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Introduction to Linear Algebra with Mathematica
Glossary
Preface
The Hankel transform of order ν of a function f(r) is given by
For Hankel transformations, we have
\[
F_{\nu} (k) = \int_0^{\infty} f(r)\,J_{\nu} (kr)\,r\,{\text d}r ,
\]
where Jν is the Bessel function of the first kind of order
ν with ν ≥ −1/2. The inverse Hankel transform of
Fν(k) is defined as
\[
f(r) = \int_0^{\infty} F_{\nu} (k)\,J_{\nu} (kr)\,k\,{\text d}k .
\]
\[
\int_0^{\infty} r\left( \frac{{\text d}^2 f}{{\text d} r^2} + \frac{1}{r} \,
\frac{{\text d} f}{{\text d} r} - \frac{\nu^2}{r^2} \, f \right) J_{\nu} (rk)\,{\text d}r = - k^2 F_{\nu} (k) = - k^2 \int_0^{\infty} f(r)\,J_{\nu} (kr)\,r\,{\text d}r .
\]
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