Bessel functions are important in many problems in mathematical physics--especially those with circular symmetry. Examples include the vibrations of a circular membrane and conduction of heat in a cylinder. Solving such problems usually requires finding the zeros of the Bessel functions and their derivatives. Like sine and cosine, the Bessel function of the first kind oscillates--as in the figure below. However, its zeros are not as easily predicted.
Find the kth zero of the Bessel function of the first kind of order ν and its derivative. For Bessel functions with ν > 0, skip the zero at z = 0. For the derivative of the Bessel function of order ν = 0, start counting with the zero at z = 0.
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