# Book:Andrew Gray/A Treatise on Bessel Functions/Second Edition

## Andrew Gray, G.B. Mathews and T.M. MacRobert: A Treatise on Bessel Functions and their Applications to Physics (2nd Edition)

Published $\text {1922}$, Dover Publications, Inc..

### Contents

PREFACE TO THE FIRST EDITION
PREFACE TO THE SECOND EDITION
NOTE ON THE ASYMPTOTIC EXPANSIONS

CHAPTER $\text {I}$: INTRODUCTORY
$\S 1$. Bernoulli's Problem. $\S 2$. Fourier's Problem. $\S 3$. Bessel's Problem. $\S 4$. Laplace's Equation -- Cylindrical Harmonics

CHAPTER $\text {II}$: SOLUTION OF THE DIFFERENTIAL EQUATION
$\S 1$. Solution by the Method of Frobenius. $\S 2$. Definition of the Bessel Function $\map {J_n} x$. $\S 3$. Definition of Neumann's Bessel Function $\map {Y_n} x$. $\S 4$. Recurrence Formulae for $\map {J_n} x$. $\S 5$. Expressions for $\map {J_n} x$ when $n$ is half an odd integer
Examples

CHAPTER $\text {III}$: OTHER BESSEL FUNCTIONS AND RELATED FUNCTIONS
$\S 1$. The Function $\map {I_n} t$. $\S 2$. The Function $\map {K_n} t$. $\S 3$. The Bessel Function $\map {G_n} x$. $\S 4$. Theorem (as to relation connecting any two solutions of Bessel's Equation). $\S 5$. The Function $\map {F_n} x$. $\S 6$. Kelvin's $\Ber$ and $\Bei$ Functions
Examples

CHAPTER $\text {IV}$: FUNCTIONS OF INTEGRAL ORDER. EXPANSIONS IN SERIES OF BESSEL FUNCTIONS
$\S 1$. The Bessel Coefficients. $\S 2$. Expansion of $x^n$ in terms of Bessel Functions -- Expansion of a Power Series in Terms of Bessel Functions -- Soninie's Expansion. $\S 3$. The Addition Theorem -- Generalization of the Addition Theorem. $\S 4$. Schlömilch's Expansion
Examples

CHAPTER $\text {V}$: DEFINITE INTEGRAL EXPRESSIONS FOR THE BESSEL FUNCTIONS. ASYMPTOTIC EXPANSIONS
$\S 1$. Bessel's Second Integral. $\S 2$. Contour Integral Expressions -- Solution of Bessel's Equation -- Expressions for $\map {J_n} x$ and $\map {K_n} x$ -- Expression for $\map {F_n} x$. $\S 3$. The Asymptotic Expansions -- Asymptotic Expansion of $\map {K_n} x$ -- Asymptotic Expansion of $\map {J_n} x$ -- Asymptotic Expansion of the $\Ber$ and $\Bei$ Functions. $\S 4$. Asymptotic Expessions for the Bessel Functions, regarded as Functions of their Orders
Examples

CHAPTER $\text {VI}$: DEFINITE INTEGRALS INVOLVING BESSEL FUNCTIONS
$\S 1$. Various Integrals. $\S 2$. Lommel Integrals. $\S 3$. Gegenbauer's Addition Formulae -- Addition Theorem for $J_n$ -- Addition Theorem for $K_n$
Examples

CHAPTER $\text {VII}$: THE ZEROS OF THE BESSEL FUNCTIONS
$\S 1$. Theorems on the Zeros of the Bessel Functions (Theorems $\text 1$.-$\text {XV}$.). $\S 2$. The Zero of $\map {J_n} x$ -- Stokes' Method of Calculating the Zeros of $\map {J_n} x$. $\S 3$. Zeros of the Bessel Functions regarded as Functions of their Orders
Examples

CHAPTER $\text {VIII}$: FOURIER-BESSEL EXPANSIONS AND INTEGRALS
$\S 1$. The Fourier-Bessel Expansions. $\S 2$. Validity of the Expansions. $\S 3$. The Fourier-Bessel Integrals
Examples

CHAPTER $\text {IX}$: RELATIONS BETWEEN BESSEL FUNCTIONS AND LEGENDRE FUNCTIONS. GREEN'S FUNCTION
$\S 1$. Bessel Functions as Limiting Cases of Legendre Functions. $\S 2$. Legendre Functions as Integrals involving Bessel Functions. $\S 3$. Dougall's Expressions for the Green's Function. --Green's Function. Case $\textit I$. Whole of Space. Case $\textit{II}$. Space bounded by two parallel planes. Case $\textit{III}$. Space bounded externally by a cylinder. Case $\textit{IV}$. Space bounded by two axial planes. Case $\textit{V}$. Space bounded externally by two parallel planes and a cylinder. Case $\textit{VI}$. Space bounded by two parallel planes and two axial planes. Case $\textit{VII}$. Space bounded by two axial planes and a cylinder. Case $\textit{VIII}$. Space bounded by two axial planes, two parallel planes, and a cylinder. Case $\textit{IX}$. Space bounded by two parallel planes, two axial planes, and two cylinders

CHAPTER $\text {X}$: VIBRATION OF MEMBRANES

CHAPTER $\text {XI}$: HYDRODYNAMICS
$\S 1$. Stokes' Current Function for Motion in Coaxial Planes. $\S 2$. Oscillations of a Cylindrical Vortex. $\S 3$. Wave Motion in a Cylindrical Tank. $\S 4$. Oscillations of a Rotating Liquid. $\S 5$. Two-Dimensional Motion of a Viscous Liquid -- Pendulum moving in a Viscous Fluid

CHAPTER $\text {XII}$: STEADY FLOW OF ELECTRICITY OR OF HEAT IN UNIFORM ISOTROPIC MEDIA
$\S 1$. Electric Potential -- Potential due to Charged Circular Disk. $\S 2$. Circular Disk Electrode in Unlimited Medium. $\S 3$. Conductor bounded by Parallel Planes. $\S 4$. Conductor bounded by Circular Cylinder and Parallel Planes. $\S 5$. Metal Plate and Conductor separated by Film -- Conductor bounded by Parallel Planes -- Cylinder of Finite Radius. $\S 6$. Finite Cylindrical Conductor with Electrodes on the same Generating Line

CHAPTER $\text {XIII}$: PROPAGATION OF ELECTROMAGNETIC WAVES ALONG WIRES
$\S 1$. Equations of the Electromagnetic Field. $\S 2$. Waves guided by a Straight Wire. $\S 3$. Diffusion of Electric Current -- Current Density at Different Distances from the Axes. $\S 4$. Hertz' Investigations

CHAPTER $\text {XIV}$: DIFFRACTION
$\text I$. Case of Symmetry round an Axis
$\S 1$. Intensity (on a Screen at Right Angles to the Axis) expressed by Bessel Functions. $\S 2$. Discussion of the Series $\tuple {U, V}$ of Bessel Functions which express the Intensity. $\S 3$. Bessel Function Integrals expressed in terms of $U$ and $V$ Functions. $\S 4$. Two Cases of Diffraction: Case $(1)$, $y = 0$. $\S 5$. Case $(2)$, $y$ not zero. $\S 6$. Graphical Method of finding Situations of Maxima and Minima. $\S 7$. Case when Orifice is replaced by an Opaque Disk. $\S 8$. Source of Light a Linear Arrangement of Point Sources. Struve's Function

$\text {II}$. Case of a Slit
$\S 9$. Diffraction produced by a Narrow Slit bounded by Parallel Edges. Fresnel's Integrals

CHAPTER $\text {XV}$: EQUILIBRIUM OF AN ISOTROPIC ROD OF CIRCULAR SECTION
$\S 1$. Solutions of the Equations of Equilibrium in Terms of Harmonic Functions. $\S 2$. The General Problem of Surface Traction for a Circular Cylinder

CHAPTER $\text {XVI}$: MISCELLANEOUS APPLICATIONS
$\S 1$. Variable Flow of Heat in a Solid Sphere. $\S 2$. Stability of a Vertical Cylindrical Rod. $\S 3$. Torsional Vibration of a Solid Circular Cylinder. $\S 4$. Oscillations of a Chain of Variable Density. $\S 5$. Tidal Waves in an Estuary

MISCELLANEOUS EXAMPLES

APPENDIX $\text {I}$: Formulae for the Gamma Function and the Hypergeometric Function
APPENDIX $\text {II}$: Stokes' Method of obtaining the Asymptotic Expansion of the Bessel Functions
APPENDIX $\text {III}$: Formulae for Calculation of the Zeros of the Bessel Functions

EXPLANATION OF THE TABLES
Table $\text{I}$. Values of $\map {J_0} x$ and $-\map {J_1} x$
Table $\text{II}$. Values of $\map {J_n} x$ for different values of $n$
Table $\text{III}$. The first forty roots of $\map {J_0} x = 0$ with the corresponding values of $\map {J_1} x$
Table $\text{IV}$. The first fifty roots of $\map {J_1} x = 0$ with the corresponding maximum or minimum values of $\map {J_0} x$
Table $\text{V}$. The smallest roots of $\map {J_n} {x_s} = 0$
Table $\text{VI}$. $\map {I_0} {x \surd i} = \operatorname {ber} x + i \operatorname {bei} x$
Table $\text{VII}$. Values of $\map {I_0} x$ for $x = 0$ to $x = 5 \cdotp 10$
Table $\text{VIII}$. Values of $\map {I_1} x$ for $n = 0$ to $x = 5 \cdotp 10$
Table $\text{IX}$. Values of $\map {I_0} x, \map {I_1} x, \map {I_2} x, \dotsc$ for $x = 0$ to $x = 6$
Table $\text{X}$. Values of $\map {K_0} x$ and $\map {K_1} x$ for $x = 0 \cdotp 1$ to $x = 11 \cdotp 0$, to $21$ places of decimals
Table $\text{XI}$. Values of $\map {K_0} x$ and $\map {K_1} x$ for $x = 6 \cdotp 1$ to $x = 12 \cdotp 0$, to a smaller number of decimals
Table $\text{XII}$. Values of $\map {K_2} x, \map {K_3} x, \map {K_4} x \dotsc \map {K_{10} } x$ for values of $x$ from $x = 0 \cdotp 2$ to $x = 5 \cdotp 0$
Table $\text{XIII}$. The first two positive zeros of $\map {J_n} x$ when $n$ is small
Bibliography
Graphs of $\map {J_0} x$ and $\map {j_1} x$
Index

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