Preface ......................................................... 7
Chapter 1. Introduction to the theory of Bessel functions ...... 15
1.1 Bessel differential equation .............................. 15
1.2 Bessel functions of first kind ............................ 18
1.3 Bessel functions of second kind ........................... 20
1.4 Bessel functions in complex domain ........................ 25
1.5 Modified Bessel functions ................................. 30
1.6 Spherical Bessel functions ................................ 32
1.7 Integral representations of Bessel functions .............. 35
1.8 Asymptotic formulas for Bessel functions .................. 39
1.9 Zeros of Bessel functions ................................. 54
1.10 Functions related to Bessel functions .................... 63
1.11 Integrals involving Bessel functions ..................... 69
Comments and references to Chapter 1 ........................... 74
Chapter 2. Bessel's series and integral representations ........ 81
2.1 Series of Fourier-Bessel and Dini ......................... 81
2.2 Series of Schlomilch ...................................... 97
2.3 Series of Neumann ........................................ 104
2.4 Integral of Hankel ....................................... 110
2.5 Integral of Hardy ........................................ 115
2.6 Integral of Meijer ....................................... 122
Comments and references to Chapter 2 .......................... 129
Chapter 3. Bessel's integral transforms ....................... 135
3.1 Mellin transform and reciprocal kernels of Fourier type .. 135
3.2 H-transforms ............................................. 141
3.3 Y-transforms ............................................. 145
3.4 K-transforms ............................................. 149
Comments and references to Chapter 3 .......................... 152
Addendum. Analytic solutions of linear second order
differential equations ................................... 153
Appendix. The Euler gamma-function ............................ 163
Exercises ..................................................... 175
Bibliography .................................................. 181
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