTIOIS A i(X) AND Bi(X) ARE THE LINEARLY INDEPENDENT SOLUTIONS OF THE DIFFERENTIAL EDUATION

(1) y(x) = xy(x), - ∞ < x < ∞,

WE WANT TO COMPUTE THESE FUNCTIONS AS WELL AS THEIR DERIVATIVES AND INTEGRALS FOR THE RANGE -∞ < X < ∞. FOR THIS REASON WE HAVE TO THEIR ASYMPTUTIC BEHAVIOUR FOR X → 0 AS WELL AS FOR X → +- ∞. THE FOLLOWING FORMULAE ARE VALID FOR THE CASE X → 0 :

(2) Ai(±x) = C10F1(; 2 ; ± 1 ξ2) C2x0F1( 4 ;± 1 ξ2), (2) Ai(±x) = C10F1(; 3 ; ± 4 ξ2) C2x0F1( 3 ;± 4 ξ2),

(3) Bi(±x) = C1√30F1(; 1 ; ± 1 ξ2)± C2√3x0F1 4 ; ± 1 ξ2), (3) Bi(±x) = C1√30F1(; 3 ; ± 4 ξ2)± C2√3x0F1 3 ; ± 4 ξ2),

(4) Ai'(±x) = -C20F1(; 1 ; ± 1 ξ2)+ 1 C1x02F1 5 ; ± 1 ξ2), (4) Ai'(±x) = -C20F1(; 3 ; ± 4 ξ2)+ 2 C1x02F1 3 ; ± 4 ξ2),

(5) Bi'(±x) = C2√3,F1(; 1 ; ± 1 ξ2)+ 1 C1x2√30F1(; 5 ; ± 1 ξ2), (5) Bi'(±x) = C2√3,F1(; 3 ; ± 4 ξ2)+ 2 C1x2√30F1(; 3 ; ± 4 ξ2),

(6) Ai(1)(±x) = C1x1F2( 1 ; 2 , 4 ; ± 1 ξ2) 1 C2X12F2 2 ; 4 , 5 ; ± 1 ξ2), (6) Ai(1)(±x) = C1x1F2( 3 ; 3 , 3 ; ± 4 ξ2) 2 C2X12F2 3 ; 3 , 3 ; ± 4 ξ2), (7) Bi(1)(±x) = C1√3x1F2(

1 ;

2 ,

4 ; ±

1 ξ2)± 1 C2√3x21F2( 2 ; 4 , 5 ; ± 1 ξ2), (7) Bi(1)(±x) = C1√3x1F2( 3 ; 3 , 3 ; ± 4 ξ2)± 2 C2√3x21F2( 3 ; 3 , 3 ; ± 4 ξ2), (8)Ai(2)(±x) =

1 C1x21F2(

1 ; 4

,

5 ; ± 1 ξ2) 1 C2x32F3( 2 ,1; 4 , 5 ,2; ± 1 ξ2), (8)Ai(2)(±x) = 2 C1x21F2( 3 ; 3 , 3 ; ± 4 ξ2) 6 C2x32F3( 3 ,1; 3 , 3 ,2; ± 4 ξ2), (9) Bi(2)(±x) =

= 1 C1√3x21F2( 1 ; 4 , 5 ; ± 1 ξ2)± 1 C2√3x32F3( 2 ,1; 4 , 5 ,2; ± 1 ξ2), = 2 C1√3x21F2( 3 ; 3 , 3 ; ± 4 ξ2)± 6 C2√3x32F3( 3 ,1; 3 , 3 ,2; ± 4 ξ2),

WHERE

C1 = 3 = 0.355028053887817, C2 = 3 = 0.258819403792807, C1 = Γ() = 0.355028053887817, C2 = Γ() = 0.258819403792807, C3 = 3

¼ = 0.448288357353830. C3 =

Γ(⅓)

= 0.448288357353830.

pFq(a1, a2,...,ar;b1,...bq;t)~ ∞ (a1)k...(ap)k tk , t→0, pFq(a1, a2,...,ar;b1,...bq;t)~ Σ (a1)k...(ap)k t

k , t→0, pFq(a1, a2,...,ar;b1,...bq;t)~ Σ (b1)k...(bp)k k! , t→0, pFq(a1, a2,...,ar;b1,...bq;t)~ k = 0 (b1)k...(bp)k k! , t→0,

(a)0 = 1, (a)k = a(a + 1)...(a + k - 1), ξ = 2 x, x > 0. (a)0 = 1, (a)k = a(a + 1)...(a + k - 1), ξ = 3 x, x > 0.

IT IS EASILY SEEN THAT FOR X > 5 THE NUMERICAL COMPUTATION OF THESE EXPRESSIONS BECOMES ALREADY VERY UNCOMFORTABLE. IN THESE CASES THE ASYMPTUTIC REPRESENTATIONS FOR X →∞ARE APPLIED. THESE ARE THE FOLLOWING:

(10) Ai(x) = 1 π-½xe-ξR(ξ), R(ξ)~2F0(

1 , 5 ;;- 1 ), (10) Ai(x) = 2 π

-½xe-ξR(ξ), R(ξ)~2F0( 6 , 6 ;;- 2ξ ),

(11) Bi(X) = πxeξS(ξ), S(ξ)~2F0( 1 , 5 ;; 1 ), (11) Bi(X) = πxeξS(ξ), S(ξ)~2F0( 6 , 6 ;; 2ξ ),

(12) Ai'(X) = -1 πxe-ξP(ξ), P(ξ)~2F0(-1 , 7 ;;- 1 ), (12) Ai'(X) = -2 πxe

-ξP(ξ), P(ξ)~2F0(- 6 , 6 ;;-2ξ ),

(13) Bi'(X) = πxeξQ(ξ), Q(ξ)~2F0(-1 , 7 ;; 1 ), (13) Bi'(X) = πxeξQ(ξ), Q(ξ)~2F0(- 6 , 6 ;; 2ξ ),

(14) Ai(-X) = π-½x-¼{cos(ξ- π )P1(ξ)+sin(ξ-π )Q1(ξ)}, (14) Ai(-X) = π-½x-¼{cos(ξ- 4 )P1(ξ)+sin(ξ-4 )Q1(ξ)},

(15) Bi(-x) = π-½x-¼{cos(ξ-Π )Q1(ξ)+sin(ξ-π )P1(ξ)}, (15) Bi(-x) = π-½x-¼{cos(ξ-4 )Q1(ξ)+sin(ξ-4 )P1(ξ)},

P1(ξ)~4F1 1 , 7 , 5 , 11 ; 1 ; -1 ), P1(ξ)~4F1 12 , 12 , 12 , 12 ; 2 ; -ξ2 ),

(16) Q1(ξ)~

5 4F1(

7 , 13 , 11 , 17 ; 3 ; -1 )' Q1(ξ)~ 72ξ 4F1( 12 , 12 , 12 , 12 ; 2 ; -ξ2 )'

( 1 7 ) Ai'(-X) = π-½x-¼{cos(ξ-π )Q2(ξ) + sin(ξ-π )P2(ξ)}, ( 1 7 ) Ai'(-X) = π-½x-¼{cos(ξ-4 )Q2(ξ) + sin(ξ-4 )P2(ξ)},

(18) Βi'(-X) = π-½x-¼{cos(ξ- π )P2(ξ) + sin(ξ-π )Q2(ξ)}, (18) Βi'(-X) = π-½x-¼{cos(ξ- 4 )P2(ξ) + sin(ξ-4 )Q2(ξ)},

P2(ξ)~4F1(-1 , 5 , 7 , 13 ; 1 ; -1 ), P2(ξ)~4F1(-12 , 12 , 12 , 12 ; 2

; -ξ2 ),

(19) Q2(ξ)~

7 4F1(

5 , 11 , 13 , 19 ; 3 ; 1 ), Q2(ξ)~ 72ξ 4F1( 12 , 12 , 12 , 12 ; 2 ; ξ2 ),

(20) Ai(1)(x) = 1 -(6πξ)-½e-ξR1(ξ), R1(ξ)~ ∞ (-

1 )k(

1 )kUk, (20) Ai(1)(x) =

1 -(6πξ)-½e-ξR1(ξ), R1(ξ)~ Σ (-1 )k(

1 )kUk, (20) Ai(1)(x) =

1 -(6πξ)-½e-ξR1(ξ), R1(ξ)~ Σ (- ξ )k( 2 )kUk, (20) Ai(1)(x) = 3 -(6πξ)-½e-ξR1(ξ), R1(ξ)~ k=0 (- ξ )

k( 2 )kUk,

(21) Bi(1)(x) = π-½x-¾eξS1(ξ), S1(ξ)~ ∞ 1

( 1 ) k (21) Bi

(1)(x) = π-½x-¾eξS1(ξ), S1(ξ)~ Σ 1 ( 1 ) k (21) Bi

(1)(x) = π-½x-¾eξS1(ξ), S1(ξ)~ Σ ξk ( 2 ) k (21) Bi(1)(x) = π-½x-¾eξS1(ξ), S1(ξ)~ k = 0 ξk ( 2 ) k

(22) Ai(1)(-x) = 2 = π-½x-¾{cos(ξ-π )Q3(ξ)-sin(ξ- π )P3(ξ)}, (22) Ai(1)(-x) = 3 = π

-½x-¾{cos(ξ-4 )Q3(ξ)-sin(ξ- 4 )P3(ξ)},

(23) Bi(1)(-x) = π-½x-¾{cos(ξ-π )P3(ξ)-sin(ξ-π )Q3(ξ)}, (23) Bi(1)(-x) = π-½x-¾{cos(ξ- 4 )P3(ξ)-sin(ξ-4 )Q3(ξ)},

P3(ξ)~ ∞ (-

4 )k(

1 )k(

3 )k Q3(ξ)~ 1 ∞

(-4 )k(

3 )k(

5 ) k (24) P3(ξ)~ ∞ (-

4 )k(

1 )k(

3 )k Q3(ξ)~ 1 Σ (-4 )k(

3 )k(

5 ) k (24) P3(ξ)~ Σ (- ξ2 )k( 4 )k( 4 )k Q3(ξ)~ 2ξ Σ (- ξ2 )k( 4 )k( 4 ) k (24) P3(ξ)~ k=0 (- ξ2 )k( 4 )k( 4 )k Q3(ξ)~ 2ξ k=0 (- ξ

2 )k( 4 )k( 4 ) k (24)

(25) = 1, = k (

1 )j : (

5 )j , k = 1,2,..., (25) = 1, =

k ( 6 )j : ( 6 )j , k = 1,2,..., (25) = 1, = Σ ( 6 )j : ( 6 )j , k = 1,2,..., (25) = 1, = Σ ( 1

) j ; , k = 1,2,..., (25) = 1, =

j=0 ( 1

) j ; , k = 1,2,..., (25) = 1, =

j=0 ( 2 ) j ; , k = 1,2,...,

(26) Ai(2)(x) = 1 x-C2 + 1 π-½x-¾e-ξR2(ξ), R2(ξ)~ ∞ (-1 )k(

3 ),

(26) Ai(2)(x) = 1 x-C2 + 1 π-½x-¾e-ξR2(ξ), R2(ξ)~ Σ (-1 )k(

3 ),

(26) Ai(2)(x) = 3 x-C2 + 2 π-½x-¾e-ξR2(ξ), R2(ξ)~ Σ (- ξ )k( 2 ),

(26) Ai(2)(x) = 3 x-C2 + 2 π-½x-¾e-ξR2(ξ), R2(ξ)~

k=0 (- ξ )k( 2 ),

(27) Bi(2)(x) = C3+π-½x-¾eξS2(ξ), S2(ξ)~ ∞ ( 1 )k(

3 )k tk,

(27) Bi(2)(x) = C3+π-½x-¾eξS2(ξ), S2(ξ)~ Σ ( 1 )k(

3 )k tk,

(27) Bi(2)(x) = C3+π-½x-¾eξS2(ξ), S2(ξ)~ Σ ( ξ )k( 2 )k tk, (27) Bi(2)(x) = C3+π-½x-¾eξS2(ξ), S2(ξ)~

k=0 ( ξ )k( 2 )k tk, (28) Ai(2)(-x) = 2 x-C2-π-½x-¾{cos(ξ- π )Ρ4(ξ) + sin(ξ- π )Q4(ξ)}, (28) Ai

(2)(-x) = 3 x-C2-π-½x-¾{cos(ξ- 4 )Ρ4(ξ) + sin(ξ- 4 )Q4(ξ)},

(29) Bi(2)(-x = C3-π-½x-¾{cos(ξ- π )Q4(ξ)-sin{ξ-π )P4(ξ)}, (29) Bi(2)(-x = C3-π-½x-¾{cos(ξ- 4 )Q4(ξ)-sin{ξ-4 )P4(ξ)}, (30) Ρ4(ξ)~ ∞ (-

4 )k(

3 )k(

5 ) k , Q4(ξ)~ 3 ∞

(-4 )k(

5 )k(

7 )k 2k+1, (30) Ρ4(ξ)~ Σ (-4 )k(

3 )k(

5 ) k , Q4(ξ)~ 3 Σ (-4 )k(

5 )k(

7 )k 2k+1, (30) Ρ4(ξ)~ Σ (- ξ2 )k( 4 )k( 4 ) k , Q4(ξ)~ 2ξ Σ (- ξ2 )k( 4 )k( 4 )k 2k+1, (30) Ρ4(ξ)~

k=0 (- ξ2 )k( 4 )k( 4 ) k , Q4(ξ)~ 2ξ k=0 (- ξ2 )k( 4 )k( 4 )k 2k+1,

(31) t0 = 1, t k = k (-

1 ) j (

7 )j , k = 1,2,…. (31) t0 = 1, t k =

k (- 6 ) j ( 6 )j , k = 1,2,…. (31) t0 = 1, t k = Σ (- 6 ) j ( 6 )j , k = 1,2,…. (31) t0 = 1, t k = Σ ( 3 ) j 2

, k = 1,2,…. (31) t0 = 1, t k = j=0 (

3 ) j 2 , k = 1,2,…. (31) t0 = 1, t k =

j=0 ( 2 ) j 2 , k = 1,2,….

IT CAN BE SEEN, THAT THE FUNCTION EXPANSIONS R,S,P,Q AS WELL AS THE Ρ1,Q1, P2,Q2 DIVERGE FOR X → ∞, WHEREBY THE MAGNITUDE OF THE SERIES COEFFICIENTS IS OF THE ORDER OR RESPECTIVELY. SINCE AND tk ARE BOUNDED ANALOG STATEMENTS HOLD FOR THE SERIES R1,S1,R2,S2 AS WELL AS THE P3,Q3,Ρ4,Q4. THEREFORE THESE SERIES ARE NOT SUITABLE FOR HIGHLY ACCURATE NUMERICAL CALCULATIONS.

3. CHEBYSHEV SERIES EXPANSIONS OF THE AIRY FUNCTIONS

WE SHALL MAKE USE OF TWO IMPORTANT ADVANTAGES OF SERIES EXPANSIONS IN CHEBYSHEV . THE FIRST ADVANTAGE COMES FROM THE FACT THAT IN THE SERIES (2)-(9) WHEN REARRANGED AS CHEBYSHEV POLYNOMIALS THE SPEED OF CONVERGENCE INCREASLS. THE SECOND COMES FROM THE FACT THAT THE FUNCTIONS R,S,P,Q ETC. EXPANDED IN CHEBYSHEV POLYNOMIAL SERIES CONVERGE (CONTRARY TO THE ASYMPTOTIC EXPANSIONS). WE TALK ABOUT SMALL ARGUMENTS AND LARGE ARGUMENTS: IN THE CASE OF 0 × IT WILL BE A SMALL ARGUMENT, IN THE CASE X IT WILL BE A LARGE ARGUMENT. HERE "a" WILL BE A CONVENIENTLY CHOSEN POSITIVE PARAMETER.

RETURNING TO THE SMALL ARGUMENTS WE SHALL USE THE CHEBYSHEV SERIES EXPANSIONS MENTIONED BELOW.

LET BΕ THE COEFFICIENT OF THE SERIES FOR Ai(X):

(32) Ai(x) = ∞ ak(1)T( X ) 0xa (32) Ai(x) = Σ ak(1)T( X ) 0xa (32) Ai(x) = Σ ak(1)T( a ) 0xa (32) Ai(x) = k=0 ak

(1)T( a ) 0xa

AND STMILARLY bk(1),ck(1),dk(1) THE SERIES COEFFICIENTS FOR Bi(x), Ai'(x),Bi'(x) AS WELL AS ak(2), bk(2), ck(2), dk(2) THE SERIES COEFFICIENTS FOR Ai(-x), Bi(-x), Ai'(-x), Bi'(-x)

FURTHERMORE LET

(33) Ai(1)(x) = x ∞ ak(3)Tk*( x , (33) Ai(1)(x) = x Σ ak(3)Tk*( x , (33) Ai(1)(x) = x Σ ak(3)Tk*( a , (33) Ai(1)(x) = x

k=0 ak(3)Tk*( a ,

(34) Ai(2)(x) = x2 ∞ ak(4)T( x ) (34) Ai(2)(x) = x2 Σ ak

(4)T( x ) (34) Ai(2)(x) = x2 Σ ak(4)T( a ) (34) Ai(2)(x) = x2 k=0 ak(4)T( a )

AMD ANALOGOUS TO EQUATIONS (33) AND (34) THE FUNCTIONS Bi(1)(x)/x, Ai(1)(-x)/x, Bi(1)(-x)/x HAVE THE COEFFICIENTS bk(3), ck(3), dk(3) AND THE FUNCTIONS Bi(2)(x)/x2, Ai(2)(-x)/x2, Bi(2)(-x)/x2 HAVE THE COEFFICIENTS bk(), ck(), dk(). WITH THE HELP OF THE SERIES (2) TO THE VALUES ak(), bk(), ck(),dk()

(l = 1,2,3,4) CAN BE EVALUATED BY THE USUAL CONVERGENT PROCEDURE (WE WILL NOT CONSIDER THIS FURTHER).

IN THE CASE OF LARGE ARGUMENTS ONE CAH NOT DERIVE THE CHEBYSHEV SERIES EXPANSIONS FROM THE CORRESPONDING ASYHPTOTIC SERIES EXPANSION. HOW WE SHALL THE DESIRED SERIES EXPANSION FROM THE INTEGRAL REPRESENTATION OF THE FUNCTIONS:

(35) R(ξ) = 2-π-∞ 1 e-ηK(η)dη =

∞ rkTk*( a ), ξ a (35) R(ξ) = 2-π- ∫ 1 e-ηK(η)dη =

∞ rkTk*( a ), ξ a (35) R(ξ) = 2-π- ∫ 1 e-ηK(η)dη = Σ rkTk*( a ), ξ a (35) R(ξ) = 2-π- ∫

η½(1 + η ) e

-ηK(η)dη = Σ rkTk*( ξ ), ξ a (35) R(ξ) = 2-π- 0 η½(1 + η ) e-ηK(η)dη =

k=0 rkTk*( ξ ), ξ a (35) R(ξ) = 2-π- 0 η½(1 + ξ

) e-ηK(η)dη = rkTk*( ξ ), ξ a

(36)

rk = εk2-½π-a½ ∞

1 ηk-1 e-aηK⅓(aη)dη, εk = { 1 k = 0

(36)

rk = εk2-½π-a½ ∫ 1 ηk-1 e-aηK⅓(aη)dη, εk = {

1 k = 0 (36)

rk = εk2-½π-a½ ∫ √1 + η (1 + √1 + η)2k e-aηK⅓(aη)dη, εk = { 2(-1)k k 1

(36)

rk = εk2-½π-a½ 0 √1 + η (1 + √1 + η)2k

e-aηK⅓(aη)dη, εk = { 2(-1)k k 1

(37) S(ξ) = ∞ SkTk*[ a ], (37) S(ξ) = Σ SkTk*[

a ], (37) S(ξ) = Σ SkTk*[ ξ ], (37) S(ξ) = k=0 SkTk*[ ξ ],

Sk(a) = εk 3 {

ΤHE LISTED INTEGRAL REPRESENTATIONS CAN BE OBTAINED FROM THE INTEGRAL REPRESENTATIONS OF THE BESSEL FUNCTIONS K ( X ) . THE EVALUATION OF THE NUMBERS S k AND q k BY MEANS OF HYPERGEOMETRIC SERIES CAN BE FOUND IN REFERENCE [4]. IN THE CASE OF 0 < X < ∞ THE FUNCTIONS Ai(-x),Bi(-x), Ai'(-x),Bi'(-x)HAVE AN INFINITE NUMBER OF ROOTS. FOR A GIVEN REPRESENTATION BY SERIES EXPANSION THE ROOTS CAN DE COMPUTED, WHICH IN ESSENCE GOES BACK TO MC . THE METHOD EXPLOITS THE FACT, THAT THESE FUNCTIONS CAN BE REPRESENTED BY COMBINATIONS OF BESSEL FUNCTIONS:

(51) Ai(-x) = x {J-⅓(ξ)+J⅓(ξ)}, (51) Ai(-x) = 3 {J-⅓(ξ)+J⅓(ξ)},

(52) Bi(-x) = ( x )½ {J-⅓(ξ)-J⅓(ξ)}, (52) Bi(-x) = ( 2 )

½ {J-⅓(ξ)-J⅓(ξ)},

(53) Ai'(-x) = - x {J-⅓(ξ)-J(ξ)}, (53) Ai'(-x) = - 2 {J-⅓(ξ)-J(ξ)},

(54) Bi'(-x) = x {J(ξ)+J(ξ)}. (54) Bi'(-x) = √3 {J(ξ)+J(ξ)}. FOR BESSEL FUNCTIONS J(ξ) WE HAVE THE REPRESENTATIONS BELOW: (55) J(ξ) = √ 2 {cos(ξ-π ν — π )P(ξ)-sin(ξ- π ν -π )Q(ξ)}, J(ξ) = √ πξ {cos(ξ-2 ν — 4 )P(ξ)-sin(ξ- 2 ν -4 )Q(ξ)}, (56) P(ξ)~1- (4ν

2-1)(4ν2-9) + (4ν2-1)(4ν2-9)(4ν2-25)(4ν2-49) ..., (56) P(ξ)~1- 2!(8ξ)2 + 4!(8ξ)4 ...,

(57) Q(ξ)~ 4ν2-1 - (4ν

2-1)(4ν2-9)(4ν2-25) + ... (57) Q(ξ)~ 8ξ - 3!(8ξ)3 + ... THIS EXPRESSION (58) Xk =( 3 ξk)⅓ (58) Xk =( 2 ξk)⅓ GIVES THE k ΤΗ ROOT. WHEREIN THE ξk ARE

(59) tg ε = Q , ξk = (k-1 )π-ε, RESP.; ξk = (k-3 )π-ε, (59) tg ε = P , ξk = (k-4 )π-ε, RESP.; ξk = (k-4 )π-ε,

(60) tg ε = Q , ξk = (k-3 π-ε, RESP., ξk = (k- 1 )π-ε (60) tg ε = P , ξk = (k-4 π-ε, RESP., ξk = (k- 4 )π-ε Αi RESP. Bi AS HELL AS Ai' RESP. Bi'SATISFY THE EQUATION. WE HAVE SOLVED THE CONCERNING ε BY RECURSION. WE HAVE TRANSFORMED THE SERIES SOLUTION INTO A CHEBYSHEV SERIES POLYNOMIAL SERIES AND HAVE GIVEN THE RESULTS IN BETWEEN THE TABLES,

FOR THE INTEGRALS OF THE AIRY FUNCTIONS WE DEFINE THE SERIES EXPANSIONS GIVEN BELOW:

(61)

r1(ξ) = (2π)-½ ∞ 1 e-ηx(η)Dη =

∞ rk(2)Tk*(

a ),

(61)

r1(ξ) = (2π)-½ ∫ 1 e-ηx(η)Dη =

∞ rk(2)Tk*(

a ),

(61)

r1(ξ) = (2π)-½ ∫ 1 e-ηx(η)Dη = Σ rk(2)Tk*(

a ),

(61)

r1(ξ) = (2π)-½ ∫ η½(1 + η ) e-ηx(η)Dη = Σ rk(2)Tk*( ξ ),

(61)

r1(ξ) = (2π)-½ ∫ η½(1 + η ) e-ηx(η)Dη =

k=0 rk(2)Tk*( ξ ),

(61)

r1(ξ) = (2π)-½ 0 η½(1 + ξ )

e-ηx(η)Dη = k=0

rk(2)Tk*( ξ ),

(61)

r1(ξ) = (2π)-½ 0 η½(1 + ξ )

e-ηx(η)Dη = k=0

rk(2)Tk*( ξ ),

α(η) = √3- 1 ∞ K½(u)du, α(η) = √3-

1 ∫ K½(u)du, α(η) = √3- π ∫ K½(u)du, α(η) = √3- π K½(u)du,

(62) rk(1) = εk( α ) ∞ 1 ηk- e-(αη)dη, (62) rk(1) = εk( α ) ∫ 1 ηk- e-(αη)dη, (62) rk(1) = εk( 2π ) ∫ √1 + η (1 + √1 + η)2k e-(αη)dη, (62) rk(1) = εk( 2π ) 0 √1 + η (1 + √1 + η)2k e-(αη)dη, (63) S1(ξ) =

∞ Sk(1) Tk* a

), (63) S1(ξ) = Σ Sk(1) Tk* a ), (63) S1(ξ) = Σ Sk(1) Tk* ξ ), (63) S1(ξ) =

k=0 Sk(1) Tk* ξ ),

(64) p3(ξ) = √ 1 ∞ η-½

e-ηα(η)dη = ∞ pk(3)T2k( a ),

(64) p3(ξ) = √ 1 ∫ η-½

e-ηα(η)dη = ∞ pk(3)T2k( a ),

(64) p3(ξ) = √ 1 ∫ η-½

e-ηα(η)dη = Σ pk(3)T2k( a ), (64) p3(ξ) = √ 2π ∫ 1 + η2

e-ηα(η)dη = Σ pk(3)T2k( a ), (64) p3(ξ) = √ 2π ∫ 1 + η2

e-ηα(η)dη = Σ pk(3)T2k( ξ ), (64) p3(ξ) = √ 2π ∫ 1 + η2

e-ηα(η)dη = k=0 pk(3)T2k( ξ ),

(64) p3(ξ) = √ 2π 0 1 + ξ2

e-ηα(η)dη = k=0 pk(3)T2k( ξ ),

(65) pk(3) =εk√ a ∞ 1 η2k-

e-aηα(aη)dη, (65) pk(3) =εk√ a ∫

1 η2k-e-aηα(aη)dη, (65) pk(3) =εk√ 2π ∫ √1 + η2 (1 + √1 + η2)2k e-aηα(aη)dη, (65) pk(3) =εk√ 2π 0 √1 + η2 (1 + √1 + η2)2k e-aηα(aη)dη,

(66) Q3(ξ) = 1 ∞ η

e-ηα(η)dη = 1 ∞

qk(3) T2k( a ),

(66) Q3(ξ) = 1 ∫ ξ e-ηα(η)dη = 1

∞ qk(3) T2k( a

), (66) Q3(ξ) = 1 ∫

ξ e-ηα(η)dη = 1 Σ qk(3) T2k( a ),

(66) Q3(ξ) = √2π ∫ 1 + η2 e-ηα(η)dη = 2ξ Σ qk

(3) T2k( ξ ), (66) Q3(ξ) = √2π

0 1 + ξ2 e-ηα(η)dη =

2ξ k=0 qk(3) T2k( ξ ),

(67) q = εk √ 2 a ∞ 1 η2k+½

e-aηα(aη)dη, (67) q = εk √ 2 a ∫

1 η2k+½ e-aηα(aη)dη, (67) q = εk √ π a ∫ √1 + η2 (1 + √1 + η2)2k e-aηα(aη)dη, (67) q = εk √ π a 0 √1 + η

2 (1 + √1 + η2)2k e-aηα(aη)dη,

R2(ξ) = (2π)-½ ∞ η½

e-ηβ(η)dη = ∞

rk(2) Tk*( a ), R2(ξ) = (2π)-½ ∫

η½ e-ηβ(η)dη =

∞ rk(2) Tk*( a ),

R2(ξ) = (2π)-½ ∫ η½

e-ηβ(η)dη = Σ rk(2) Tk*( a ), R2(ξ) = (2π)-½ ∫ 1 + η

e-ηβ(η)dη = Σ rk(2) Tk*( ξ ), R2(ξ) = (2π)-½

0 1 + ξ e-ηβ(η)dη =

k=0 rk(2) Tk*( ξ ),

(68)

β(η) = 3√3 + ∞ Κ(u) du, β(η) = 3√3 + ∫ Κ(u) du, β(η) = 2 + ∫ u du, β(η) = 2 + η u

du,

(69) rk(2) = εk a ∞ 1 ηk+½ e-aηβ(aη)dη, (69) rk(2) = εk a ∫ 1 ηk+½

e-aηβ(aη)dη, (69) rk(2) = εk

√2π ∫ √1 + η (1 + √1 + η)2k e-aηβ(aη)dη, (69) rk

(2) = εk √2π 0 √1 + η (1 + √1 + η)2k e

-aηβ(aη)dη,

(70) S2(ξ) = ∞ Sk(2) Tk*( a ), (70) S2(ξ) = Σ Sk

(2) Tk*( a ), (70) S2(ξ) = Σ Sk(2) Tk*( ξ ), (70) S2(ξ) = k=0 Sk(2) Tk*( ξ ),

(71) P(ξ) = 1 ∞ η e-ηβ(η)dη =

∞ Pk(4) T2k(

a

), (71) P(ξ) = 1 ∫

η e-ηβ(η)dη = ∞

Pk(4) T2k( a

), (71) P(ξ) = 1 ∫

η e-ηβ(η)dη = Σ Pk(4) T2k( a ),

(71) P(ξ) = √2π ∫ 1 + η2

e-ηβ(η)dη = Σ Pk(4) T2k( ξ ), (71) P(ξ) =

√2π ∫ 1 + η2 e-ηβ(η)dη =

k=0 Pk(4) T2k( ξ ),

(71) P(ξ) = √2π 0 1 + ξ

e-ηβ(η)dη = k=0

Pk(4) T2k( ξ ),

(72) Pk(4) = a ∞ 1 η2k+½ e-aηβ(aη)dη, (72) Pk(4) = a ∫ 1 η2k+½ e-aηβ(aη)dη, (72) Pk(4) = √2π ∫ √1 + η2 (1 + √1 + η2)2k e-aηβ(aη)dη, (72) Pk(4) =

√2π 0 √1 + η2 (1 + √1 + η2)2k e-aηβ(aη)dη,

(73) Q4(ξ) = 1 ∞ η

e-ηβ(η)dη = 3 ∞

qk(4) T2k( a ), (73) Q4(ξ) = 1 ∫

ξ e-ηβ(η)dη = 3 ∞

qk(4) T2k( a ), (73) Q4(ξ) = 1 ∫

ξ e-ηβ(η)dη = 3 Σ qk(4) T2k( a ), (73) Q4(ξ) =

√2π ∫ 1 + η2 e-ηβ(η)dη = 2ξ Σ qk

(4) T2k( ξ ), (73) Q4(ξ) =

√2π ∫ 1 + η2 e-ηβ(η)dη = 2ξ k=0 qk

(4) T2k( ξ ), (73) Q4(ξ) =

√2π 0 1 + ξ e-ηβ(η)dη = 2ξ k=0 qk

(4) T2k( ξ ),

(74) qk(4) = 3∙2½π-½.a½ ∞ 1 η2k+ e-aηβ(aη)dη. (74) qk(4) = 3∙2½π-½.a½ ∫ 1 η2k+

e-aηβ(aη)dη. (74) qk(4) = 3∙2½π-½.a½ ∫ √1 + η2 (1 + √1 + η2)2k e

-aηβ(aη)dη. (74) qk(4) = 3∙2½π-½.a½

0 √1 + η2 (1 + √1 + η2)2k e-aηβ(aη)dη.

THE CONNVERGENCE OF THE SERIES (35), (37), (39), (41), (43), (45), (47), (49) AS WELL AS (61), (64), ( ) , (68), (71), (73) CAN BE WITH THE HELP OF THE INTEGRAL EXPRESSION OF THE COEFFICIENTS. APFLYING THE LAPLACE TRANSFORM METHOD TO THE INTEGRALS WE OBTAIN FOR k → ∞ THE EXPRESSIONS BELLO:

(75) rk = exp {-3(2a) ⅓k}0(k), (76) pk = exp { - 3 ( 2 a ) ⅓ k } 0 ( k ) , (77) pk(1) = exp { - 4 a ½ k ½ } 0 ( k ) , l = 1,2,3,4. (78) qk(1) = exp {-4a½k½}0(k), l = 1,2,3,4. (79) rk(1) = exp {-3(2a)⅓k}0(1), l = 1,2. (80) Sk = exp { } 0 ( k ) , λ = 3 · 2 a ⅓ ( 1 - i √ 3 ) . (81) qk = exp { } 0 ( k ) .

FOR THE COEFFICIENTS (Sk(1),Sk(2)) OF THE SERIES (63) AND (70) WΕ DO NOT GIVE EXPLICIT EXPRESSIONS BECAUSE OF THEIR COMPLEXITY, ALTHOUGH THEIR BEHAVIOUR FOR k → ∞ GIVES SIMPLE EXPRESSIONS SIMILAR TO (80)-(81).

FOR THE COMPUTATION OF THE VALUES Sk(l) (l = 1.2) HE HAVE USED THE FOLLOWING PROCEDURE. IT CAN BE SHOWN THAT A TRANSFORMATION EXISTS SUCH THAT

φ(ξ) = 2ξ½e-ξ ξ½

edu, φ(ξ) = 2ξ½e-ξ ∫ edu, φ(ξ) = 2ξ½e-ξ 0 edu,

ψ(ξ) = 4ξe-ξ ξ

edu-2ξ ψ(ξ) = 4ξe-ξ ∫ edu-2ξ ψ(ξ) = 4ξe-ξ 0

edu-2ξ

TRANSFORMS THE FUNCTION S1 (ξ) AND S2(ξ)RESPECTIVELY.

WE OBSERVE THAT THE ASYMPTOTIC SERIES OF S1(ξ) AND φ(ξ) RESPECTIVELY S2(ξ) AND ψ(ξ) DIFFER ONLY BECAUSE OF THE EXISTANCE OF THE FACTURS νk RESPECTIVELY tk. BASED ON THIS WE PROCEED IN SUCH A WAY THAT IN THE REPRESENTATION

S1(ξ) = T1(ξ.φ(ξ))

WE MULTIPLY THE COEFFICIENTS OF THE APPROXIMATION POLYNOMIAL φ(ξ) BY THE NUMBER νk, THEN REARRANGE THIS POLYNOMIAL AS A CHEBYSHEV POLYNONIAL AND THNS RESULT THE VALUES Sk(1). IN AN ANALOGOUS WAY RESULT THE VALUES Sk(2). FOR AN ERROR ESTIMATION OF THE METHOD ONE CAN GIVE THE TRANSFORM OF T1 IN AN EXPLICIT WAΥ (T2 CAN BE OBTAINED IN A SIMILAR WAY, HENCE WE DEAL ONLY WITH T 1).

WE SHALL PROVE THAT

(82) S1(ξ) = Γ( 1 ) 1 u-¼(1-u) 1 1 u - ¼(1-u) (82) S1(ξ) = Γ( 2 ) ∫ u-¼(1-u) 1 ∫ u - ¼(1-u) (82) S1(ξ) = Γ( 1 )Γ( 1 ) ∫ u-¼(1-u)

Γ( 1 )Γ( 5 ) ∫ u

- ¼(1-u) (82) S1(ξ) = Γ( 6 )Γ( 3 ) 0 u-¼(1-u)

Γ( 6 )Γ( 6 ) 0 u- ¼(1-u)

(83) φ(ξ)- 1 2ξ ) du . (83) φ(ξ)- 2 ) du . (83)

1- 1 du . (83)

1- 2 du .

THE EXPRESSION (83) CAN BE PROVED FOR INSTANCE EASILY BY A LAPLACE TRANSFORMATION. LET X = λ/, 0 < 1 AND

(84) ∞

λ-½e-λpφ( λ )dλ = √ π 1 , (84) ∫ λ-½e-λpφ( λ )dλ = √ π 1 , (84) ∫ λ-½e-λpφ( σ )dλ = √ p 1 + pσ , (84) 0

λ-½e-λpφ( σ )dλ = √ p 1 + pσ , (85)

∞ λ-½e-λpS1 λ dλ = H(p). (85) ∫ λ-½e-λpS1 λ dλ = H(p). (85) ∫ λ-½e-λpS1 σ dλ = H(p). (85)

0 λ-½e-λpS1 σ dλ = H(p).

USING THE FACT THAT

S1(ξ) = ( π ξ)½ e [1-⅓(x) + 1⅓(x)] dx. S1(ξ) = ( π ξ)½ e ∫ [1-⅓(x) + 1⅓(x)] dx. S1(ξ) = ( 2 ξ)½ e ∫ [1-⅓(x) + 1⅓(x)] dx. S1(ξ) = ( 2 ξ)½ e

0 [1-⅓(x) + 1⅓(x)] dx.

H(p) CAN BE GIVEN EXPLICITLY:

(86) H(p) = √ π 1 2F1

1 , 5 ; 1 ; -pσ ). (86) H(p) = √ p 1 + pσ 2F1 6 , 6 ; 2 ; -2 ).

FOR THE HYPFRGEOMETRIC FUNCTION H(p) WHICH APPEARS HERE WE OBTAIN WRITTEN OUT IN INTEGRAL FORM THE FOLLOWING:

H(p) = Γ( 1 )

1

u(1-u) 1 1

(1-) H(p) = Γ( 2 ) ∫ u(1-u) 1 ∫ (1-) H(p) =

Γ( 1 )Γ( 1 ) ∫

u(1-u)

Γ 1 )Γ( 5 ) ∫ (1-) H(p) =

Γ( 6 )Γ( 3 ) 0 u(1-u)

Γ 6 )Γ( 6 ) 0 (1-) (87)

1 {√ π 1

- 1 uν√ π 1 }dν du. 1- uν {√ p 1 + pσ - 2 uν√ p 1 + pσ 1 uν }dν du. 1- 2 {√ p 1 + pσ - 2 uν√ p 1 + pσ 2 uν }dν du.

EXECUTING THE INVERSE TRANSFORMATION IN EXPRESSION (87) TERM BY TERM AND OBSERVING (84) AND (85) WE OBTAIN THE DESIRED EXPRESSION (83). FROM THIS IT CAN BE SEEN THAT IN THE CASE ξ α

|S1(ξ)| < C4|φ(ξ)|, C4 < 3 (konst.), |S1(ξ)| < C4|φ(ξ)|, C4 < 2 (konst.),

AND THIS MEANS THAT IN THE APPROXIMATION S1 THE ERROR OF φ CAN APPEAR IN THE EXPRESSION S1 (ξ) MAGNIFIED 3/2 TIMES AT MUST.

4. TABLES

WE HAVE CALCULATED NUMERICALLY THE SERIES EXPANSION COEFFICIENTS WHICH ARE DISCUSSED ABOVE FOR THE GENERATION OF AIRY FUNCTIONS AND GIVE THEM IN TABULAR FORM. AS A VALUE FΟR THE PARAMETER "a" FOR DISTINCTION BETWEEN SMALL AND LARGE ARGUMENTS WE HAVE CHOSEN FOR THE AIRY FUNCTIONS AND THEIR DERIVATIVES (36)⅓ (ξ = 4) AND FOR THE FIRST AND SECOND INTEGRALS OF THE AIRY FUNCTIONS a=(144)⅓, (ξ = 8 ) :

x1 = (36)⅓ = 3,301927248894627, X2 = (144)⅓ = 5,241482788417746. THE TABULATED NUMBERS MAY BE IN ERROR BY ONE UNIT IN THE LAST

DIGIT. THE COMPUTATIONS HAVE BEEN CARRIED OUT ON THE ICT TYPE 1905 COMPUTER OF THE KFKI (CENTRAL RES.INST.F.PHYSICS) BY MEANS OF A PROGRAM NAMED WRITTEN IN DOUBLE PRECISION FORTRAN.

REFERENCES

[1] MILLER, J. C. P.: The Airy integral. British Assoc. Adv. Sct. Math. Tables, Part-Vol. B., Cambridge (1946). [2] SMIRNOV, A. D.: Tables of Airy functions. Moscow (1955). [3] LUKE, Y. L.: Integrals of Bessel Functions. McGraw-Hill Book Company. New York (1962). [4] NÉMETH GÉZA: Bessel függvények Csebisev sorfejlése, II. KFKI Közl. 299-309. (1966), 14.

(Beérkezett: 1968. 1. 8.)

NÉMETH G.

MTA III. Osztály Közleményel 20 (1971)

AIRY-FÜGGVÍNYEK CSEBISEV-SORFEIRÉSE

MTA III. Osztály Közleményel 20 (1971)

NÉMETH G.

MTA III. Osztály Közleményel 20 (1971)

AIRY-FÜGGVÍNYEK CSEBISEV-SORFEIRÉSE

MTA III. Osztály Közleményel 20 (1971)

NÉMETH G.

MTA III. Osztály Közleményel 20 (1971)

MTA III. Osztály Közleményel 20 (1971)

AIRY-FÜGGVÍNYEK CSEBISFV-SORFEIRÉSE

M T A III. Osztály Közleményel 20 (1971)