FEB 2 3 1965

^ecUnic&l vtete ^Uo. 228

A TABULATION OF AIRY FUNCTIONS

H. T. Dougherty and M. E. Johnson

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards is a principal focal point in the Federal Government for assuring

maximum application of the physical and engineering sciences to the advancement of technology in industry and commerce. Its responsibilities include development and maintenance of the national stand- ards of measurement, and the provisions of means for making measurements consistent with those standards; determination of physical constants and properties of materials; development of methods

for testing materials, mechanisms, and structures, and making such tests as may be necessary, particularly for government agencies; cooperation in the establishment of standard practices for incorporation in codes and specifications; advisory service to government agencies on scientific and technical problems; invention and development of devices to serve special needs of the Government; assistance

to industry, business, and consumers in the development and acceptance of commercial standards and

simplified trade practice recommendations: administration of programs in cooperation with United

States business groups and standards organizations for the development of international standards of

practice; and maintenance of a clearinghouse for the collection and dissemination of scientific, technical, and engineering information. The scope of the Bureau's activities is suggested in the following

listing of its four Institutes and their organizational units. Institute for Basic Standards. Electricity. Metrology. Heat. Radiation Physics. Mechanics. Applied Mathematics. Atomic Physics. Physical Chemistry. Laboratory Astrophysics.* Radio Standards Laboratory: Radio Standards Physics; Radio Standards Engineering.** Office of Standard Ref- erence Data. Institute for Materials Research. Analytical Chemistry. Polymers. Metallurgy. Inorganic Materials. Reactor Radiations. Cryogenics.** Office of Standard Reference Materials. Central Radio Propagation Laboratory.** Ionosphere Research and Propagation. Troposphere

and Space Telecommunications. Radio Systems. Upper Atmosphere and Space Physics.

Institute for Applied Technology. Textiles and Apparel Technology Center. Building Research.

Industrial Equipment. Information Technology. Performance Test Development. Instrumentation.

Transport Systems. Office of Technical Services. Office of Weights and Measures. Office of Engineering Standards. Office of Industrial Services.

* NBS Group, Joint Institute for Laboratory Astrophysics at the University of Colorado. ** Located at Boulder, Colorado.
NATIONAL BUREAU OF STANDARDS

technical cYlote 22S

Issued September 18, 1964

A TABULATION OF AIRY FUNCTIONS

H. T. Dougherty and M. E. Johnson

Central Radio Propagation Laboratory

National Bureau of Standards

Boulder, Colorado

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Oifice Washington D.C. 20402

Price: 20 cents

TABLE OF CONTENTS

Page No,

1. INTRODUCTION 1

2. COMPUTATION OF THE TABLES 2

3. APPLICATION 4

4. TABULATION FOR REAL VALUES OF THE

ARGUMENT 5

5. REFERENCES 6

A TABULATION OF AIRY FUNCTIONS

H. T. Dougherty and M. E. Johnson

A tabulation is provided for Wait's formulation of the

Airy function and its first derivative. The argument covers

the range of real values from -6.0 to +6.0 in intervals of

0.1. As an aid for their application the tabulations are also

graphed.

1. INTRODUCTION

A series of recent papers has indicated a renewed interest in Airy

Functions and Airy Integrals [Fock, 1945; Rice, 19 54; Wait and Conda

1958 and 1959; Wait, 1959; Spies and Wait, 1961] specifically for the

diffraction of radio waves by convex surfaces. There is also a variety

of formulations for these Airy Functions and Integrals whose histories

and interrelations are described thoroughly by Logan [19 59] in an unpublished report. For three of the formulations, that of Furry, h(-t), of

Miller, Ai(t) and Bi(t), and of Smirnov, U, ^(-t, 1), there are extensive tabulations available [Furry and Arnold, 1945; Miller, 1946;

Smirnov, I960] . Extensive tabulations are not available however for

the formulation, W(t), of Fock or of Wait. The purpose of this report

is to provide a tabulation of Wait's formulation of the Airy Functions and

Integrals for real values of the argument between -6. and +6. at

intervals of 0. 1. As an aid for their application, the tabulations are

also graphed.

The Airy functions u(t) and v(t) are linearly independent solutions of the differential equation:

f"(t) - tf(t) =0, (1)
which arises, for example, in certain cases of the diffraction of radio

waves by convex surfaces. For the formulation of Spies and Wait [19 61] , the Airy integrals are given by

w i (t) = ~3T~ Jexp

(

tz

" t) dz = u(t) " jv(t) (2)

c

l and

W o(t) = -T^- J

exp (tz -- -^ ) dz = u(t)+jv(t), (3) zTM TT Lexp

V

tz

" T

C

2

where the contour of integration, C,, for z extends from 00. e

to zero and from zero to infinity along the real axis. The contour C ? is the complex conjugate of C,. These integral representations differ

from those of Fock [1945] . The corresponding formulations of these

integrals by Fock are given by the complex conjugates of (2) and (3).

This is a result of Wait's choice of the exp (jcot) time dependency in

the wave equation as contrasted to Fock's choice of exp (-jwt).

2. COMPUTATION OF THE TABLES

Note that for real values of t, W ?

(t) = W.*(t) where * indicates

"complex conjugate of." Because of this relationship W,(t) maybe

obtained from the tabulations of W,(t) which are given in terms of

W

x (t) = |W(t)| exp[-j9(t)] (4)

and

u(t) = |W(t)| cos 9(t), v(t) = |W(t)| sin 0(t). (5)
Rather than recomputing u(t) and v(t) directly from their series

representation, as given by Spies and Wait [1961] for example, the

authors have simply transformed the tabulations given by Furry and

Arnold [1945] . There, the real and imaginary components, R[h? (-t)]

and I[h? (-t)] , of the modified function are tabulated. Note that

|W(t)| = ± jR2[h

2

(-t)] +I2

[h

2

("t)] (6)

and

2„ /I[h("t)]

9(t) = - - arctan (jjp-^jj

where a = 0. 8536 6721 8838 952. The determination of u(t) and v(t)

from (5) and (6) and Furry's tabulation of l^-t) was carried out by a double precision program on an IBM 7090 computer. The accuracy of

the input was checked by means of the Wronskian for the modified Hankel

function of the second kind [Furry and Arnold, 1945] :

I[h (-t)] R[h'

2

(-t)] - I[h» (-t)] R[h (-t)] = 0.7287 4772 0520, (8)

where the prime indicates differentiation with respect to the argument.

Similarly, the accuracy of the output was checked by means of the Airy

function Wronskian [Spies and Wait, 1961] given by

u'(t) v(t) - u(t) v'(t) = Unity. (9)

This procedure was intended to maintain the accuracy within a unit in

the eighth significant figure.
3. APPLICATION

Recent examples of the application of this formulation of the Airy

integrals are given by Spies and Wait [1961] for VLF ionospheric propagation and by Dougherty and Maloney [1964] for VHF propagation over

irregular terrain. For such applications, integral expressions for the

electromagnetic fields are often encountered which are readily evaluated

by numerical integration. The numerical integration involving tabulated

values of the Airy functions are generally carried out by means of computer programs. However in some instances sufficient accuracy is

achievable even by graphical integration techniques. For example, the

integrals

I = n

6 Jtt/12 ~

r

*: at

w(t)T

-jir/4 r

W (t) v(t) + ^-7=^ [-jtf — J

dt (10)

** oJ

|w(t)| 2

have been evaluated for n = 0, 1, 2, 3 by computer programs based

upon the series expansions of the Airy functions [Wait, 1959; Dougherty

and Maloney, 1964] . Graphical integration, which involves "counting

squares" under the curve obtained by a point-by -point plot of the integral,

will also provide an evaluation of (10). This is feasible due to the rapidity with which v(t) decreases to very small values requiring plotting

only for t < 6.0. The graphical evaluation of (10) from tabulations of

u(t) and v(t) for t equal to multiples of 0.5 and for n = 0, 1, 2, 3 has

provided values which agree with the computer evaluation within 2% or

less. J. R. Wait reported similar experience with the graphical
integration technique at the August 1963 VLF Symposium at the National

Bureau of Standards, Boulder, Colorado.

4. TABULATION FOR REAL VALUES

OF THE ARGUMENT

Table I is a tabulation of the values of u(t), v(t), | W(t) J , and

9(t) for t in the range from -6.0 to +6.0 at intervals of 0. 1. For

larger values of (t) see asymptotic expressions given by Spies and Wait

[1961] . Table II is a similar tabulation for their first derivatives with

respect to t, u'(t), v'(t), etc. Figure 1 is a plot of v(t) versus u(t)

for the tabulated values. The Airy function v(t) is plotted versus t in

Figure 2. Figures 3 and 4 are plots of log |u(t)| and log

n |W(t)|,

respectively. The corresponding plots for the first derivative of the

Airy functions are presented in Figures 5, 6, 7, and 8.
5. REFERENCES

Dougherty, H„ T., and L. J„ Maloney (1964), The application of diffraction by convex surfaces to irregular terrain situations, J. Res.

NBS-USNC-URSI 68D (Radio Science), No. 2, 239-250.

Fock, V. A. (1945), Diffraction of radio waves around the earth's surface, Journal of Physics, USSR, 9, 255-266.

Furry, W. H. , and H. A. Arnold (1945), Tables of the modified Hankel

functions of order one-third and of their derivatives, Annals of

the Computation Laboratory of Harvard University, Vol. II (Harvard University Press, Cambridge, Mass.).

Logan, N. A. (1959), General research in diffraction theory, Vol. 1,

Tech. Report LMSD-288087, Lockheed Missiles and Space Division, California.

Miller, J. C. P. (1946), The Airy integral, giving tables of solutions

of the differential equation y" = xy (University Press, Cambridge,

Mass. ).

Rice, S. O. (1954), Diffraction of plane radio waves by a parabolic

cylinder, BSTJ3_3, No. 2, 417-504.

Spies, K. P., and J. R. Wait (1961), Mode calculations for VLF propagation in the earth-ionosphere waveguide, NBS Technical Note

No. 114.

Smirnov, A. D. (I960), Tables of Airy functions and special confluent

hypergeometric functions (Pergamon Press, Macmillan Co.,

New York).

Wait, J. R. (1959), Electromagnetic radiation from cylindrical structures

(Pergamon Press, New York).

Wait, J. R. , and A. M. Conda (1958), Pattern of an antenna on a curved

lossy surface, IRE Trans, on Antennas and Propagation, AP-6, No. 4, 348-359.

Wait, J. R. , and A. M. Conda (1959), Diffraction of electromagnetic

waves by smooth obstacles for grazing angles, J. Res. NBS 63D

(Radio Prop. ), No. 2, 181-197.
TABLE I

u(t) v(t)

-6,

-5, 9

-5. 8

-5. 7

-5. 6

-5. 5

-5. 4

-5. 3

-5.,2

-5..1

-5«,0

-4.,9

-4,,8

-4,,7

-4,,6

-4.,5

-4,,4

-4,,3

-4,,2

-4,,1

-4«.0

-3, 9

-3.8

-3.7

-3.6

-3, 5

-3.4

-3, 3

-3,.2

-3,.1

-3,.0

-2,.9

-2,.8

-2,.7

-2,.6

-2,.5

-2 .4

-2,.3

-2,.2

-2 .1

0.26001610

•0.39495534

0.50672720

0.58926690

0.63838864

•0.65193237

0.62978312

•0.57377180

0.48747274

0.37591820

0.24525291

•0.10235310

0.04556587

0.19133098

0.32816231

0.44997757

0.55163833

0.62913083

0.67967819

0.70178491

0.69521791

0.66093133

0.60094422

0.51818151

0.41628969

0.29943806

0.17211623

0.03893726

-0.09554546

-0.22700109

-0.35145921

-0.46541978

-0.56593368

-0.65065382

-0.71785848

-0.76644888

-0.79592443

-0.80633929

-0.79824406

-0.77261679

0.58339462

0.50536697

0.39765342

0.26696727

0.12111301

0.03151696

0.18244682

0.32359323

-0.44768699

-0.54862054

-0.62170769

-0.66384840

-0.67359747

-0.65114190

-0.59819604

-0.51782732

-0.41422823

-0.29245100

-0.15812196

-0.01715200

0.12454242

0.26129497

0.38791212

0.49985514

0.59337763

0.66561637

0.71463652

0.73943395

0.73989920

0.71674883

0.67143086

0.60601100

0.52304686

0.42545646

0.31638732

0.19909100

0.07680776

-0.04733574

-0.17041324

-0.28976875

W(t) e(t)

0.63871564

0.64139340

0.64412786

0.64692118

0.64977566

0.65269375

0.65567798

0.65873110

0.66185596

0.66505563

0.66833334

0.67169254

0.67513688

0.67867026

0.68229686

0.68602110

0.68984772

0.69378181

0.69782878

0.70199448

0.70628519

0.71070760

0.71526901

0.71997724

0.72484075

0.72986870

0.73507099

0.74045843

0.74604273

0.75183666

0.75785419

0.76411054

0.77062245

0.77740826

0.78448819

0.79188453

0.79962187

0.80772751

0.81623174

0.82516824

245.97771

231.99160

218.12300

204.37290

190.74231

177.23226

163.84382

150.57807

137.43615

124.41919

111.52838

98.76494

86.13010

73.62515

61.25142

49.01027

36.90309

24.93133

13.09648

1.40007

349.84368

338.42896

327.15759

316.03131

305.05195

294.22136

283.54148

270.01431

262.64194

252.42652

242.37028

232.47554

222.74472

213.18030

203.78490

194.56122

185.51206

176.64034

167.94911

159.44153

-2.0 -0.73078731 -0.40306918 0.83457465 151.12087
u(t) v(t) W(t) e(t)

-2.0 -0.73078731 -0.40306918 0.83457465 151.12087

-1.9 -0.67435821 -0.50833986 0.84449299 142.99055

-1.8 -0.60512608 -0.60398413 0.85497041 135.05412

-1.7 -0.52500553 -0.68878803 0.86605991 127.31525

-1.6 -0.43595842 -0.76191229 0.87782121 119.77776

-1.5 -0.33992982 -0.82287335 0.89032177 112.44557

-1.4 -0.23879217 -0.87151588 0.90363800 105.32278

-1.3 -0.13429820 -0.90797848 0.91785671 98.41357

-1.2 -0.02804264 -0.93265524 0.93307673 91.72223

-1.1 0.07856683 -0.94615450 0.94941091 85.25317

-1.0 0.18433057 -0.94925694 0.96698837 79.01084

-0.9 0.28827003 -0.94287445 0.98595732 72.99976

-0.8 0.38963412 -0.92801066 1.0064882 67.22444

-0.7 0.48789985 -0.90572461 1.0287776 61.68935

-0.6 0.58276837 -0.87709794 1.0530525 56.39884

-0.5 0.67415753 -0.84320609 1.0795762 51.35708

-0.4 0.76219226 -0.80509394 1.1086538 46.56799

-0.3 0.84719366 -0.76375584 1.1406402 42.03507

-0.2 0.92966805 -0.72011996 1.1759487 37.76133

-0.1 1.0102969 -0.67503668 1.2150615 33.74914

-0.0 1.0899290 -0.62927084 1.2585416 30.00000

0.1 1.1695743 -0.58349735 1.3070475 26.51449

0.2 1.2504027 -0.53829983 1.3613499 23.29194

0.3 1.3337459 -0.49417163 1.4223515 20.33038

0.4 1.4211056 -0.45151906 1.4911105 17.62630

0.5 1.5141666 -0.41066622 1.5688681 15.17450

0.6 1.6148177 -0.37186092 1.6570808 12.96803

0.7 1.7251801 -0.33528163 1.7574584 10.99811

0.8 1.8476451 -0.30104475 1.8720098 9.25413

0.9 1.9849228 -0.26921235 2.0030961 7.72382

1.0 2.1401026 -0.23979956 2.1534955 6.39336

1.1 2.3167287 -0.21278206 2.3264798 5.24767

1.2 2.5188938 -0.18810301 2.5259075 4.27075

1.3 2.7513520 -0.16567953 2.7563359 3.4460 5

1.4 3.0196582 -0.14540865 3.0231571 2.75690

1.5 3.3303370 -0.12717266 3.3327643 2.18685

1.6 3.6910893 -0.11084379 3.6927532 1.72009

1.7 4.1110427 -0.09628818 4.1121702 1.34173

1.8 4.6010586 -0.08336952 4.6018138 1.03807

1.9 5.1741062 -0.07195173 5.1746065 0.79672

2.0 5.8457211 -0.06190141 5.8460488 0.60670
u(t) v(t) W(t) e(t)

2.0 5.8457211 -0.06190141 5.8460488 0.,60670

2.1 6.6345669 -0.05308960 6.6347793 0«,45847

2.2 7.5631253 -0.04539326 7.5632616 o.,34389

2.3 8.6585462 -0.03869619 8.6586326 0«,25607

2.4 9.9536942 -0.03288982 9.9537485 0« 18933

2.5 11.488444 -0.02787347 11.488478 0«,13902

2.6 13.311283 -0.02355463 13.311304 0«,10139

2.7 15.481298 -0.01984888 15.481311 0«,07347

2.8 18.070646 -0.01667968 18.070653 0.,05289

2.9 21.167625 -0.01397812 21.167630 0«,03784

3.0 24.880517 -0.01168249 24.880520 0«,02691

3.1 29.342372 -0.00973786 29.342374 0*,01902

3.2 34.717010 -0.00809557 34.717011 0,,01337

3.3 41.206544 -0.00671279 41.206544 0.,00934

3.4 49.060830 -0.00555193 49.060830 0.,00649

3.5 58.589360 -0.00458019 58.589360 0.,00449

3.6 70.176259 -0.00376908 70.176259 0.,00308

3.7 84.299229 -0.00309394 84.299229 0.,00211

3.8 101.55350 -0.00253353 101.55350 0i,00144

3.9 122.68224 -0.00206961 122.68224 0.,00097

4.0 148.61506 -0.00168660 148.61506 0«,00066

4.1 180.51713 -0.00137122 180.51713 0.,00044

4.2 219.85164 -0.00111220 219.85164 0,,00029

4.3 268.45959 -0.00090002 268.45959 0.,00020

4.4 328.66176 -0.00072666 328.66176 0,,00013

4.5 403.38936 -0.00058535 403.38936 0«,00009

4.6 496.35159 -0.00047028 496.35159 0«,00006

4.7 612.25098 -0.00037728 612.25098 0.,00004

4.8 757.06070 -0.00030189 757.06070 o.,00003

4.9 938.38201 -0.00024104 938.38201 0.» 00002

5.0 1165.9060 -0.00019203 1165.9060 0«,00001

5.1 1452.0113 -0.00015266 1452.0113 0<,00000

5.2 1812.5381 -0.00012110 1812.5381 0.,00000

5.3 2267.7937 -0.00009587 2267.7937 0<,00000

5.4 2843.8598 -0.00007573 2843.8598 0«,00000

5.5 3574.2950 -0.00005970 3574.2950 0«.00000

5.6 4502.3568 -0.00004697 4502.3568 0..00000

5.7 5683.9044 -0.00003687 5683.9044 o.00000

5.8 7191.1997 -0.00002889 7191.1997 0.» 00000

5.9 9117.8891 -0.00002259 9117.8891 0.,00000

6.0 11585.548 -0.00001762 11585.548 0.,00000
TABLE II

u'(t) v'(t) W(t) e'(t)

-6.0 -1.4408255 -0.61315469 1.5658662 156.94745

-5.9 -1.2451125 -0.93874220 1.5593402 142.98584

-5.8 -0.98017057 -1.20426100 1.5527327 129.14279

-5.7 -0.66362273 -1.39636967 1.5460411 115.41935

-5.6 -0.31520893 -1.-50664347 1.5392632 101.81658

-5.5 0.04450912 -1.53174968 1.5323962 88.33559

-5.4 0.39539008 -1.47330453 1.5254374 74.97754

-5.3 0.71883096 -1.33745010 1.5183842 61.74359

-5.2 0.99870504 -1.13420236 1.5112335 48.63495

-5.1 1.2220805 -0.87663086 1.5039822 35.65289

-5.0 1.3796989 -0.57993416 1.4966271 22.79867

-4.9 1.4662075 -0.26047526 1.4891648 10.07364

-4.8 1.4801580 0.06516445 1.4815917 357.47917

-4.7 1.4237931 0.38106017 1.4739042 345.01668

-4.6 1.3026551 0.67270634 1.4660982 332.68765

-4.5 1.1250558 0.92763593 1.4581697 320.49359

-4.4 0.90145110 1.13587737 1.4501142 308.43610

-4.3 0.64376354 1.29024149 1.4419273 296.51682

-4.2 0.36469463 1.38644107 1.4336042 284.73745

-4.1 0.07706401 1.42305452 1.4251396 273.09977

-4.0 -0.20679320 1.40135266 1.4165283 261.60563

-3.9 -0.47554647 1.32501219 1.4077648 250.25696

-3.8 -0.71929000 1.19974311 1.3988429 239.05577

-3.7 -0.92985342 1.03285813 1.3897565 228.00418

-3.6 -1.1010001 0.83281220 1.3804990 217.10439

-3.5 -1.2285166 0.60873763 1.3710633 206.35871

-3.4 -1.3102007 0.36999805 1.3614420 195.76958

-3.3 -1.3457624 0.12577974 1.3516276 185.33956

-3.2 -1.3366509 -0.11526471 1.3416116 175.07136

-3.1 -1.2858258 -0.34531025 1.3313855 164.96782

-3.0 -1.1974897 -0.55758521 1.3209401 155.03198

-2.9 -1.0767981 -0.74652710 1.3102659 145.26705

-2.8 -0.92956412 -0.90787032 1.2993529 135.67644

-2.7 -0.76196966 -1.03867070 1.2881904 126.26381

-2.6 -0. 58029637 -1.13727371 1.2767675 117.03305

-2.5 -0.39068455 -1.20323513 1.2650728 107.98836

-2.4 -0.19892669 -1.23720397 1.2530943 99.13426

-2.3 -0.01029982 -1.24077734 1.2408200 90.47561

-2.2 0.17056179 -1.21633728 1.2282376 82.01771

-2.1 0.33975394 -1.16687849 1.2153345 73.76631

-2.0 0.49415156 -1.09583558 1.2020987 65.72769

10
u'(t) V(t) W'(t) e'(t)

-2.0 0.49415156 -1.09583558 1.2020987 65.72769

-1.9 0.63142341 -1.00691634 1.1885184 57.90875

-1.8 0.75001628 -0.90394726 1.1745829 50.31709

-1.7 0.84911411 -0.79073518 1.1602831 42.96113

-1.6 0.92857712 -0.67094856 1.1456122 35.85021

-1.5 0.98886637 -0.54801963 1.1305672 28.99478

-1.4 1.0309588 -0.42506832 1.1151498 22.40655

-1.3 1.0562579 -0.30484754 1.0993693 16.09872

-1.2 1.0665034 -0.18970852 1.0832446 10.08622

-1.1 1.0636844 -0.08158455 1.0668086 4.38600

-1.0 1.0499584 0.01800913 1.0501128 359.01735

-0.9 1.0275788 0.10796193 1.0332347 354.00226

-0.8 0.99883197 0.18754333 1.0162862 349.36581

-0.7 0.96598578 0.25637000 0.99942689 345.13651

-0.6 0.93124929 0.31436702 0.98287937 341.34658

-0.5 0.89674415 0.36172535 0.96695156 338.03202

-0.4 0.86448753 0.39885779 0.95206419 335.23232

-0.3 0.83638591 0.42635513 0.93878649 332.98939

-0.2 0.81423923 0.44494382 0.92787959 331.34546

-0.1 0.79975486 0.45544638 0.92034735 330.33925

-0.0 0.79457042 0.45874545 0.91749090 330.00000

0.1 0.80028529 0.45575180 0.92095942 330.33907

0.2 0.81850019 0.44737703 0.93278550 331.33981

0.3 0.85086475 0.43451060 0.95539013 332.94799

0.4 0.89913335 0.41800139 0.99154725 335.06659

0.5 0.96522973 0.39864353 1.0443108 337.55919

0.6 1.0513212 0.37716623 1.1169291 340-26433

0.7 1.1599042 0.35422701 1.2127880 343.01768

0.8 1.2939035 0.33040819 1.3354235 345.67521

0.9 1.4567867 0.30621593 1.4886220 348.12927

1.0 1.6526996 0.28208150 1.6765995 350.31414

1.1 1.8866258 0.25836423 1.9042345 352.20213

1.2 2.1645762 0.23535595 2.1773338 353.79457

1.3 2.4938174 0.21328621 2.5029216 355.11162

1.4 2.8831463 0.19232824 2.8895541 356.18359

1.5 3.3432241 0.17260512 3.3476768 357.04454

1.6 3.8869831 0.15419623 3.8900404 357.72828

1.7 4.5301250 0.13714342 4.5322004 358.26598

1.8 5.2917319 0.12145701 5.2931255 358.68517

1.9 6.1950201 0.10712141 6.1959462 359.00937

2.0 7.2682696 0.09410025 7.2688787 359.25825

11
u'(t) v'(t) W'(t) e«(t)

2.0 7.2682696 0.09410025 7.2688787 359.25825

2.1 8.5459747 0.08234110 8.5463713 359.44797

2.2 10.070270 0.07177953 10.070526 359.59162

2.3 11.892705 0.06234273 11.892869 359.69966

2.4 14.076448 0.05395264 14.076552 359.78040

2.5 16.699038 0.04652848 16.699102 359.84036

2.6 19.855817 .03998888 19.855857 359.88461

2.7 23.664237 0.03425366 23.664261 359.91707

2.8 28.269239 0.02924510 28.269254 359.94073

2.9 33.850027 0.02488895 33.850036 359.95788

3.0 40.628568 0.02111519 40.628573 359.97023

3.1 48.880315 0.01785850 48.880319 359.97907

3.2 58.947736 0.01505844 58.947738 359.98537

3.3 71.257403 0.01265973 71.257404 359.98983

3.4 86.341650 0.01061207 86.341650 359.99296

3.5 104.86602 0.00887009 104.86602 359.99516

3.6 127.66417 0.00739313 127.66417 359.99669

3.7 155.78225 0.00614499 155.78225 359.99774

3.8 190.53549 0.00509359 190.53549 359.99847

3.9 233.58048 0.00421070 233.58048 359.99897

4.0 287.00757 0.00347160 287.00757 359.99931

4.1 353.45932 0.00285473 353.45932 359.99954

4.2 436.28240 0.00234140 436.28240 359.99970

4.3 539.72298 0.00191549 539.72298 359.99980

4.4 669.17814 0.00156312 669.17814 359.99987

4.5 831.52027 0.00127238 831.52027 359.99992

4.6 1035.5159 0.00103319 1035.5159 359.99995

4.7 1292.3681 0.00083692 1292.3681 359.99997

4.8 1616.4187 0.00067632 1616.4187 359.99998

4.9 2026.0605 0.00054523 2026.0605 359.99998

5.0 2544,9230 0.00043853 2544.9230 359.99999

5.1 3203.4169 0.00035189 3203.4169 360.00000

5.2 4040.7486 0.00028171 4040.7486 360.00000

5.3 5107.5521 0.00022502 5107.5521 360.00000

5.4 6469.3341 0.00017934 6469.3341 360.00000

5.5 8210.9876 0.00014261 8210.9876 360.00000

5.6 10442.720 0.00011315 10442.720 360.00000

5.7 13307.847 0.00008959 13307.847 360.00000

5.8 16993.055 0.00007077 16993.055 360.00000

5.9 21741.942 0.00005580 21741.942 360.00000

6.0 27872.904 0.00004389 27872.904 360.00000

12
oz>

UJI

u.O<

QCO<Q<

cr< o

II

ind© • mo

Po '

rvj o l

to

in Oro

(00 OtO

(\j"

U)'

UJ

cr

ITS T

13
GRAPH OF v(t) , THE IMAGINARY PART

OF THE AIRY FUNCTION ,W(t)

2 1-

-£

o o

-2*—

6 OOQ •OQQpO OGQOOOO( i

-4

FIGURE 2 14
GRAPH OF u(t) , THE REAL PART

OF THE AIRY FUNCTION W(t)

o

o

-2

«*

• e

,°°o>* o^°^e

Jf

jf

<*

-6 -4 -2

t

FIGURE 3

15
MAGNITUDE

GRAPH OF THE

OF THE AIRY FUNCTION , W(t)

o

e^°

„«0°

npoooooooaQpc

soooeo *

-2A-

-6 -4 -2

FIGURE 4

16
>

I-I

ccUJo3

U.

LUX

U.o<<<o

CE< 5 £

IT)I > m

iriI o

inI

roI I e

mo

in in b ° 1 m l o o

in

toio m

UJ

ccO

•r>

T

UM

17
GRAPH OF V' (t), THE IMAGINARY PART

OF THE AIRY FUNCTION DERIVATIVE , W' (t)

-~>

-2

•

•

•

•

• o

'

• • O

e

• •

o

•

•

o o

•

•

•

•

• •

•

•

•

o

• e

•

•

• o* y

•

•

•

•

•

-4 -2 t

FIGURE 6

18
GRAPH OF u'(t) , THE REAL PART

OF THE AIRY FUNCTION DERIVATIVE, W'(t)

o

Ofc _pp. -7f°«sr ^•"•Oooo,

.£-

"v°<#

-2 *- -6 -4 -2

FIGURE 7

19
MAGNITUDE OF

GRAPH OF THE

THE AIRY FUNCTION DERIVATIVE, W'(t)

FIGURE 8

20 GPO 84