This material represents a collection of integral tra- forms involving Bessel (or related) functions as kernel. The following types of inversion formulas have been singled out. k I. g(y) = f (x) (xy) 2J (xy) dx J V 0 k I' . f (x) g (y) (xy) 2J (xy) dy J V 0 II. g(y) f(x) (XY)~K (xy)dx J v 0 c+ioo k 1 II'. f (x) = g (y) (xy) 2 [Iv (xy) + I_v(xy)]dy J 27fT c-ioo or also c+ioo k 1 II". f(x) = g (y) (xy) 2Iv (xy) dx J rri oo c-i k III. g(y) f(x) (xy) 2y (xy) dx + J v 0 k III' . f(x) g(y) (xy) "1lv (xy) dy J 0 k IV. g(y) f (x) (xy) "Kv (xy) dx J 0 k g(y) (xy) 2Y (xy)dy IV' f(x) J v 0 V Preface V. g(y) f(X)Kix(y)dx J 0 -2 -1 sinh (7TX) V'. f(x) 27T x g(y)y Kix(y)dy J 0 21-~[r(~~+~-~v)r(~~+~+~v)]-1 VI. g(y) . J f (x) (xy) ~s (xy) dx o ~,v l-~ -1 VI' . f(x) 2 [r (~~+~-~v) r (~~+~+~v) ] J -5 (xy)]dy g(y) (XY)~[S~,v(xy) ~,v 0 [xy)~]dX VII. g(y) f(x) ~ J 0 0 VII' f(x) g(y) [(xy) lz]dy ~ f 0 0 with (z) o (For notations and definitions see the appendix of this book. ) The transform VII is also known as the divisor transform. I. Hankel Transforms.- 1.1 General Formulas.- 1.2 Transforms of Order Zero.- 1.3 Transforms of Order Unity.- Transforms of General Order.- 1.4 Algebraic Functions and Powers with Arbitrary Index.- 1.5 Exponential and Logarithmic Functions.- 1.6 Trigonometric and Inverse Trigonometric Functions.- 1.7 Orthogonal Polynomials.- 1.8 Miscellaneous Functions.- 1.9 Legendre Functions.- 1.10 Bessel Functions of Argument x.- 1.11 Bessel Functions of Other Arguments.- 1.12 Modified Bessel Functions of Argument x.- 1.13 Modified Bessel Functions of Other Arguments.- 1.14 Functions Related to Bessel Functions.- 1.15 Parabolic Cylinder Functions.- 1.16 Whittaker Functions.- 1.17 Gauss' Hypergeometric Function.- II. Integral Transforms with Modified Bessel Functions as Kernel.- 2.1 General Formulas.- 2.2 Transforms of Order Zero.- Transforms of General Order.- 2.3 Elementary Functions.- 2.4 Higher Transcendental Functions.- III. Integral Transforms with Neumann Functions as Kernel.- 3.1 General Formulas.- 3.2 Transforms of Order Zero.- Transforms of General Order.- 3.3 Elementary Functions.- 3.4 Higher Transcendental Functions.- IV. Integral Transforms with Struve Functions as Kernel.- 4.1 General Formulas.- 4.2 Transforms of Order Zero.- 4.3 Elementary Functions.- 4.4 Higher Transcendental Functions.- V. Kontorovich-Lebedev Transforms.- VI. Transforms with Lommel Functions as Kernel.- VII. Divisor Transforms.- Appendix. List of Notations and Definitions.
