A TREATISE ON

BESSEL FUNCTIONS AND

THEIR APPLICATIONS TO PHYSICS

BY

A N D R E W GRAY, F.R.S. PROFESSOR OF .NATURAL PHILOSOPHY IN T H E UNIVERSITY OF GLASGOW

AND

G. B. M A T H E W S , F.R.S. SOMETIME FELLOW OF ST. JOHN'S COLLEGE, CAMBRIDGE

SECOND EDITION PREPARED BY

A. GRAY AND

T. M. MACROBERT, D.Sc. LECTURER IN MATHEMATICS IN T H E UNIVERSITY OF GLASGOW

MILLAN AND CO., LIMITED"

ST. MARTIN'S STREET, LONDON

1922

CONTENTS

CHAPTEE I

INTRODUCTORY

i 1. Bernoulli's Problem. § 2. Fourier's Problem. § 3. Bessel's Prob- PAGE lem. § 4. Laplace's Equation—Cylindrical Harmonics - I

CHAPTER I I

SOLUTION OF THE DIFFERENTIAL EQUATION.

} 1. Solution by the Method of Frobenius. § 2. Definition of the Bessel Function Jn(x). § 3. Definition of Neumann's Bessel Function Yn(x). § 4. Recurrence Formulae for Jn(x). § 5. Ex­pressions for Jn(x) when n is half an odd integer - 9

EXAMPLES 18

CHAPTER I I I

OTHER BESSEL FUNCTIONS AND RELATED FUNCTIONS

i 1. The Function In(t). § 2. The Function Kn{t). § 3. The Bessel Function Gn(x). § 4. Theorem (as to relation connecting any two solutions of Bessel's Equation). § 5. The Function Fn(x). § 6. Kelvin's Ber and Bei Functions 20

EXAMPLES - 27

CHAPTER IV

FUNCTIONS OF INTEGRAL ORDER. EXPANSIONS IN SERIES OF BESSEL FUNCTIONS

j 1. The Bessel Coefficients. § 2. Expansion of xn in terms of Bessel Functions—Expansion of a Power Series in Terms of Bessel Functions—Sonine's Expansion. § 3. The Addition Theorem— Generalization of the Addition Theorem. § 4. Schlömileh's Expan­sion 31

EXAMPLES 42

X CONTENTS

C H A P T E R V

D E F I N I T E INTEGRAL EXPRESSIONS FOR T H E BESSEL FUNCTIONS. ASYMPTOTIC EXPANSIONS

§ 1. Bessel's Second Integral. § 2. Contour Integral Expressions— PAGE

Solution of Bessel's Equation—Expressions for Jn(x) and Kn(x)— Expression for F.n(x). § 3. The Asymptotic Expansions—Asymptotic Expansion of Kn(x)—Asymptotic Expansion of Jn(x)—Asymptotic Expansions of the Ber and Bei Functions. § 4. Asymptotic Expressions for the Bessel Functions. § 5. Asymptotic Expres­sions for the Bessel Functions, regarded as Functions of their Orders » 4 5

EXAMPLES - - 62

C H A P T E R V I

D E F I N I T E INTEGRALS INVOLVING BESSEL FUNCTIONS

§ 1. Various Integrals. § 2. Lommcl Integrals. § 3. Gegenbauer's Addition Formulae—Addition Theorem for Jn—Addition Theorem forKn 64

EXAMPLES 75

C H A P T E R V I I

T H E ZEROS OF T H E BESSEL FUNCTIONS

§ 1. THEOREMS ON THE ZEROS OF THE BESSEL FUNCTIONS (Theorems

I.-XV.). § 2. The Zeros of Jn(x)—Stokes' Method of Calculating the Zeros of Jn(x). § 3. Zeros of the Bessel Functions regarded as Functions of their Orders * 79

EXAMPLES 89

C H A P T E R V I I I

FOURIER-BESSEL EXPANSIONS AND INTEGRALS

§ 1. The Fourier-Bessel Expansions. § 2. Validity of the Expansions. § 3. The Fourier-Bessel Integrals 91

C H A P T E R I X

RELATIONS B E T W E E N BESSEL FUNCTIONS AND L E G E N D R E FUNCTIONS. GREEN'S FUNCTION

§ 1. Bessel Functions as Limiting Cases of Legendre Functions. § 2. Legendre Functions as Integrals involving Bessel Functions. § 3. Dougall's Expressions for the Green's Function.—Green's Function. Case I. Whole of Space. Case II. Space bounded by two

CONTENTS XI

parallel planes. Case III. Space bounded externally by a cylinder, PAGE Case IV. Space bounded by two axial planes. Case V. Space bounded externally by two parallel planes and a cylinder. Case VI. Space bounded by two parallel planes and two axial planes. Case VII. Space bounded by two axial planes and a cylinder. Case VIII. Space bounded by two axial planes, two parallel planes, and a cylinder. Case IX. Space bounded by two parallel planes, two axial planes, and two cylinders 98

CHAPTER X

VIBRATIONS OF MEMBRANES - - - 111

CHAPTER XI

HYDRODYNAMICS

§ 1. Stokes' Current Function for Motion in Coaxial Planes. § 2. Oscil­lations of a Cylindrical Vortex. § 3. Wave Motion in a Cylindrical Tank. § 4. Oscillations of a Rotating Liquid. § 5. Two-Dimen­sional Motion of a Viscous Liquid—Pendulum moving in a Viscous Fluid 118

,-;, / CHAPTER XI I

STEADY FLOW OF ELECTRICITY OR OF HEAT IN UNIFORM ISOTROPIC MEDIA

j 1. Electric Potential—Potential due to Charged Circular Disk. § 2. Circular Disk Electrode in Unlimited Medium. § 3. Conductor bounded by Parallel Planes. § 4. Conductor bounded by Circular Cylinder and Parallel Planes. § 5. Metal Plate and Conductor separated by Film—Conductor bounded by Parallel Planes— Cylinder of Finite Radius. 6 §. Finite Cylindrical Conductor with Electrodes on the same Generating Line . . . . 13g

CHAPTER X I I I

PROPAGATION OF ELECTROMAGNETIC WAVES ALONG WIRES

§ 1. Equations of the Electromagnetic Field. § 2. Waves guided by a Straight Wire. § 3. Diffusion of Electric Current—Current Density at Different Distances from the Axes. § 4. Hertz's Investigations 157

CHAPTER XIV

DIFFRACTION I. Case of Symmetry round an Axis

§ 1. Intensity (on a Screen at Right Angles to the Axis) expressed by Bessel Functions. § 2. Discussion of the Series (U, V) of Bessel

Xll CONTENTS

Functions which express the Intensity. § 3. Bessel Function Inte- FAGE grals expressed in terms of U and V Functions. § 4. Two Cases of Diffraction : Case (1), у =0. § 5. Case (2), у not zero. § 6. Graphical Method of finding Situations of Maxima and Minima. § 7. Case when Orifice is replaced by an Opaque Disk. § 8. Source of Light a Linear Arrangement of Point Sources. Struve's Function - .- 178

II. Case of a Slit § 9. Diffraction produced by a Narrow Slit bounded by Parallel Edges.

Fresnel's Integrals 218

CHAPTER XV

EQUILIBRIUM OF AN ISOTROPIC ROD OF CIRCULAR SECTION § 1. Solutions of the Equations of Equilibrium in Terms of Harmonic

Functions. § 2. The General Problem of Surface Traction for a Circular Cylinder 222

CHAPTER XVI

MISCELLANEOUS APPLICATIONS § 1. Variable Flow of Heat in a Solid Sphere. § 2. Stability of a

Vertical Cylindrical Rod. § 3. Torsional Vibration of a Solid Circular Cylinder. § 4. Oscillations of a Chain of Variable Density. § 5. Tidal Waves in an Estuary 229

MISCELLANEOUS EXAMPLES - - - 241

APPENDIX I

Formulae for the Gamma Function and the Hypergeometrio Function 254

APPENDIX II

Stokes' Method of obtaining the Asymptotic Expansions of the Bessel Functions 257

APPENDIX III

Formulae for Calculation of the Zeros of Bessel Functions - - - 260

EXPLANATION OF THE TABLES - - - 264 TABLE I. Values of J0(x) and -Jt(x) 267 TABLE II. Values of Jn(x) for different values of и . . . 286 TABLE III. The first forty roots of J0(x) = 0 with the corresponding

values of Jt(x) 300

CONTENTS Xl l l

TABLE IV. The first fifty roots of J1(x)=0 with the corresponding PA«E

maximum or minimum values of J0(x) - - - 301

TABLE V. The smallest roots of Jn(xs) = 0 302

TABLE VI. /„(a^/i) = ber x+i bei ж 302

TABLE VII. Values of Ia(x) for x = 0 to x = 5-10 . . . - 303

TABLE VIII. Values of 1г{х) for n = 0 to x = 5'10 - . . . 306

TABLE IX. Values of 1Лх), I^x), I2{x), ... for , r = 0 to ж = 6 - - 309

TABLE X. Values of -Ко(ж) and ЛТх(ж) for ж = 0 ' 1 to ж = 114), to 21 places of decimals . . . . . . 313

TABLE XI. Values of КЛх) and ^ ( ж ) for ж = 6-1 to ж = 12'0, to a smaller number of decimals 315

TABLE XII . Values of К%(х)., Kz(x), К±(х) ... Kw(x) for values of ж from ж=0 - 2 to ж=5 - 0 316

TABLE XI I I . The first two positive zeros of Jn(x) when n is small - 317

BIBLIOGRAPHY 318

GRAPHS OF J„(X) AND JX(X) 323

INDEX 324