Dataset of Bessel function maxima and minima to 600 orders and 10000 extrema

Area of Science:

• Mathematical physics and applied mathematics.
• Essential for solving cylindrical problems in electrostatics, heat flow, and quantum mechanics (Schrödinger equation).

Background:

• Bessel functions of the first kind are fundamental in numerous scientific and engineering fields.
• While Bessel function roots are well-documented, their extrema (maxima and minima) are not extensively tabulated, hindering applications requiring Neumann boundary conditions.

Purpose of the Study:

• To compute and tabulate a comprehensive set of extrema for Bessel functions of the first kind.
• To provide accurate data for the first 600 orders, extending existing literature.

Main Methods:

• Utilized an adaptive root solver algorithm.
• Bounded the solver by the known roots of the Bessel function for precision.
• Achieved a high accuracy of 10^-15 for the computed extrema.

Main Results:

• Successfully computed 10,000 extrema for Bessel functions up to the 600th order.
• Validated results against existing literature for lower orders (up to 30), showing exact agreement.
• Generated a valuable dataset of Bessel function maxima and minima.

Conclusions:

• The computed extrema data fill a critical gap in the literature for Bessel function analysis.
• These data are vital for orthogonal function expansions and numerical methods.
• Applications include calculating geometric correction factors in material resistivity measurements.