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# ch09 - Piessens R The Hankel Transform The Transforms and...

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Piessens, R . “ The Hankel Transform .” The Transforms and Applications Handbook: Second Edition. Ed. Alexander D. Poularikas Boca Raton: CRC Press LLC, 2000

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© 2000 by CRC Press LLC 9 The Hankel Transform 9.1 Introductory Deﬁnitions and Properties 9.2 Deﬁnition of the Hankel Transform 9.3 Connection with the Fourier Transform 9.4 Properties and Examples 9.5 Applications The Electriﬁed Disc • Heat Conduction • The Laplace Equation in the Halfspace z > 0, with a Circularly Symmetric Dirichlet Condition at z = 0 • An Electrostatic Problem 9.6 The Finite Hankel Transform 9.7 Related Transforms 9.8 Need of Numerical Integration Methods 9.9 Computation of Bessel Functions Integrals over a Finite Interval Integration between the Zeros of J v ( x ) • Modiﬁed Clenshaw–Curtis Quadrature 9.10 Computation of Bessel Function Integrals over an Inﬁnite Interval Integration between the Zeros of J v ( x ) and Convergence Acceleration • Transformation into a Double Integral • Truncation of the Inﬁnite Interval 9.11 Tables of Hankel Transforms ABSTRACT Hankel transforms are integral transformations whose kernels are Bessel functions. They are sometimes referred to as Bessel transforms. When we are dealing with problems that show circular symmetry, Hankel transforms may be very useful. Laplace’s partial differential equation in cylindrical coordinates can be transformed into an ordinary differential equation by using the Hankel transform. Because the Hankel transform is the two-dimensional Fourier transform of a circularly symmetric function, it plays an important role in optical data processing. 9.1 Introductory Deﬁnitions and Properties Bessel functions are solutions of the differential equation (9.1) where p is a parameter. Equation (9.1) can be solved using series expansions. The Bessel function J p (x) of the ﬁrst kind and of order p is deﬁned by xy x y x p y 22 2 0 ′′ + +− = () Robert Piessens Katholieke Universieit Leuven
© 2000 by CRC Press LLC . (9.2) The Bessel function Y p (x) of the second kind and of order p is another solution that satisFes . Properties of Bessel function have been studies extensively (see References 7, 22, and 26). Elementary properties of the Bessel functions are 1. Asymptotic forms. (9.3) 2. Zeros. J p (x) and Y p (x) have an inFnite number of real zeros, all of which are simple, with the possible exception of x = 0. ±or nonnegative p the s th positive zero of J p (x) is denoted by j p,s . The distance between two consecutive zeros tends to π : ( j p,s+ 1 j p,s ) = π . 3. Integral representations. . (9.4) If p is a positive integer or zero, then . (9.5) 4. Recurrence relations. (9.6) (9.7) (9.8) . (9.9) Jx x x kp k p p k k () !( ) = ++ = 1 2 1 4 1 2 0 Γ Wx Yx x pp () d e t = ′′ = 2 π x xp x p ()~ c o s , .

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ch09 - Piessens R The Hankel Transform The Transforms and...

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